domain: qfin stat math

text:
Lee Dinetan

Paul Sabatier Toulouse

Stochastic processes governed by Markovian processes

We introduce a new class of processes aiming at modelling random fluctuations of an asset value more efficiently than traditional Lévy processes. In this study, we consider that the object value C is a real discrete random process (N → R), whose increments are subject to the present state of a "market", described by a Markovian process M : as the successive market states are not pairwise independent, C's fluctuations are not independent either, so C cannot be assimilated as a Lévy process. We call this structure a C-process : we present methods to analyze it, mainly extending the notion of Lundberg's parameter of a diffusion Lévy process, to take M into account during the computations. Once done, we aim more specifically at controlling C's default time T0 = min ({t ∈ N|C(t) < 0}) : we achieve it through its log-Laplace transform to get some of its properties, like C's default probability.

Part I Study

 focused on ruin problems for a company investing capital in risky assets, raising the question of how the financial market influences the ruin probability.

Model

In this study, we address the question of ruin theory under some non-stationary behaviour of the market. More specifically, we consider an actor in a purely speculative economy, whose assets are deemed to be an amount of cash C, randomly fluctuating over time : at time t, the assets amount to value C(t). The aim of this study is to introduce a new model for C's fluctuations and control C's risks of default in it, for example

• Its default probability (C(t) falling down to negative values for some time t) ;

• The distribution of its default time, or some properties about it, given some properties of the process C ; in particular, we want to emphasize correlation between C's successive variations and an exogenous "market" M , whose configuration may have an influence on C.

Introduction

This paragraph explains the model that will be used throughout this study. We begin with the contents of our universe, and then we shall introduce some useful tools to the analysis.

Motivation for a new model

After the basics of speculation theory came with Bachelier ( [START_REF] Bachelier | Théorie de la spéculation[END_REF]), introducing the notions of Brownian motions to model C, several refinements of Brownian motions have been suggested, eventually leading to the theory of Lévy processes used by most traditional models ( [START_REF] Bertoin | Lévy processes[END_REF]) : they introduce the notion of sudden jumps for C, as well as the ideas behind "fat tails" for the distribution of C's increments, to improve the fit between the model and real observations. Hence, C is commonly chosen to be a Lévy process (and in particular, a drifted Brownian motion or a variation thereof) with no need to define a market M , like in [START_REF] Tankov | Financial modelling with jump processes[END_REF] ; indeed, default theory was introduced by the model of Cramér and Lundberg, and one may refer to [START_REF] Blom | [END_REF] to investigate on the risks of bankruptcy for an insurance company. When C is a Lévy process, ruin theory has already been investigated by [START_REF] Delbaen | Classical risk theory in an economic environment[END_REF], using Lundberg's exponent to evaluate the risks of default with accurate results ; as in [START_REF] Kyprianou | Introductory Lectures on Fluctuations of Lévy Processes with Applications[END_REF], we shall extend this tool through scale functions to get results about the Laplace transform of the default time. For example, in McKean's selling problem ( [START_REF] Jr | A free boundary problem for the heat equation arising from a problem of mathematical economics[END_REF]), one eventually finds out that Lundberg's exponent governs the final result.

Unfortunately, models relying on Lévy processes are likely too restrictive, because they ignore the possible changes of the market environment (the market is not stationary). In particular, independence of successive variations is not realistic, be it because of endogenous behaviour of investors towards momentum ( [START_REF] Cont | Herd behavior and aggregate fluctuations in financial markets[END_REF][START_REF] Timothy | Rational momentum effects[END_REF]), or even an exogenous consequence of price volatility ( [START_REF] Lewellen | Momentum and autocorrelation in stock returns[END_REF]). For example, speculative bubbles or busts are often the results of • Positive feedback effects : one is encouraged to buy because prices are likely to increase, and more demand means higher prices ;

• Exogenous effects : financial markets may exhibit cycles due to some political decisions, or regulatory policies ; therefore, the modelled company may not be able to cover possible short-term losses by selling its assets because the market is suddenly becoming illiquid (or conversely, may benefit from these policies and should take them into account to extract maximum profits) ;

• States of the market : in a so-called "boom-bust" economy, the prices are driven one way or the other depending on whether they were in a boom or bust phase, so one should always beware of the looming danger of the market suddenly turning around.

As a consequence, estimations by means of Lévy processes severely underestimate the risks underlying behind positive correlation between fluctuations of the market, due to the persistence of an underlying market effect. As we shall see during this study, this may eventually lead to disasters like the financial crisis of 2008. Hence, the aim of this study is to tackle the default problem for a broader class of discrete cash processes, that exhibits some persistence in C's fluctuations.

Time dependency

To improve accuracy, we focused on taking into account some time dependency between fluctuations of C, as well as the "state of the market" affecting the random variables of its successive variations : hence, we need to define a new process M indicating the market configuration, and whose successive states govern C's variations. The most natural model allowing some short-term dependency is of course the Markovian process : information on the state of the market M is required to be Markovian, and it governs the values of C's fluctuations in turn.

More specifically, we want to exhibit the effects of market local trends on C, and interpret these results in order to help computing trading decisions in a speculative and volatile market, e.g. deciding whether to buy/sell a risky asset or not in order to avoid eventual bankruptcy, well-known examples being

• The investment dilemma, where the investor must risk an immediate liquidity shortage to benefit from a long-term income ;

• McKean's problem ( [START_REF] Jr | A free boundary problem for the heat equation arising from a problem of mathematical economics[END_REF]), where one aims at optimizing the selling time of an option.

In particular, we are interested in C's risks of eventual default when it is subject to a market M . For this purpose, we introduce a class of discrete Markov processes that extends the class of Lévy processes in discrete time, involving :

• An auxilliary Markovian process M , describing the state of the market ;

• Multiple possible random increments for C, being random variables whose distributions depend on the evolution of the market. The actual increment of C between time t and time t + 1 will be the one associated to the market evolution, i.e. the states M (t) and M (t + 1).

It should be noted that M is exogenous and describes the whole market : whereas M affects C's increments, C does not affect M in return. One picks any of these possible interpretations for this assumption :

• Atomistic market : the actor has negligible incidence on the market.

• Fully described market : one assumes that M suffices at describing the state of the market, so in a sense it "incorporates" C.

Giving M and C leads to the definition of what will be called a C-process in this study.

Concept of the study

In particular, a similar example of risk analysis in such a model where C describes the assets of an insurance company (under specific constraints on C's increments) is named "Markov-modulated risk model" and has already been investigated, for example ( [START_REF] Asmussen | Risk theory in a markovian environment[END_REF][START_REF] Schmidli | Risk theory[END_REF]). Despite yielding powerful results on default risks, we remarked that the Markov-modulated risk model failed to exhibit some specific facts of general C-processes, because corner stones of its behaviour are required to state the desired properties, that are not guaranteed in our work, like

• Continuous time : piecewise continuous trajectories of C allow it to hit every positive value, which is a necessary hypothesis for the default asymptotics to work. Using discrete time allows us to introduce the issue of cycles and periodicity, as well as explaining why the given asymptotic is optimal.

• Purely negative jumps : in standard Markov-modulated risk models, C's only jumps called "claims" are alwalys negative, which allows for a fairly easy study through the use of a renewal equation ( [START_REF] Schmidli | Risk theory[END_REF]). In this study, we will not make assumptions on the jumps' direction. Consequently, one needs additional work to obtain such a renewal equation that gives the desired ruin probability estimation. Alternatively, one may refer to [START_REF] Yu | On the expected discounted penalty function for a markov regime-switching insurance risk model with stochastic premium income[END_REF] if interested in jumps only.

We shall also focus on the interpretations of the main results in terms of econometrics, solving the elementary investment problem with comments on the corresponding solution, emphasizing on the differences between Lévy-like processes and C-processes in terms of default risks. This study is thus organised as follows :

• The rest of paragraph 1 presents the notion of C-processes and studies several of its features ;

• Paragraph 2 shows the theory : we extend the notion of Lundberg's exponent from Lévy processes to C-processes, allowing then to state the main theorems ;

• Paragraph 3 shows some examples of application, indroducing and using some links with Lévy-like risk processes ;

• Paragraph 4 discusses and concludes.

Main definitions

We begin by defining the object we study, i.e. the C-process itself and its default time.

Universe

We consider the following discrete-time universe with a real-valued random process C : (N → R).

Model 1 C-process A C-process is the determination of

• A Markovian time-homogeneous process (M (t)) t∈N with -A finite state space (Ai) i≤A with A ∈ N * ; -Transition probabilities : for any t ∈ N, we have ∀i, j ≤ A, Pi→j = P (M (t + 1) = Aj|M (t) = Ai)

The set of transitions (i → j) happening with probability Pi→j > 0 will be called Γ ⊆ [|1, A|] 2 . -Some deterministic initial distribution over (Ai) i≤A : there is an i ≤ A such that

P (M (0) = Ai) = 1
This Ai is called M 's (or C's) starting state.

• Random variables called transition payoffs, such that -For every i, j ≤ A, there is a probability distribution over R∪{+∞} defining a random variable Di→j with respect to this distribution ; -For every t ∈ N * , we define the family (Di→j(t)) i,j≤A to be an i.i.d. copy of the family (Di→j) i,j≤A , with respect to the time variable t ∈ N * .

For every i, j ≤ A and t ∈ N, Di→j(t) is called the transition payoff between states Ai and Aj at time t, and D(t + 1) = D M (t)→M (t+1) (t + 1) is called the active increment.

• A process C, satisfying -C(0) = C0 ∈ R + is deterministic, called C's starting point ; -The increments of C are the successive active increments ∀t ∈ N, C(t + 1) = C(t) + D(t + 1)

We say that C is a C-process whose :

• Underlying Markovian process is M ;

• Transition payoffs are (the distributions of ) the random variables Di→j, for each state numbers i, j ≤ A.

Moreover, we shall say that C is • Bounded iff ∀(i, j) ∈ Γ, Di→j is an almost surely bounded random variable ;

• Strongly exponentially integrable (henceforth abbreviated as "sEI") iff

∀(i, j) ∈ Γ, ∀α ∈ R, E e -αD i→j < ∞
When there is no possible confusion, we will commonly abbreviate for the sake of simplicity :

• For i ≤ A, i for Ai or the reverse, especially when noting (vector) indices ;

• For X a random variable identically equal to x almost surely, X for x or the reverse.

Our study aims at controlling C's default time, i.e. the first time t ∈ N when C(t) < 0, noted by the random variable T0.

Definition 1.1 Default time

The default time of a C-process C is the random variable

T0 = min ({t ∈ N; C(t) < 0})
If this set is empty, we set T0 = min (∅) = +∞.

Among several properties, we chose to look for its Laplace transform, because this simplifies the computations, and making a = 0 allows us to recover C's default probability.

Definition 1.2 Laplace transforms

The Laplace transform of a random variable X ∈ R ∪ {+∞} is defined by

LX = R → R + ∪ {+∞} a → E e -aX 1X<+∞
Its log-Laplace transform is

ΛX = R → R ∪ {±∞} a → ln (LX (a))
where ln(0) = -∞.

Since a Laplace transform is a holomorphic function on its convergence domain, imposing X < ∞ in the definition has the same effect as considering its analytic continuation from R * + over R -whenever possible.

Choice and interpretation of the model

The first thing to notice regarding the model is that the only item preventing C from being a Lévy process is the Markovian process M , introducing short-term dependency between successive increments of C. We are going to characterize this idea by the following proposition.

Proposition 1.1 Conditional independence

We consider

• C a random discrete-time process, valued over R ∪ {∞} ;

• M a discrete-time, finite state space, time-homogeneous Markovian process, whose transition matrix is given by its entries Pi→j for i, j ≤ A ;

• For every t ∈ N, F(t) the natural filtration associated with the joint process (C, M ) up to time t.

The following statements are equivalent :

1. C is a C-process whose underlying Markovian process is M .

We have simultaneously

• C(0) and M (0) are deterministic ;

• Markovian property : for every t ∈ N, H a mesurable subset of R and i ≤ A, P (C(t + 1) -C(t) ∈ H ∧ M (t + 1) = Ai|F(t))

= P (C(t + 1) -C(t) ∈ H ∧ M (t + 1) = Ai|M (t))

• Time-homogeneous property : for every s, t ∈ N, H a mesurable subset of R and i, j ≤ A,

P (C(t + 1) -C(t) ∈ H ∧ M (t + 1) = Aj|M (t) = Ai)
= Pi→jP (C(s + 1) -C(s) ∈ H|M (s) = Ai ∧ M (s + 1) = Aj)

It follows from this property that C-processes are natural continuations of Lévy processes, since if A = 1 then M is trivial and this definition is exactly the definition of a Lévy process, which means that there are no effects of an exogenous market, and increments have a stationary distribution. Another useful hypothesis to ensure "permanent" behaviour of C is positive recurrence of M . Indeed, the states A i≤A of Markovian process M we are going to analyze may split between several communicating classes, some of which are terminal (closed) ; as our goal lies in asymptotical considerations, we will often deem that M itself already lies in some closed communicating class A , which is tantamount to deem that M is positive recurrent (over A ). Nevertheless, this is not sufficient to avoid issues of transience for C, because of the transition payoffs amounting to +∞ that "push" C to +∞ without any possible return : C's transtion payoffs may be such that C(t) is automatically driven to +∞ if M (t) hits some specific state Aa, voiding the default time (T0 = ∞). These +∞ payoffs translate to situations where the player is safe from default : in real economy, they may stand for retirement from the market, or the fulfillment of some goal, or whatever is the final aim of the player ; when looking at C's default risks, a realization of such a payoff indicates that default will not ever happen. In this case, it is natural to say that Aa is not "interestingly accessible" from the other states, and to remove M 's property of positive recurrence. This is done by the following definitions.

Definition 1.3 Positive recurrence

Let C be a C-process, whose underlying Markovian process is M .

1. M is said positive recurrent iff it holds the positive recurrence property for all states : for any i, j ≤ A, there is n ∈ N such that for any t,

P (M (t + n) = Aj|M (t) = Ai) > 0
Positive recurrence of M implies existence and unicity of M 's invariant distribution without any coordinates equal to 0, that shall be noted

µ [i] i≤A = µ ∈ (0, 1] A 2.
C is said positive recurrent iff one has the stronger condition

P (M (t + n) = Aj ∧ C(t + n) < ∞|M (t) = Ai ∧ C(t) < ∞) > 0
Positive recurrence of C implies positive recurrence of M by construction.

Finally, we will often deem C to be sEI or even bounded throughout this study. In particular, we chose to avoid the concern of fat tails because

• Fat tails lead to integrability issues as well as wild behaviour for C's trajectories, so one only gets poor results on C's default time ;

• They do not suppress the concern about the dependency between successive increments of C ;

• Similar behaviour to fat tails appears when considering a suitable M , as we shall see during the study.

Indeed, we will find out that apparently high moments of increments distributions (when looking at the market, or as prescribed in [START_REF] Mandelbrot | The variation of certain speculative prices[END_REF]) are actually a consequence of time dependency, so of M 's structure.

Example

Throughout this study, we shall illustrate this theory of C-processes by the following example process, hereby named "boom-bust process". To clarify the numeric values, let us deem that C denotes a cash amount and the time step is one second.

• It has A = 2 states, where A1 is named the "good" state and A2 the "bad" state.

• The transition matrix governing M is given by P = .89 .11 .09 .91 where M 's starting state is M (0) = A1.

• The transition payoffs have these distributions :

-D1→1 is Gaussian, of mean 3$/s and variance 1$ 2 /s ; -D1→2 is Bernouilli, amounting to either 1 or 0$/s with even probability ; -D2→1 is a constant -2$/s ; -D2→2 is Gaussian, of mean -2$/s and variance 16$ 2 /s. with C's starting point to be defined in each example.

A typical random realization for this process C with C(0) = 6$ may be found on figure 1. The colors of • Long sequences of transitions (1 → 1) (red, "booms") and transitions (2 → 2) (blue, "busts") happen, as the respective transition probabilities are high.

• These sequences are roughly drifted as stated by the expectancies of the associated transition payoffs, with the blue one being less "smooth" because of the higher variance.

In this example, the default time (first t ∈ N when C(t) crosses 0) comes at T0 = 62s. The aim for our study is to control (find bounds from above and below for) the Laplace transform of T0.

Cycles of a C-process

Some Lévy processes are monotonous almost surely, which is equivalent to saying that their increments D are either non-negative or non-positive almost surely, i.e. with supports supp(D) ⊆ R + or supp(D) ⊆ R -. An analoguous definition holds for C-processes, but it requires analysis of cumulative increments over several time periods rather than one, to ensure that monotonicity is taken relatively to identical states of M (one does not want the specificities of M 's states to hamper monotonicity properties) : hence, we consider the "possible" values of increments of a C-process. This leads us to look at the process defined as follows.

Definition 1. [START_REF] Blom | [END_REF] Restricted Lévy process Let C be a C-process, whose underlying Markovian process is M deemed positive recurrent. For every n ∈ N, let T (n) be the random n th hitting time of M (0) by M (so T (0) = 0). Then the process given by

C = N → R n → C(T (n))
is a Lévy process, named C's restricted Lévy process.

Considerations on the increments of C will be of prime interest to state numerous properties of C.

Notions of paths and cycles

It turns out that C's restricted Lévy process succeeds at characterizing the notions of monotonicity. However, direct considerations on C's support lead to interesting and useful definitions for the proofs of the main result.

Definition 1.5 Paths and cycles

Let C be a C-process whose underlying Markovian process is M , and T ∈ N.

• A path of length T is the determination of Its value is the sum of its payoffs values, i.e.

-

T t=1 xt.

• A cycle is a path whose starting and finishing states are identical.

We define the cycle support of a C-process as the set of values that its cycles may take. We also define the minimum and maximum drifts δ -(C) and δ + (C) along cycles as the average increment by time period, i.e.

δ -(C) = inf ∞ n=1 1 n Vn(C) δ + (C) = sup ∞ n=1 1 n Vn(C)
We may notice that

• supp(C) ∩ R is never empty, because since C is positive recurrent, there is n ∈ N * such that

P (M (n) = M (0) ∧ C(n) -C(0) < ∞) = p > 0
which means that there is a cycle of finite value going from Ai to Ai of length n.

• We excluded trivial paths of zero length (and zero value) from the definitions of supp(C) and δ ± (C), so that each path lasts at least one time period.

• When C is a Lévy process, V1(C) is the support of any of its increments, so this notion of cycle support extends the notion of support of a random variable, as supp(C) is the support of all multiperiod increments.

Defining paths and cycles will be useful both to

• Seize a probability of C taking any "possible" specific path over a short amount of time (force C's short-term behaviour) ;

• Assure that C cannot drift faster than δ ± (C) over time (control C's long-term behaviour).

Monotonicity of a C-process

The condition "existence of positive [negative] values in the support of increments" for Lévy processes translates to "existence of positive [negative] cycles" for C-processes. If a Lévy process only has nonnegative [non-positive] increments, then it will be monotone non-decreasing [non-increasing] ; C-processes benefit from similar properties, as stated here.

Definition 1.7 Global monotonicity A C-process C is said to be

• Globally increasing iff supp(C) ⊆ R + ∪ {∞} ;

• Globally decreasing iff supp(C) ⊆ R -;

• Globally monotone iff either globally increasing or globally decreasing ;

• Globally constant iff both globally increasing and globally decreasing, i.e. supp(C) = {0}.

Globally monotone processes indeed hold properties of no-return.

Proposition 1.2 Consequences of global monotonicity

Let C be a positive recurrent C-process. The following properties are equivalent :

1. C is globally increasing.

2. C's restricted Lévy process is almost surely non-decreasing.

3. There is Q ∈ R + such that for any s, t ∈ N with s < t,

P (C(t) -C(s) ≥ -Q) = 1
4. There are C-processes C + and C = , whose underlying Markovian processes are C's one, with :

• C + such that C + (0) = 0 and almost surely non-decreasing ;

• C = globally constant ;

• C rewrites as C = C + + C = .

We notice that

• For C globally decreasing, the symmetric properties hold : the restricted Lévy process is nonincreasing, C(t) -C(s) ≤ Q almost surely, and we may take C -non-increasing instead of C + . However, we shall not need this symmetric lemma in the sequel.

• Positive recurrence of C is required. A counter-example where M and not C is positive recurrent, yet we have statement 2 and not statement 1, is provided after the proof.

C is actually said globally monotone because faults (values of C(t) -C(s) of the wrong sign) in its monotonicity are always bounded by the value Q.

One expects globally constant C-processes to hold both bounds (by -Q and Q) and be bounded over N. It turns out that an even stronger control holds : a globally constant C-process is actually no more than its underlying Markov chain, as its values C(t) are completely determined by M (t).

Proposition 1.3 Globally constant C-processes

Let C be a positive recurrent C-process. The following properties are equivalent :

1. C is globally constant.

2. For each state number i ≤ A, there is ci ∈ R such that almost surely,

∀t ∈ N, C(t) = c M (t)
3. There is Q ∈ R + such that almost surely,

∀t ∈ N, C(t) ∈ [C(0) ± Q]
Moreover, every globally constant and positive recurrent C-process is bounded.

Looking at a random realization for a globally constant C-process yields the figure 2. In this graph, each color refers to the state hit by M at present time. We see that points of a same color are at the same value for C, referring to the "height" of the corresponding state in the proof ; in particulal, C is stuck in an interval whose width is the maximal discrepancy between these heights. Likewise, a similar graph for globally increasing C-processes appears in figure 3 : points of a same color are oriented upwards, corresponding to non-negative values of cycles. The properties of global monotonicity will be used in the proof to recognize and exclude globally increasing processes easily from the study, because they void the definition of martingale parameters that we will see later.

Periodicity of a C-process

In the proof, the key step ensuring the convenient control of T0's Laplace transform only works when C's increments lie "evenly" on reals. It so happens that the only phenomenon preventing convergence to take place in the general case is the periodicity of C's cycle support, described here.

Definition 1.8 Form of a cycle support Let C be a positive recurrent C-process, whose cycle support is supp(C) deemed not contained in {0, ∞}.

• If there is p ∈ R * + such that supp(C) ⊆ pZ ∪ {∞}, then C will be said p-periodic. There is a largest such p holding this property, it will be called C's fundamental period.

• If not, then C will be said aperiodic.

We should notice that :

• The case C globally constant is excluded from this definition, as it would be p-periodic for any p ∈ R * + , so there would not be a largest p.

• The set of values q such that C is q-periodic is p/N * where p is C's fundamental period, and we may refer to them as harmonics of the fundamental frequency (1/p).

The constraint of periodicity is very stringent to C's transition payoffs, e.g. if C is a Lévy, it means that its increments D hold D ∈ pZ almost surely ; in particular, this holds for C's restricted Lévy process. Since we shall consider bounded C-processes, then D can take only finitely many values ; this idea will be used when solving the case where C is periodic.

Laplace matrix function

As we aim at controlling the Laplace transform of C's default time T0, it is natural to define Laplace transforms for the random variables governing it ; this is the purpose of this section.

Definition of the Laplace matrix function

While a Lévy process is characterized by the Laplace transform of its only increment, we chose to define C's "Laplace transform" as a matrix of Laplace transforms for all increments. The reasons behind this definition are multiple, mainly because we will often need the structure of matrices to express main properties of a C-process.

Definition 1.9 Laplace matrix function Let C be a C-process. We define its Laplace matrix function as

LC = R → MA R + α → Pi→jE e -αD i→j 1D i→j <∞ i,j
whenever ∀(i, j) ∈ Γ, the expectancies converge. If (i, j) / ∈ Γ, then we set (LC (α)) i,j = 0 disregarding exponential integrability of Di→j.

Requiring the condition Di→j < ∞ is tantamount to considering the analytical continuation of Di→j's natural Laplace transform from R + so that the values +∞ are eliminated (this is done on purpose, as we shall see). This Laplace matrix function will play the part of the Laplace transform of C's increments.

We know that giving the Laplace transform of a sEI random variable fully describes its distribution ; if C is a Lévy process, for every n ∈ N * , the joint distribution of (C(t)) t≤n ∈ R n may be recovered using the Laplace transform of the increment D, as is gives D's distribution function, thanks to

P (∀t ≤ n, C(t) -C(t -1) ≤ xt) = n t=1 P (D ≤ xt)
So in a sense, one Laplace transform characterizes a whole Lévy process. Likewise, giving the Laplace matrix function of a C-process fully characterizes the distribution of its trajectories. As a matter of fact, it does not characterize the distribution of its underlying Markovian process in general, because of the cases where an increment is allowed to be +∞.

Proposition 1.4 Characterization by the Laplace matrix function

Let C1 and C2 be two C-processes whose

• Underlying Markovian processes M1 and M2 share the same state space (Ai) i≤A ;

• Laplace matrix functions L1 and L2 coincide over an interval I containing 0 and a positive value ;

• Starting points and starting states are identical.

Then for every n ∈ N * , for every xt ∈ R ∪ {∞} (with t ≤ n),

P (∀t ≤ n, C1(t) ≤ xt) = P (∀t ≤ n, C2(t) ≤ xt)
Moreover, if additionally L1(0) 1 = 1 , the same holds for their underlying Markovian processes, i.e.

for every n ∈ N * , for every at ≤ A (with t ≤ n),

P (∀t ≤ n, M1(t) = Aa t ∧ C1(t) ≤ xt) = P (∀t ≤ n, M2(t) = Aa t ∧ C2(t) ≤ xt)
so the whole C-process is recovered.

One might say that when P (Di→j = ∞) = p > 0, there is a loss of p between the sum of L1(0) over row number i and the one for M1 (which is always 1), and we do not know where it comes from in general. Even if one cannot determine for which k ≤ A the transition (i → k) allows for a probability p/P i→k of being +∞ looking only at L1 (thus we cannot recover M1), all possibilities have the same effect on C, i.e. driving it to +∞ with probability p (thus it is not an obstacle to the recovery of C1).

Concatenation of Laplace matrices

When C is a Lévy process, the Laplace transform of the concatenation of n increments, i.e. the random variable C(t + n) -C(t), is simply given by the n th power of the original Laplace transform. When C is a C-process, this multiplicative property does not hold as is because increments are not independent, but thanks to the construction of its Laplace matrix function, it translates to the matrix power. More specifically, let us take CT n the process whose successive values are CT n (t) = C(nt) ; as explained in the proof, it may be seen as a C-process whose

• Underlying Markovian process is MT n , such that ∀t ∈ N, MT n (t) = M (nt), with transition proba- bilities P i n -→j ; • Transition payoffs are D i n -→j ; • Starting point is C(0).
Then CT n 's Laplace matrix function is C's one to the n th power.

Proposition 1.5 Powers of LC (α)

Let n ∈ N * and α ∈ R such that LC (α) is well-defined (converges). The n th power of the matrix LC (α) corresponds to the Laplace matrix function of

CT n at point α, i.e. ∀i, j ≤ A, (LC (α) n ) i,j = P i n -→j E e -αD i n -→j
Incidentally, we notice that making n = 0 still works : the unit transition matrix is the identity, and empty transition payoffs are 0, so we still get

P i 0 -→j E e -αD i 0 -→j i,j = LC (α) 0 = Id
Nevertheless, the above property will become useful when considering long-range expectancies of C.

Differentiation of a Laplace matrix function

A useful link between expectancies and Laplace transforms appears when computing their local derivatives at point 0 : for X an exponentially integrable random variable, we have

-E(X) = dLX (α) dα (α = 0)
It is a well-known fact that the behaviour of a Lévy process tremendously depends on the expectancy and variance of each of its increments, but that they roughly suffice to describe its asymptotical behaviour. Indeed, the central-limit theorem proves that asymptotical computations on Lévy-kind processes "dump" any information on the distributions of increments but mean and variance. Some kind of variance term may be defined for C a C-process, however it requires some deep analysis (as explained in paragraph 2.1.4), so we will focus on the notion of "average increment" for now, that also governs C's asymptotical behaviour : we call it C's mean expectancy. To get this average drift, we are going to translate the above characterization for a single random variable (the only increment of a Lévy process) to C's Laplace matrix function, in order to get an analogous item for C-processes.

Definition 1.10 Diff-Laplace matrix function Let LC be the Laplace matrix function of a C-process C, deemed well-defined over an opened interval I ⊆ R. Its diff-Laplace matrix function is defined by

RC = I → MA (R) α → -dL C dα (α)
When X is a real random variable deemed exponentially integrable over I, we may define RX likewise.

When C is not sEI but only integrable, RC may not be defined anywhere. However, we shall still name RC (0) the matrix whose entries are given by ∀i, j ≤ A, (RC (0)) i,j = Pi→jE (Di→j) as this is the expression we find when C is sEI.

Definition 1.11 Mean expectancy

Let C be a sEI C-process, whose underlying Markovian process is M deemed positive recurrent, and transition payoffs are Di→j. By hypothesis, let µ be its invariant measure. The mean expectancy of C is the value

E(C) = A i=1 A j=1 µ [i] Pi→jE (Di→j) = µRC (0) 1
A possible interpretation for the mean expectancy E(C) is that it describes the average increment of C, as it is the mean of expected increments of transition payoffs, with weights being the expected amount of time M will spend on each transition. Hence, the law of large numbers expects C to drift along a line of slope E(C). In particular, if C is actually a Lévy process, then

E(C) = E (C(t + 1) -C(t)) for any t ∈ N is C's drift.

Theory of C-processes

Now that the definitions are complete, we get on with controlling ΛT 0 for a C-process C : we eventually aim at proving an equivalent form of Schmidli's Cramér-Lundberg approximation from [START_REF] Schmidli | Risk theory[END_REF] E e -aT 0 = (Z + o( 1))e -αC(0) when C is a discrete-time C-process. In this idea, we want to find an equivalent term to this α for a C-process ; however, we also expect the sought value to depend on M 's starting state M (0), so we shall first define several characteristics of a C-process :

• An exponential parameter governing the decay of C's default risks ; • A vector indicating the specificity of each starting state on these risks.

As we know thanks to the proposition 1.4 that LC completely describes C's distribution, we shall use LC to find these items. After this, we shall describe how they lead to the sought asymptotics of the default risks.

Martingale parameter

Specific calculations on default probabilities for Lévy processes (of random increments D) involve a parameter usually named "scale parameter" for diffusion processes (or Lundberg's exponent in [START_REF] Kyprianou | Introductory Lectures on Fluctuations of Lévy Processes with Applications[END_REF]), the non-trivial value α ∈ R for which E e -αD = 1

The scale parameter exists (when D is sEI) iff neither D ≥ 0 almost surely nor D ≤ 0 almost surely, and controls the default probability thanks to the martingale property. Results about their default times derive from the fact that the exponential process defined below is a martingale :

N → R + t → e -αC(t)
C-processes have analoguous terms to scale parameters ; however, like in [START_REF] Kyprianou | Introductory Lectures on Fluctuations of Lévy Processes with Applications[END_REF] we will extend this notion to other values for E e -αD rather than 1. Indeed, we shall look at an exponential equation like E e -αD = e a where a ∈ R * + (and after this, a ∈ R + ) is a parameter, that we will use later to find the Laplace transform of T0 at any point. When C is a sEI Lévy process, this equation in α ∈ R * + involves D's Laplace transform ; for a sEI C-process, it becomes an equation involving "C's Laplace transform", which is its Laplace matrix function LC . This equation is then solved for its dominant eigenvalue, as it will be the one governing C's behaviour.

Main properties of the martingale parameter

A milestone for this study is the definition of C's martingale parameter and dominant eigenvectors, as they govern its behaviour and will be required in the rest of this study.

Proposition 2.1 Martingale parameter

Let C be a positive recurrent and sEI C-process, deemed not globally increasing. Let a ∈ R + be a Laplace parameter.

1. There is a single α ∈ R + such that e a is the dominant (sometimes called "Perron-Frobenius") eigenvalue of LC (α).

2. For this α, the column eigenspace associated with the dominant eigenvalue e a of LC (α) has dimension one, and may be directed by w (a) a positive vector.

3. The associated row eigenspace has dimension one, and may be directed by a positive vector µ (a) .

4. We may choose the scalings of w (a) and µ (a) such that they hold

∀a ∈ R + , µ (a) 1 = 1 ∧ µ (a) w (a) = 1
These conditions, named "equations of scaling", ensure unicity of w (a) and µ (a) .

5.

In particular, for a = 0, (a) When E(C) ≤ 0, their limits exist and are α(0) = 0, w (0) = 1 , and

µ (0) = µ (M 's invariant distribution) ; (b) When E(C) > 0, we have α(0) > 0.
For every a ∈ R + , we name these items at point a :

• α(a) is C's (positive) martingale parameter ; • w (a) is C's dominant (column) eigenvector ; • µ (a) is C's dominant row eigenvector.
These items have a smooth behaviour :

1. α(a) is an increasing and concave expression of a ∈ R + .

2. Viewed as functions of a ∈ R + , the expressions α(a), w (a) and µ (a) are continuous over R + and C ∞ over R * + . It is also possible to look for a negative value of α that solves the eigenvector equation. We name it the negative martingale parameter, and it holds the same properties as its positive counterpart.

Proposition 2.2 Negative martingale parameter

Let C be a positive recurrent and sEI C-process, deemed not globally decreasing.

• For every a ∈ R + , we name β(a) ∈ R + the negative martingale parameter of C, defined as -C's martingale parameter.

• In particular, for a = 0,

1. When E(C) ≥ 0, we have β(0) = 0 ; 2. When E(C) < 0, we have β(0) > 0.
We notice that the definition of α(a) fails when C is globally increasing, while β(a) fails when C is globally decreasing. This is no surprise, since a Lévy process with non-negative or non-positive increments cannot have a scale parameter.

For now, we are going to investigate on what happens to α(a) and β(a) when a goes in the negative region. Throughout this study, this idea will be analyzed at some steps of the work, with their respective implications on its results.

• If E > 0, we find some martingale parameters α(a) ∈ R * + when a is not too large a negative value, and we will discuss about that in similar remarks. To get an idea, let us consider that C is a Lévy whose increments are D. Recalling that α is the reciprocal function to ΛD, it stops being defined when a hits min (ΛD), where α hits a double solution : indeed, we get α(a) = -β(a).

• If E = 0, a cannot become negative, as in the proof we get that D is a martingale and then e 0 = 1 is the minimal value of D's Laplace transform.

• If E < 0, we find again some martingale parameters α(a) ∈ R * -when a is not too large a negative value ; the domain where α(a) may be continued is a finite interval (stopping at ΛD's minimum), unless D ≤ 0 almost surely, which case means that C is globally decreasing. As it turns out, we will find out later that T0's Laplace transform may be prolongated for these negative values of a, using the maringale parameter found this fashion.

Martingale process

The usefulness of martingale parameters lies in the upcoming martingale property, as it transforms a C-process into a martingale.

Definition 2.1 Martingale process

Let C be a positive recurrent and sEI C-process, deemed not globally increasing, and a ∈ R + . Let α(a) be its martingale parameter at point a and w (a) its dominant eigenvector. We define the process :

X (a) C = N → R + t → w (a) [M (t)] e -α(a)C(t) e -at
This process is a martingale, named C's martingale process at point a.

In contrast with [START_REF] Schmidli | Risk theory[END_REF], we recover Schmidli's lemma 9.3 where g is the dominant eigenvector and r the Laplace parameter, and the function θ given in there coincides with α -1 . We notice that when C is globally increasing, such exponential-based processes (for α > 0) will also have a decreasing shape, so cannot be martingales (unless C is globally constant and for a = 0, when X (0) C becomes trivial). Proceeding with the thoughts when a becomes negative, we get :

• If E > 0, then 0 < α(a) < α(0). For X (a) C
to keep the martingale property, then e -α(a)C(t) needs to be "smaller" than e -α(0)C(t) , which can be done to some extent ; in the cases of Lévy processes, it is until E e -αD encounters its positive minimum, where the martingale parameter stops being defined.

• If E = 0, the point a = 0 associated with α(0) = 0 may be regarded as a "double" solution (the local derivative of D's Laplace transform is zero and α(0) = β(0) = 0). A common link between double solutions of a Laplace equation and solution functions appears : there is a martingale process whose form is a polynomial (affine) multiplied by an exponential, but we shall not discuss about it further.

• If E < 0, we get α(a) < α(0) = 0. For X (a)

C to keep the martingale property, then e -α(a)C(t) again needs to be "smaller" than e -α(0)C(t) , which can be done to some extent. However, we could use this martingale property with the default time as stopping time, which leads to interesting results (negative values of α for which E e -αT 0 < ∞, indicating fast default events).

On a side note, since differentiation is a linear operator and the processes X (a)

C are martingales for every a, then differentiating them with respect to a still yields a martingale by linear stability of martingales. This idea may be used to get a martingale equation involving C itself as a linear term, but once again we shall not proceed in this way.

Spread

The discrepancy between w (a) 's coordinates measures the specificities of M 's states : as we shall see with the main theorem 1, higher values indicate that the corresponding state is detrimental to C compared to other states, for an identical value of present C(t). We may measure these discrepancies to quantify the magnitude of M 's effects on C's default probability : this leads to the following definition.

Definition 2.2 Spread

Let v ∈ (R * + ) A . Its spread is defined by

δ(v) = ln max i,j≤A v [i] v [j]
A spread is always a non-negative real, being 0 iff v is proportional to 1 , so it is a satisfying measure of the discrepancies between v's coordinates.

• If w (a) has a high spread, one should take extra care of M 's present state, as C(t) and α may fail to represent default risks. Rewriting

w (a) [M (t)] e -α(a)C(t) = e -α(a)   C(t)- ln w (a) [M (t)] α(a)   
one sees that w (a) 's spread may lead to an error of up to δ w (a) /α(a) on the "true value" of the cash reserves C(t). In the state of the market M (t), one should reevaluate the capital value C(t) as described by this equation to correctly assess the default risks.

• Conversely, when w (a) has a low spread, M 's effects are limited and one can safely ignore it, simplifying the model to a Lévy process. In an extreme case, we have indeed the equivalence between δ w (a) = 0 and the fact that C is actually a Lévy process, as explained below.

Thus, the maximal correction δ w (a) /α(a) is a measure of C's distance from being a Lévy.

Proposition 2.3 Zero spread

Let C be a positive recurrent, bounded, and not globally increasing C-process. For every a ∈ R * + , let w (a) be C's dominant eigenvector at point a.

If ∀a ∈ R * + , δ w (a) = 0, then there is a Lévy process C such that ∀t ∈ N, C(t) = C (t) almost surely.

We remark that M 's states need not be equivalent for C to have a Lévy-like distribution.

Lévy processes and variance terms

When C is a Lévy process whose increments are D, the function α is actually reciprocal to ΛD over its domain. The above results now translate to simplified forms :

• w (a) and µ (a) disappear, since they amount identically to the 1-dimensional vector whose only coordinate is 1.

• Their spread is identically zero, which characterizes Lévy processes after the proposition 2.3.

• The martingale process becomes

X (a) C = N → R + t → e -α(a)C(t) e -at
In particular, let us make a = 0 :

• When E(C) > 0, we get α(0) = Λ -1 D (0) is C's natural scale parameter, β(0) = 0, and
dβ(a) da (a = 0) = 1 E(C)
• When E(C) < 0, we get α(0) = 0, -β(0) is C's natural scale parameter, and

dα(a) da (a = 0) = -1 E(C)
Notice that this is not for Lévy processes only : differentiating the martingale equation for the martingale process around a = 0 yields

A j=1 Pi→j dw (a) [j] da (a = 0) - dα(a) da (a = 0) A j=1 Pi→jE (Di→j) dw (a) [j] da (a = 0) = dw (a) [i] da (a = 0)
thus by definition of E(C), one gets

dw (a) [i] da (a = 0)E(C) 1 = (P -Id) dw (a) da (a = 0)
and thanks to the definition of µ, left multiplication by µ leads to the result. In particular, E(C) = E(D) when C is a Lévy process. Now, let us name γ the one among α and β that is 0 at point a = 0. It is C 2 around 0 as soon as D is sEI, and one finds out that

- d 2 γ(a) da 2 (a = 0) = V(D) |E(D)| 3
This is, to some extend, linked with the relationship between moments of the random variable D and its log-Laplace transform, as for a Lévy process α = Λ -1

D . An interesting idea might be to reverse this equality, getting some "variance term" for C-processes that we define as

V (C) = -d 2 γ(a) da 2 (a = 0) dγ(a)
da (a = 0)

3
As γ is concave, one finds out that V (C) ≥ 0 ; moreover, V (C) = 0 iff α has no concavity around zero. However, using the notations Dτ and τ of the proof, V (C) = 0 leads to

τ + 1 = Dτ E(C)
almost surely, which is tantamount to saying that there are a drift E (it is E(C)) and a globally constant C-process C = such that ∀t ∈ N, C(t) = tE+C = (t) almost surely. While a zero variance for Lévy processes is the translation of C(t) = tE + C(0) almost surely, the term C = that plays the part of the constant C(0) is now a globally constant C-process. In a sense, the "kernel" of Lévy processes is constituted by identically constant processes, while this notion translates to globally constant C-processes.

Main theorems

To simplify the proofs, we shall deem henceforth that C is bounded and positive recurrent. However, we firmly believe that only C sEI is required, even if we shall state our results only for C bounded.

Method

The natural method used to approximate LT 0 (a) comes from the definition of the Laplace transform.

Considering the general problem of finding LT 0 (a) from any starting state Ai and starting point C0, named L

[i] (C0) hereafter, the proof follows these generic steps : 1. By virtue of of T0's definition and the fact that C is a C-process, write down an equation describing the relationship between E L

(a) [M (t+1)] (C(t + 1)) and E L (a) [M (t)] (C(t)) .

Use time concatenations to scale this equation in functions L into an equation in auxiliary functions

K, involving decreasing transition payoffs.

3. Prove that, when these functions K behave "well", they converge to a common limit at infinity.

4.

Get back to the initial question about LT 0 (a), using all previous results. This methods works for most bounded and positive recurrent C-processes, but fails in specific cases :

• When C is globally increasing, it has no martingale parameter, and the Laplace equation cannot be scaled. However, the proposition 1.3 indicates that it cannot drop lower than the no-return property states ; as a consequence, there is Q ∈ R + such that the default probability is exactly 0 once C passes Q, so we are not interested in this case anymore.

• When C's cycle support is not "evenly" distributed over R * -, the functions K do not converge. This is a serious failure (and not a flaw of our study), which leads to the issues of periodicity, dealt with in this section. On a side note, present litterature (e.g. [START_REF] Schmidli | Risk theory[END_REF][START_REF] Yu | On the expected discounted penalty function for a markov regime-switching insurance risk model with stochastic premium income[END_REF]) avoided these considerations thanks to the continuous nature of the premiums in Cramér and Lundberg's risk model ; however, we decided to emphasize on them because they will allow us to explain the optimality of L (a)

[i] (C0)'s asymptotics.

We first deal with the latter case, where the general method of proof fails. Hence, we deem that C is periodic, and we shall note by p ∈ R * + its fundamental period.

Statement for periodic C-processes

S k n-1 →kn = {0}
This means that all sets S k n-1 →kn must be singletons, e.g. {pi,j} for some pi,j ∈ R/pZ. When C is positive recurrent, it is possible to link all values pi,j through a Chasles-like identity, creating values p k modulo p such that pi,j = pj -pi for every i, j. If we choose p M (0) ≡ C(0) modulo p, it follows that for every t ∈ N, we get C(t) ≡ p M (t) modulo p. To get a result when C is p-periodic, our idea is to "split" the transition payoffs between

• A principal part in pZ, indicating the main effects of transition payoffs in C ;

• A residual part in [0, p), indicating the residual effects of the transition payoffs, not hampering the study.

In particular, the residual part has no effect on C's default. One may imagine that, for p = 1 in this example,

• The principal part is C's integer part ;

• The residual part is C's fractional part.

The idea behind this decomposition is that C(t) < 0 iff C(t) < 0, so we may dump the residual part and find out C's default only observing its principal part, that lies in Z.

Definition 2.3 Regular process

Let C be a positive recurrent, periodic C-process. We assume that C is not globally constant, so p is its fundamental period.

• The values (pi) i≤A such that ∀t ∈ N, C(t) ≡ p M (t) modulo p are called C's natural offsets.

• The process C defined by

C = N → Z t → 1 p C(t) -p M (t) is well-defined in Z,
and is a C-process with integer increments, whose fundamental period is 1. It shall be named C's regular process. Now, let us take C a bounded, p-periodic C-process. As C lies in Z, the Laplace transform of its default time at point a ∈ R + may be written through numerical sequences (L k,c ) k≤A,c∈Z solving induction-like equations

∀k ≤ A, c ∈ N, L k,c = e -a A j=1 ∞ d=-∞ P k→j P (D k→j = d) L j,c+d
where L k,c = 1 whenever c < 0. This system may be solved analytically, which yields a solution for the process C, that eventually translates to a solution for C itself.

Proposition 2.4 Laplace transform of T0 for a periodic C-process

Let C be a positive recurrent, bounded and not globally increasing C-process. We deem that C is periodic, so it has p ∈ R * + as fundamental period. For every a ∈ R * + , let α(a) be C's martingale parameter at point a, and w (a) the associated column eigenvector. There is a function K :

• Defined over [0, p) × R + onto R ;
• Being piecewise continuous over its domain ;

• Being continuous of its second variable at any fixed point for the first, such that for every i ≤ A, the log-Laplace transform of T0 given M (0) = Ai and C(0) = C0 ∈ p0 + pN (with p0 ∈ [0, p) being any congruence modulo p) holds

-ΛT 0 (a) ∈ α(a)C0 -ln w (a) [i] + K(p0, a) ± e(C0, a)
where e is a non-negative error function, uniformly exponentially convergent to 0 over any compact subset of R + in a when C0 goes to infinity. It means that

∀a0 ∈ R + , ∃e1(a0) ∈ R + , e2(a0) > 0, ∀C0 ∈ R + , ∀a ≤ a0, e(C0, a) < e1(a0)e -e 2 (a 0 )C 0
As most remarks on this proposition are similar to observations on the main result of this study, we shall only focus on the differences between those.

• Because of C's periodicity, the default times are piecewise constant of C0.

• For this reason, since the martingale parameter α(a) has a linear (in C0) effect on the main term in the theorem, the approximation of a constant function over a non-trivial interval by an affine (multiplicative factor α(a)) function of C0 on this interval yields an incompressible error, no matter how remote is C0. It follows that the requirement stating C0 ∈ p0 + pN cannot be removed.

• The guaranteed convergence is exponential. The main idea behind this assertion comes from the fact that p is C's fundamental period, rather than an harmonic np for some n ∈ N greater than 1, so C has cycles whose value are -kp for any k ≥ k0 for some large enough k0. It follows that L

C 's "second" eigenvalue λ2(a) is strictly less than 1, so residual terms in the recursive scheme converge exponentially (at speed o(λ n ) for every λ > λ2). However, this idea does not work if C is aperiodic, for reasons that we shall see in a later section ; indeed, the convergence may not be exponential.

Main theorem

Now that the special cases have been removed from the study, we move on to the general case, where C is not "simple" : neither globally increasing, nor restricted to a discrete subset of Z. The main theorem in this study controls asymptotics of the ΛT 0 's log-Laplace transform.

Theorem 1 Laplace transform of T0 for an aperiodic C-process

Let C be a positive recurrent, bounded and not globally increasing C-process, deemed aperiodic. For every a ∈ R + , let

• α(a) be C's martingale parameter at point a,

• w (a) be the associated column eigenvector.

There is a continuous function K : R + → R such that for every i ≤ A, the log-Laplace transform of T0 giving M (0) = Ai and C(0) = C0 holds

-ΛT 0 (a) ∈ α(a)C0 -ln w (a) [i] + K(a) ± e(C0, a)
where e is a non-negative error function, uniformly convergent to 0 over any compact subset of R + in a, i.e.

∀a0 ∈ R + , ∀ > 0, ∃Ca 0 ∈ R + ; ∀C0 > Ca 0 , ∀a ≤ a0, e(C0, a) < Moreover, if C is a Lévy process, then K(a) ∈ R * + .
The strength of this theorem appears in the interpretation of its error terms.

• The error goes to 0 when C0 increases. Indeed, the main idea in the proof of this main theorem is that LT 0 's dependency on C(T0) and M (T0) eventually disappears when C0 goes to infinity, because the distribution of (C(T0), M (T0)) conditionally to defaulting converges. This "limit" distribution, represented by the random couple (C f , M f ) ∈ R * -× {Ai} i≤A , is used by the martingale property : neglecting dependency between T0 and the final values C(T0) and M (T0), one should get

w (a) [M (0)] e -α(a)C 0 = E w (a) [M (T 0 )] e -α(a)C(T 0 ) e -aT 0 ≈ E w (a) [M f ] e -α(a)C f E e -aT 0 Hence, considering K(a) ≈ ln E w (a) [M f ] e -α(a)C f then one gets ΛT 0 (a) ≈ ln w (a)
[M (0)] e -α(a)C 0 -K(a) which is the term given by the theorem. The convolution equation in the proof "merges" uncertainty on the severity of default and the state of the market when defaulting into the term K(a), and this holds independently of C(0).

• The error term e(C0, a) comes from the "speed" of merging uncertainties, and e(C0, a) converges faster when the transition payoffs are evenly distributed among the convex hull of their supports. Let us take extreme examples for C being a Lévy process, whose increments are represented by the random variable D :

-If -D has an exponential distribution, then C(T0)'s distribution will be exactly identical no matter C0 and independent of T0, because exponential distributions hold the "memory loss" property. The approximate computation above then holds rigourously and there is no error term. As a matter of fact, this corresponds to the special case of Cramér and Lundberg's risk model ( [START_REF] Blom | [END_REF]) with exponential claims, where we eventually get an exact ruin probability as Ke -α(0)C 0 ( [START_REF] Rolski | Stochastic Processes for Insurance and Finance[END_REF]). -If the Lebesgue measure is absolutely continuous with respect to D's distribution over the convex hull of its negative support, for example if

∃y ∈ R + ; ∀x ∈ (-Q, 0) , ∀η ∈ (0, -x), P (D ∈ (x, x + η)) ≥ yη
then the convergence will be at least exponential, with a better exponential parameter when y is larger. This is because D "greatly mixes" the possible values hit by C through its way to default, so (C(T0), M (T0)) closes to (C f , M f ) faster, and the error term will decrease faster. -Contrariwise, if C is periodic, the theorem fails because repetitions of D do not mix the values hit by C, but concentrate them (e.g., if C is a regular process, on integers). As stated before, the function associating C0 with the sought Laplace transform at any fixed point a will be piecewise constant, and cannot behave asymptotically like a non-trivial exponential.

-If repetitions of D hardly mix the values hit by C, we are driven back to periodicity problems, which happen with a severe impact if t successive time concatenations of D leave C(t) "close to" a periodic set. Indeed, the upcoming proposition 2.6 creates such processes, whose error terms converge arbitrarily slowly.

For example, doing a = 0 leads us to the asymptotical default probability.

Proposition 2.5 Default probability

Let C be a positive recurrent and bounded C-process, deemed aperiodic.

1. If E ≤ 0, then P (T0 < ∞) = 1 unless C is globally constant. 2. If E > 0, there is some constant X1 ∈ R * + such that P (T0 < ∞) = X1w (0) [i] e -α(0)C 0 (1 + o(1))
where the o(1) refers to C(0) going to +∞.

Slow or quick convergence

The mixing involved in the proof constitutes the key step for the convergence speed of the error term e(C0, a) in the main theorem. The convergence may thus be

• Arbitrarily slow if this mixing is ill-distributed ;

• Exponentially fast if this mixing is well-distributed.

To enlighten this, we build the following Lévy processes, whose increments may only take two distinct values, chosen such that the mixing is slow.

Definition 2.4 Liouville processes Let f : (N → N) be an increasing function, with f (0) = 0. We define

• The Liouville number associated with f , named f -Liouville number, as

L f = ∞ k=0 10 -f (k)
• The f -Liouville process is the Lévy process defined by its increments D such that

P (D = -1) = P (D = -L f ) = 1/2 and C(0) = C0 ∈ R + is deterministic.
Approximating at order n ∈ N * gives the truncated f -Liouville number at order n

(L f ) n = n k=0 10 -f (k)
This states that C's cycle support is "almost" 10 -f (n) -periodic, as L -Ln < 1.2 * 10 -f (n+1) , but is aperiodic as soon as L f is not a rational number. In particular, if ∀n ∈ N * , f (n) = n!, then L f is the common Liouville number

L f = 1 + ∞ k=1 10 -k!
Now, the mixing involved during the computations (descending process, convolution process) features a convolution equation like

K(x) = i piK(x -xi)
with i pi = 1 and values xi ∈ R * + . However, the construction of L f states that these values xi are concentrated, at order n, in intervals around points of 10 -f (n) N whose length have order 10 -f (n+1) . The terms K(x -xi) will not mix until the number of time concatenations exceeds something like 10 f (n+1)-f (n) , which is the C0 required to have a precision like 10 -f (n) on the error term : the convergence may be made arbitrarily slow.

Proposition 2.6 Slow convergence for the Liouville process

Let g : R + → R + be any function that converges to 0 towards ∞. There is f an increasing function (N → N) with f (0) = 0, defining some aperiodic f -Liouville process C such that any error function e that suits the main theorem 1 for C holds

∀a ∈ R * + , ∀y ∈ R * + , ∃x > y; e(x, a) > g(x)
Moreover, if E(C) > 0, x may be chosen uniformly :

∀y ∈ R * + , ∃x > y; ∀a ∈ R + , e(x, a) > g(x)
Thus, convergence may be very slow compared with the "expected" exponential form found for periodic C-processes.

On the other hand, the standard Markov-modulated risk model holds exponential convergence thanks to the continuity of C's premiums : this is illustrated by the following proposition.

Proposition 2.7 Quick convergence

Let C be a positive recurrent and bounded C-process, whose underlying Markovian process is M , and T be a stopping time for C such that almost surely,

• Either T = ∞ ; • Or ∀t < T, C(T ) < C(t).
We deem that there are u ∈ R * + , η > 0 and i, j ≤ A such that

∀x < y ∈ [0, u] , P (T < ∞ ∧ C(T ) -C(0) ∈ [-y, -x] ∧ M (T ) = Aj|M (0) = Ai) > (y -x)η
Then the error term e(C0, a) from the theorem 1 is actually uniformly exponentially convergent in the sense of the proposition 2.4.

We notice that periodicity voids this statement because the "mass" of C(T ) will be concentrated on pj + pZ as given by the offsets. As a matter of fact, this proposition should also work with continuoustime processes, so it solves the Markov-modulated risk model (from [START_REF] Schmidli | Risk theory[END_REF]) ; however, looking at the same model in discrete time (provided that claim sizes are commensurable with the unit drift) yields periodicity, so one should take extra care with discretization.

Applications

In this paragraph, we shall use the previous main theorems to solve the example presented in paragraph 1.2.3, and discuss about how positive correlation of increments is dangerous to Lévy-like estimations of default risks.

Solution for the example process

The boom-bust C-process from paragraph 1.2.3 is sEI rather than bounded, however we shall admit that the theorems still work for it, since it is simple to present the ideas behind the main values when using it.

Laplace matrix function

When C is the boom-bust process, computations lead to the values of LC (α) and RC (α) for every α ∈ R.

LC (α) = 1 200 178e α 2 /2-2α 11 + 11e -α 18e 2α 182e 8α 2 +2α
In particular for α = 0, one recovers M 's transition matrix in LC (0).

RC (α) = 1 200 178(3 -α)e α 2 /2-2α 11e -α -36e 2α -182(16α + 2)e 8α 2 +2α
The matrix RC (0) indicates the weighted expectancies associated with each transition. Therefore, the good state is given positive terms while the bad state is given negative terms, which indicates that as expected by choice of the transition payoffs, the good state pulls C upwards while the bad state pushes C downwards. One also observes that diagonal terms are large relatively to other terms, as they have a greater effect because of the time spent by M on these transitions.

Martingale parameter

We drew the solution set in (a, α) to the equation det (LC (α) -e a Id) = 0 getting the martingale parameter for the boom-bust process in figure 4. This calls for some remarks : • For a positive a, the martingale parameter α(a) is the first (here, only) point of the curve above the point (a, 0). As it is increasing, this is indeed tantamount to looking for the rightmost (i.e. dominant) curve.

• Likewise, the negative martingale parameter β(a) is the first portion of the curve below the horizontal axis. Other curves correspond to non-dominant eigenvalues.

• The tangent at point (0, 0) is not vertical : the slope is α (0) = -1/E(C) whenever E(C) = 0. This also means that α(0) > 0 when E(C) > 0, and illustrates the existence of martingale parameters for negative values of a, when proceeding along the branches until hitting the minimum a, where a vertical slope is found.

• Despite the appearances, f does not have any double solution on either dominant branch. The seemingly breaking slope is fairly common when looking at convex combinations of exponentials, like ln e x + e -x .

• For high values of a, the final slopes are roughly given by -1/δ ± (C), as these values come from the highest possible drifts for C. When C is not bounded (as it is the case here), these slopes go to 0.

Default probability

One finds for example at point a = 0 the values

α(0) ≈ .00398$ -1 ∧ µ (0) ≈ .429 .571 ∧ w (0) ≈ .947 1.040
Hence, for a high starting point like C(0) = 200$, one gets P (T0 < ∞) ≈ .427. We remark that

• The martingale parameter is extremely low (regarding the relative values of the transition payoffs), so the Lévy-like estimation of C's default probability through e -α(0)C(0) decays really slowly to 0. This is a consequence of positive correlation between transition payoffs, increasing the risks of default : high cash levels are required to be prepared against a bad sequence of transitions (this translates to "a crisis" in econometrics).

• Despite A2 appearing really worse than A1 through the transition payoffs, w (0) 's spread is only .094. However, we recall that the "corresponding correction" on C amounts to δ w (0) /α(0) ≈ 23.5$, a high value compared with C's transition payoffs, so it is not safe to approximate C as a Lévy process.

• The fact that w (0) [START_REF] Asmussen | Risk theory in a markovian environment[END_REF] < 1 and w (0)

[2] > 1 is not a surprise either : as A2 is a bad state, the default probability is higher than normal when starting from it, and conversely for A1.

Unfortunately, the exact default probability is extremely tedious to compute, requiring lookalikes to Pollaczek and Khintchine's formula ( [START_REF] Kyprianou | Introductory Lectures on Fluctuations of Lévy Processes with Applications[END_REF]). For this reason, we shall look at its behaviour later, through the case of a continuous-time C-process ; one might also refer to [START_REF] Dong | Numerical method for a markov-modulated risk model with twosided jumps[END_REF] for methods of numerical solving.

Alternate variance term

Recalling that we "called" the variance of a C-process to be

V (C) = -d 2 α(a) da 2 (a = 0) dα(a)
da (a = 0) 3 in paragraph 2.1.4, we may compare C's default probability to the default probability of a Brownian motion of drift E(C) and variance V (C). As we get V (C) ≈ 63.631$ 2 /s, we have

2E(C) V (C) ≈ .00397$ -1
which comes remarkably close to the actual α(0). This stresses the accuracy of defining V (C) as such, with the small discrepancy coming from the fact that C is not a Brownian motion. Interestingly, let us look at what happens if one computes V (C) while neglecting time dependency, assuming that C is a Lévy process. The random distribution of C's increment becomes either Di→j with probability given by the amount of time spent on each transition, i.e. µ [i] Pi→j : this yields the same expectancy E(C) but now the variance becomes V l (C) ≈ 14.222$ 2 /s, a largely underestimated value compared with V (C). As an immediate consequence of this estimation error, the estimated martingale parameter is overvalued by a factor greater than 4, which means that the default probability is largely undervalued. Taking C(0) = 200$, this leads to an estimated default probability around 2.87% instead of the correct 42.7%, illustrating that one cannot safely deem the transitions to be independent without risking huge errors while looking at market safety.

Quick default

When looking at a high value of a, e.g. a = 9, one gets α(9) ≈ .9485 ∧ µ (9) ≈ 7.4 * 10 -5 1 -7.4 * 10 -5 ∧ w (9) ≈ 9.417 * 10 -6 1.00007

This time,

• The martingale parameter is low compared with the value a = 9, because of the simple possibility of M getting stuck on the bad state : the probability of "quick" default is indeed headed by the worst sequences for C. Actually, when C is bounded, it may in turn be related with the minimal drift δ -(C), finding

E e -aT 0 ≈ e aC(0)/δ -(C)
When C is not bounded, thinking that δ -(C) = -∞ is a rough explanation for the low α(a).

• Computations lead to a maximal correction on C of δ w (9) /α(9) ≈ 12.2$, which still indicates that C does not look like a Lévy process. However, the Lévy-like estimation is almost correct for M (0) = A2 with a correction of ln w (9) [START_REF] Bachelier | Théorie de la spéculation[END_REF] /α(9), below 10 -4 $. Once again, this is because A2 is the most likely state to trigger a default, while starting from A1 "ensures" some time in this good state and delays the sequence of default for long enough to drop the probability of quick default.

Links with Lévy processes

As C-processes are a natural continuation of Lévy processes, one may find interesting links between these.

Hidden Lévy processes

First, if A = 1, i.e. the market always lies in the same state, the underlying Markovian process is trivial, the C-process becomes a pure Lévy process ; and reciprocally, any Lévy process is a C-process with A = 1. It follows that all results proved throughout this study also hold for Lévy processes, with a simplification : the dominant eigenvectors w (a) and µ (a) amount to the constant 1 and disappear from the computations. We also proved earlier an equivalence between zero spread and C being a Lévy : indeed, zero spread means w (a) ∈ R * + 1 , and then the scaling equations for w (a) and µ (a) lead to

µ (a) w (a) = µ (a) 1 = 1 ⇒ w (a) = 1
Hence, this simplification holds again when w (a) has zero spread for every a ; indeed, w (a) 's influence disappears from the statement of the theorem. An item used in the definition of the martingale parameter is the restricted Lévy process, being the process C (τ k ) k∈N where the times τ k are the successive hitting times of M 's starting state, which is a Lévy thanks to the property of canonical time sequences. One may remark that this trick of computing increments on several time periods, to compare identical states of the market, is commonly used by economists for instance when calculating price variations on a year from and to identical calendar dates, as the market undergoes changes due to yearly periodicity : this corresponds to identical states of M .

Continuous-time Lévy processes

In this paragraph, we imagine some extension of the study when C is a continuous time process. For example, a continuous-time C-process may be defined as such :

• M is a continuous-time Markovian process, changing of states with exponential waitings (like a non-explosive Poisson process, similarly to the Markov-modulated risk model) ;

• Transition payoffs are continuous-time Lévy processes, whose parameters are given by the present state M (t).

We notice that we may define additional random jumps for C when M changes of states (from Ai to Aj), as a random variable Di→j acting as a transition payoff between these states, but we will not in order to simplify the further explanations. As a consequence, the previous transition payoffs that were defined by means of transitions (between states) are now more accureately described as "state payoffs" (in a given state). For example, taking them to be the appropriate compounded Poisson processes, one recovers the Markov modulated risk model from [START_REF] Asmussen | Risk theory in a markovian environment[END_REF]. It is possible to define martingale parameters for such C-processes, and we shall compare the built items with [START_REF] Schmidli | Risk theory[END_REF].

1. One takes an infinitesimal time period dt, and then computes the transition probabilities, giving M 's transition matrix P dt over this time interval ∀i, j ≤ A, P dt i,j = P (M (t + dt) = Aj|M (t) = Ai)

Differentiation when dt goes to 0 ultimately leads to an log-transition matrix

P ∈ MA (R) such that ∀i, j ≤ A, t ∈ R + , P (M (t) = Aj|M (0) = Ai) = e tP
This matrix P is called η in [START_REF] Schmidli | Risk theory[END_REF].

2. Likewise, the Laplace matrix function of C over a time interval dt is reached, eventually getting a log-Laplace matrix function ΛC of C. This matrix is illustrated by Θ though the lemma 9.2 in [START_REF] Schmidli | Risk theory[END_REF].

The eigenvector equation should become

∀t ∈ R + , e tΛ C (α(a)) w (a) = e ta w (a) yielding α(a) (called R in [START_REF] Schmidli | Risk theory[END_REF]) and w (a) (g(r) in [START_REF] Schmidli | Risk theory[END_REF], where r plays a's part).

Thus one obtains martingale parameters for continuous-time C-processes. One now gets a result similar to the main theorem 1 for C-processes whose jumps are bounded by Q : assuming that C is right-continuous, a descending process C is defined through the binary determination sequence ρ given by

ρ(t) = 1 ⇔ (∀s < t, ρ(s) = 1 ⇒ C(t) ≤ C(s) -h)
where h ∈ R * + is an arbitrarily small parameter, ensuring that the time sequence τ defined by ρ is discrete. One continues as in the proof for the main theorem, with special care taken when C hits the region [0, h], as the convolution equation loses its validity here.

When C is a non-jump process, another interisting idea is to build a discrete "grid" of interesting positions for C (say, when C(t) ∈ Z), then use that C's trajectories are continuous to ensure that C must cross C0 -1 on its way to default. This allows us to get a descending equation for the functions

K (a) i , given by ∀x ∈ N, K (a) i (x + 1) = A j=1 P (a) i→j K (a) j (x)
where the terms P (a) i→j express the probability of having gone from Ai to Aj while C loses one unit. As P (a) is ergodic as soon as M is, we shall get an exponential convergence for the theorem 1. Finally, we notice that

• Problems related with periodicity do not appear here, as the values of K (a) i in (x, x + 1) are controlled by the values of K (a) j (x) because C has continuous trajectories (which is false for discrete, periodic C-processes). This is the reason why the standard Markov-modulated risk model dodges periodicity issues.

• Incidentally, exact expressions for K Therefore, there is a link between jumps and discrete time, as taking C over a discrete-time universe creates its jumps (the values C(t + 1) -C(t)). As a consequence, one is encouraged to use discrete C-processes whenever possible, unless continuous time allows C to have continuous trajectories. We suggest referring to [START_REF] Schmidli | Risk theory[END_REF] or [START_REF] Asmussen | Risk theory in a markovian environment[END_REF] for the main example of a Markov-modulated risk process for an insurance company.

Example of default probability in continuous time

Such a non-jump continuous-time C-process has a default probability shown in figure 5, and we shall admit that a typical discrete-time C-process yields a similar graph. The default probability of this process is given when starting from one of its 3 states (color), and the starting point C(0) (x-axis), in a log-plot fashion.

• The (negated) asymptotical main slope of each graph indicates the martingale parameter ; we see that it is common to each state (parallel asymptotes).

• Existence of these asymptotes indicates that the residual term (1 + o(1)) is correct.

• The vertical distance between the asymptotes refers to the dominant eigenvector w (0) : in this example, the red state is associated with a coordinate of w lower than 1, while the green one is higher. In particular, the vertical width containing all asymptotes is w (0) 's spread.

• Likewise, the horizontal distance δ w (0) /α(0) is the maximal correction to C(0) required to compare default probabilities between different states of M . Notice that in general, with discrete time (or with jumps), the default probabilities are not necessarily continuous of C(0), or well-ordered : there may be x, y ∈ R + and i, j ≤ A such that

P (T0 < ∞|M (0) = Ai ∧ C(0) = x) < P (T0 < ∞|M (0) = Aj ∧ C(0) = x) but still P (T0 < ∞|M (0) = Ai ∧ C(0) = y) > P (T0 < ∞|M (0) = Aj ∧ C(0) = y)
However, asymptotical considerations still work under the hypotheses of the theorem 1.

Links with Brownian motions

We briefly look at this example through a simple case, i.e. when C itself is a non-jump Lévy process (A = 1 and M is trivial). Infinite divisibility of a Lévy process means that one may decompose the time period into shorter and shorter subdivisions, so that the width of increments is progressively driven to 0. In particular, C(T0) = 0, and as dominant eigenvectors of Lévy processes have no spread, the martingale process X (a)

C allows for the equation E e -aT 0 = e -α(a)C 0

We recall that the martingale parameter α(a) of a Lévy process of increments D (with P (D < 0) > 0) is defined by the only α ∈ R * + solution to LD(α) = e a , i.e.

∀a ∈ R * + , α(a) = Λ -1 D (a)
where Λ -1 D is the reciprocal function to the restriction of ΛD on the set where it is increasing. The log-Laplace transform of T0 thus holds

∀a ∈ R * + , -ΛT 0 (a) = Λ -1 D (a)C0
However, when C is a non-jump Lévy process with homogeneous increments, by Lévy-Ito decomposition it is also a Brownian motion (with a drift) ; setting E ∈ R its drift (that coincides with its mean expectancy) and σ 2 its variance, we know from the definition that

∀w ∈ R, ΛD(w) = σ 2 2 w - E σ 2 2 - E 2 2σ 2
Then we get

∀a ≥ -E 2 2σ 2 , Λ -1 D (a) = E + √ 2aσ 2 + E 2 σ 2
So, the previous computation allows us to recover the formula for the log-Laplace transform of a drifted Brownian's default time from this expression :

∀a ∈ R * + , ΛT 0 (a) = -C(0) E + √ 2aσ 2 + E 2 σ 2
This allows us some observations :

• If E ≤ 0, then ΛT 0 (0) = 0 and we recover the fact that P (T0 < ∞) = 1. If E has a large negative value compared to σ 2 , then first-order approximation leads to ΛT 0 (a) ≈ C(0)a/E, which is indeed the expected behaviour of a deterministic process (E dominates the variance).

• If E > 0, then ΛT 0 (0) = -C(0) 2E/σ 2 < 0. The above comparison allows us to recover here the characteristic parameter 2E/σ 2 of the equivalent diffusion process' scale function.

• If σ goes to 0, then the process behaves as deterministic : we get again ΛT 0 (a) ≈ C(0)a/E if E < 0, but now P (T0 < ∞) goes to 0 if E > 0.

• If σ goes to infinity, then on the contrary ΛT 0 (a) ≈ -C(0) √ 2a/σ goes to 0, because high volatility is detrimental to survival.

We ultimately recover the generic ideas about stochastic processes : drift enhances the suvival probability, while variance hampers it.

Conclusion

The model of C-processes provides interesting thoughts about apparent volatility of financial processes. For example,

• Fat tails of increments are not required to yield high risks of default ;

• Positive correlation however really is a concern ;

• The present "state of the market" has a quantifiable effect on default risks (provided that the market is known).

We present these ideas in this conclusion.

Discussion

In this section, we discuss about and justify the hypotheses we assumed throughout this study, and the effects they have on the main theorem.

Need for positive recurrence of M

We deemed M to be positive recurrent ; actually, the analysis only needs the starting state M (0) to be positive recurrent, since if it is, then we only need to restrict M to its accessible states, which leads to a positive recurrent Markov process. When the starting state is not recurrent, a problem arises when computing the successive reductions of the Markov chain in the proof : for example, if all recurrent states have merged into a single A k , it is impossible to define transition payoffs "through" A k because there is no way out of A k . This has an effect on the equivalence between real solutions α ∈ R for M and its reduced matrix, as now a viable real solution disappears (this corresponds to the case (LC (α)) k,k = 1 of the proof). The alternate solution we are going to present if M is not positive recurrent has the effect of leading the study back to cases where it is. For the sake of simplicity, we shall only look at the case of the default probability (a = 0), and note α = α(0), as the general case is similar. It is possible to reorganize the matrix LC (α) as to merge separately transient and groups of recurrent states (the "closed communicating classes" of the proof), and eventually getting a reduced L C (α) as a (g + 1) × (g + 1) matrix, with row and column 1 referring to the transient state, and rows and columns 2 to g + 1 referring to the g ∈ N * recurrent states : we shall get some sub-geometrical increments D1,i for any i ≤ g + 1 and Di for any i from 2 to g + 1 such that

L C (α) =      p1LD 1 (α) p2LD 1,2 (α) • • • pg+1LD 1,g+1 (α) 0 LD 2 (α) • • • 0 . . . . . . . . . . . . 0 0 • • • LD g+1 (α)     
with pi the corresponding transition probabilities of the reduced process M . As p1 < 1, either D1 is nonnegative almost surely (in which case we set β = ∞), or it is not and then there is a single β ∈ R * + such that LD 1 (β) = 1/p1. Now we get

• β > 0, which may be regarded as the martingale parameter of the probability of default before hitting the recurrent part of M (i.e. at a = 0) ;

• Several values αi for any i from 2 to g + 1, each one being the martingale parameter of the default probability in the closed communicating class number i of M . This case will not be analyzed any further, but one eventually gets that the smallest of values αi and β governs the exponential behaviour of the default probability : for instance, if it is zero, then the default probability converges to a positive constant when C0 goes to infinity, correponding to the probability of M eventually hitting a closed communicating class over which C's mean expectancy is negative. However, if one of M 's states fails to be positive recurrent, removing inaccessible states is mandatory, else the main result fails. For example with C defined by the following M and transition payoffs :

(Pi→j) i,j = 1/2 1/2 0 1
hence A1 is not positive recurrent ;

P (D1→1 = -4) = 1 P (D1→2 = 0) = 1 P (D2→2 = π) = P (D2→2 = -1) = 1/2
Notice that we arbitrarily took π as a value to avoid C to be periodic, but any irrational number greater than 3 also worked. When M (0) = A1, the estimate given by the martingale parameter is correct : the default probability is well-controlled by the probability of defaulting before leaving state A1, which is (1/2) 1+ C 0 /4 , leading to a martingale parameter of β = ln(2)/4 ≈ .173. However, when M (0) = A2, C turns out to be a Lévy process, whose computations lead to α ≈ .619. This is because the "bad" state A1 actually has no effect on C, while still contributing to decreasing α, as it was not removed from the study.

Need for Q

The transition payoffs are deemed to be bounded by Q ∈ R + in the main theorem. The usefulness for this hypothesis lies

• When ensuring that C is automatically positive recurrent provided once M is ;

• When defining C's Laplace matrix function, ensuring both that it will be well-defined and that the Laplace transform of the restricted payoffs explodes at its boundary ;

• To technical lemmata, ensuring that C's descending process drops increments no larger than Q.

However, we expect the theorems to hold even when C is not sEI, provided that LC explodes at the boundary α0 of its convergence domain : some reasons behind this idea lie in the construction of α and LC , then of the functions K (a)

i .

• A martingale parameter α(a) is still found for every value of a ∈ R + , with the additional property α < α0 over R + .

• The definition of C is the same, and the convolution process Φ (a)

C is still well-defined ; • As exponential expectancies of C's values only appear as E e -wC(T ) for T a hitting time and some w ∈ α R + , which implies w < α0, the expectancies are always integrable.

When LC does not explode at its boundary, for example when C is the Lévy process whose increments D have a density ψ defined by

∀x < -1, ψ(x) = Ke x
x 2 where K is the normalization constant, the martingale parameter is no more defined once a passes a0 = ΛD(α0) (here computations lead to α0 = 1 and a0 = K). The best we can do now is to use the desired property for a0 and then say that -ΛT 0 's derivative is bounded from below by 1, indeed

-dΛT 0 (a) da = E T0e -aT 0 E (e -aT 0 )
and T0 ≥ 1 almost surely by construction, so we get

∀x ∈ R + , -ΛT 0 (a) ≥ α0C(0) + (a -a0)
However, we have the following interesting idea for the processes whose increments have fat tails (when α = α0, one gets φ (a)

[α] with the quadratic tail) : default times for large values of x are well-controlled by the probability of defaulting in one step. We know that for every random variable X whose Laplace transform stops being defined at point α0, for every β > α0, there are arbitrarily large values of x ∈ R * + such that

P (X < -x) > e -βx
x 2 because the opposite would lead to well-defined values of LX for x ∈ (α0, β) through integration over (-∞, -x). If C(0) is one of these values of x, the default probability in one step is P (D < -C(0)), which leads to E e -aT 0 ≥ e -a P (T0 = 1) ≥ e -a e -βC(0) C(0) 2 and so we get -ΛT 0 (a) ≤ a + βC(0) + 2 ln (C(0)) for these values. Noticing the similarity between this expression (when β goes to α0) and its counterpart, we remark a strictly sub-linear discrepancy in C(0). This means that

• We cannot improve further the quality of this control only through the change of linear terms (like the martingale parameter) ;

• More accurate knowledge of D is required in the general case.

Finally, we may remark that this should also work without any assumption on C's increments with α0 = 0 and a0 = 0, but then the study is of little interest, except showing why processes with fat tails (like the Cauchy distribution) have a "wild" behaviour, with high default probabilities. Hence, for our Cauchy example, -ΛT 0 is strictly sub-linear in C(0).

No fat tails

During the introduction, we chose to exclude the issues of fluctuations having fat tails or fractal distributions. Here are some lines of thought to explain why we could.

• Common sense : the world is finite, thus all modelled values should remain bounded.

• Discrete time : a common issue when looking at continuous-time processes lies in their nature, creating unbounded variations and jumps. This is not a concern here, as we only look at increments over well-defined, fixed intervals.

• In this model, fat tails are an illusion, which may look anticlimatic contrasted with "new" research ( [START_REF] Mandelbrot | The variation of certain speculative prices[END_REF]). When underestimating the variance V (C), it is common to assert that C has fat tails ; however, it appears that this phenomenon comes solely from the time dependency between successive increments, creating long sequences of positive feedback and pushing C far away from its mean expectancy.

To clarify this latter statement, let us take C to be the process (N → N) whose

• Transition probabilities are given by the transition matrix

P = 1 -z z z 1 -z
where z < 1/2 (positive feedback) ;

• Transition payoffs are deterministic D1→1 = 1, D1→2 = 1, D2→1 = -1 and D2→2 = -1.
We compute the martingale parameter, leading to the variance term being 1/z -1 > 1 when the feedback is positive (and < 1 when the feedback is negative), compared with V l = 1 when dependency is omitted (z = 1/2). Worse yet, the variance is arbitrarily large when the dependency goes to z = 0. Although we could proceed with the definitions of higher moments of a C-process, the main effect is that the measurement of "tails" (moments) thanks to C's martingale parameter is high despite the low width of each increment. For this reason, when applying this model to real data, the observed fat tails are explained by the positive feedback rather than the nature of the transition payoffs, and assuming C to be bounded should not be an issue.

Differences between periodicity and aperiodicity of C

Perhaps the most shocking observation about the main theorem 1 is that the convergence in K

∞ + o(1) fails for the most natural C-processes, where the transition payoffs are expressed as commensurate quantities : one instead gets the other result, the proposition 2.4, specific to periodic C-processes, where the convergence of the error function does not hold on the whole R. This is not a flaw of the study, as the default probabilty cannot be K (a) ∞ + o(1) everywhere, as we saw in the section dealing with periodicity issues.

On the bright side, the form K (a) 1) is recovered if one imposes C to remain commensurable with its increments, which is the case if one follows C's trajectory through transition payoffs, so the theorem may be applied "on-line", while analysing C's expected behaviour with respect to time.

∞ + o(

Interpretations

We shall end this study with several interpretations and examples of use for these results, like the effects of capital investment or liquidity issues.

Dimensional analysis

To understand the parameters given during this study, we give their respective units of measurement. For example,

• The time t expresses in seconds s ;

• The capital value C expresses in dollars $.

We chose to read the units of the continuous-time model, to avoid hidden terms like ×1s appearing in the equations involving transitions, as in C(t + 1) = C(t) + D(t + 1) × 1s.

• The transition payoffs Di→j and the mean expectancy E(C) express in $/s ;

• The Laplace parameter a, the differential transition matrix P and the differential log-Laplace matrix function express in 1/s ;

• Probabilities and the dominant eigenvectors w (a) and µ (a) express in 1 (without unit), as well as the maringale process X (a)

C , whereas the differentiated martingale is in $ ; • The martingale parameter α expresses in 1/$ ; in particular, its derivative α is in s/$ and its second derivative α in s 2 /$, so the "variance" -α (0)/ (α (0))

3 is in $ 2 /s.
These units explain the parameters :

• Items expressed in $ are related to an amount of money, and refer to a quantity of cash related with them. For example, ln w

(a) [i]
/α(a) is the corrective term one must substract to the cash C(t) to get the "true" value of the assets when in state Ai, cancelling the effect of present-state overestimations.

• Items in $/s are cash flows (incomes), indicating the profit/loss balance affecting C over a time period, and may be regarded as drifts. For instance, E(C) is the main drift of C.

• Items in 1/s act as frequencies ; hence, the differential P is the probability of M switching states over a time period, and may be related with the exponential parameter of a Poisson process.

• Quantities with unit 1 are coefficients and indicate a ratio between a "specific" event and the "generic" configuration. Comparing w (a) 's coordinates with 1 yields the correction between default event with and without knowledge of M (t), e.g. for X

C the probability of default, it indicates how the state M (t) increases default risks.

• Being in 1/$, α indicates the marginal effect of a single dollar on the default event. If one aims at avoiding default, looking at α(0) measures how being richer decreases the default probability.

On a side note, we recover that the "variance" term really expresses in $ 2 /s, as well as the quadratic variation of a stochastic process, which is an additional reason to name it C's variance.

Nature of the result

We consider the main result of this study, discussing about its implications in terms of economy. The Laplace transform of the default time at point a (and in particular the default probability, when a = 0) is written as

E e -aT 0 |M (0) = Ai ∧ C(0) = C0 = e -K(a) w (a)
[i] e -α(a)C 0 (1 + o(1)) One should notice that the martingale parameter α(a) does not depend on the state of the market and is an in-built parameter of the model. The only impact of the market state lies in the vector w (a) : high values of w (a)

[i] indicate that the market suffers from "bad" short-term situation, because of • Incoming transition payoffs likely to decrease C (adverse mean), thus shortening the default time ;

• By vertue of the martingale equation with α(a) > 0, leading by convexity Jensen-like inequalities to a risk-adverse-shaped expected exponential default time, high uncertainty on the immediate future (adverse variance).

However, considering the market for a long-term analysis, then one should refer to the martingale parameter α(a), a better indicator of the expected outcome of the market than the "classical" mean-variance characterization, the latter being blind to the structure of a C-process.

In the main theorem, the constant term K is related to the way C may default : for additional considerations on the severity of default, one is encouraged to refer to [START_REF] Hans U Gerber | On the probability and severity of ruin[END_REF]. Considering the martingale process

X (a) C = N → R * + t → w (a)
[M (t)] e -α(a)C(t) e -at then we know thanks to the martingale property of proposition 2.1 that for every a ≥ 0,

w (a) [i] e -α(a)C 0 = E w (a) [M (T 0 )] e -α(a)C(T 0 ) e -aT 0 |M (0) = Ai ∧ C(0) = C0
As we saw earlier, an interpretation of this is that successive time periods eventually "mix" the conditionnal probabilities P (M (t) = Ai|C(t) ∈ (x, x + )) (for i ≤ A) when t increases, which is automatic as C0 increases, regardless of the value x. So, the final value of X (a) C will keep pretty much the same distribution for any (large) C0 and any M (0). Interestingly, we see that this mixing does not occur when C has a period p ∈ R * + , as for any n ∈ N and p k the natural offsets, then P (M (t) = Ai|C(t) = pn + p k ) is 1 iff i = k and 0 otherwise (provided that several states A k do not share the same offset). This is yet another reason why the main theorem fails for periodic processes.

Examples of use

Interpretations of the results found in this study are useful when computing some buying/selling decisions in markets where liquidity issues are a concern (see also McKean's problem, [START_REF] Jr | A free boundary problem for the heat equation arising from a problem of mathematical economics[END_REF]). For example, with an intuitive model where investment of some cash enhances the transition payoffs (increases the martingale parameter α(a) to α (a)) at the expense of some illiquidity (decreases the cash level C to C = C -I for I ∈ R + the investment costs), one observes that investment is often beneficial for a rich enough buyer, as the negative logarithmic default probability is in α (a)(C -I) > α(a)C, since α (a) > α(a), for I/C small enough. We may also notice that the decrease in ΛT 0 (a) should be lesser for high values of a, because ΛT 0 (a) involves short-term default, whose probability is actually raised by the investment costs : if the market goes dramatically wrong for the investor, then the asset costs will cause liquidity shortage quicker (and before the increase in the transition payoffs amount to these costs). Conversely, we also find out that selling a valuable asset to avoid short-term default may be a viable strategy if liquidity issues are severe (C close to 0), because the immediate risk of default outweights long-term profits.

As stated earlier, long-term behaviour is accurately expressed using the martingale parameter, which may lead to seemingly counter-intuitive facts. Hence, improving C's mean expectancy does not automatically reduce the default probability, for example if the spread of increments now greatly exceeds the previous one. However, higher mean and lower variance do not guarantee a better martingale parameter in general, despite this being true specifically for Brownian motions (α(0) = 2E/σ 2 , obtained in the calculation of ΛT 0 for a Brownian motion). This constatation may seem puzzling, as general theorems like the central-limit theorem state that long-term behaviour of a Lévy process only relies on mean and variance. An explanation for this is that the central-limit theorem only takes into account the asymptotical future of C, whereas its default time (if it arises) is likely to happen early compared with C0. The main idea behind this previous sentence is that for large values of C0 and E > 0, the law of large numbers ensures that time concatenation of increments greatly decrease the probability of a cumulative negative payoff ; then, for example, the Lévy L with ±1 increments

P (D = -1) = = 1 -P (D = 1)
has a probability of "early" (as soon as possible) default in

P (T0 = 1 + C0 ) = 1+ C 0
whereas the subsequent probabilities of default at posterior times are in a greater power of , at least 2 + C0 since C went +1 somewhere. If is small, the central-limit theorem fails to represent C's early defaults ; those being in a majority of defaults, it fails at accurately computing C's default probability.

Part II Proofs

The rest of this content will be constituted by the proofs to the statements given during the study.

Proofs for auxiliary statements

This section is devoted to the proof for all statements and propositions used as basics to the definition of the martingale parameter of a C-process. The proof for the martingale parameter itself will be given in the next section.

Proofs for the definitions

We aim at proving all definitions and subsequent propositions provided in the beginning of the study. For this purpose, except when noted otherwise or when we aim at proving it, we consider throughout this part that C is a C-process whose underlying Markovian process is M .

Conditional independence

We start with the proposition 1.1, useful for further observations regarding independence between C's increments.

1. Forward implication : let C be a C-process whose underlying Markovian process is M .

• By definition, M (0) and C(0) are deterministic.

• To prove the Markovian property, let t ∈ N, H be a measurable subset of R, and

i ≤ A. Let us compute P1(t, H, i) = P (C(t + 1) -C(t) ∈ H ∧ M (t + 1) = Ai|F(t))
By definition of a C-process, since

C(t + 1) -C(t) is the active increment D M (t)→M (t+1) (t + 1), then P (t, H, i) is equal to P1(t, H, i) = P D M (t)→i (t + 1) ∈ H ∧ M (t + 1) = Ai|F(t)
However, D M (t)→i (t+1) is independent of F(t) conditionally to M (t), which is F(t)-measurable, so we get the product

P1(t, H, i) = P D M (t)→i (t + 1) ∈ H|M (t) P (M (t + 1) = Ai|F(t))
Since M is Markovian, the rightmost probability is

P (M (t + 1) = Ai|F(t)) = P (M (t + 1) = Ai|M (t))
Finally, as D M (t)→i (t + 1) and M (t + 1) are independent conditionally to M (t), the product yields

P1(t, H, i) = P D M (t)→i (t + 1) ∈ H ∧ M (t + 1) = Ai|M (t) As D M (t)→i (t + 1) is C(t + 1) -C(t) when M (t + 1) = Ai, we have the desired property. • To prove time-homogeneity, we compute for every s, t ∈ N, H a mesurable subset of R and i, j ≤ A, P2(t, H, i, j) = P (C(t + 1) -C(t) ∈ H ∧ M (t + 1) = Aj|M (t) = Ai)
Like we saw before, C(t + 1) -C(t) is Di→j(t + 1) when M (t) = Ai and M (t + 1) = Aj, so P2(t, H, i, j) is by definition

P (Di→j(t + 1) ∈ H|M (t) = Ai ∧ M (t + 1) = Aj) P (M (t + 1) = Aj|M (t) = Ai)
However, the family (Di→j(t + 1)) t∈N is i.i.d. and independent of M , and M is Markovian and time-homogeneous, so the terms in the product may rewrite, for any s ∈ N, as

P2(t, H, i, j) = P (Di→j(s + 1) ∈ H) Pi→j
Since for any s ∈ N ,

P (Di→j(s + 1) ∈ H) = P (Di→j(s + 1) ∈ H|M (s) = Ai ∧ M (s + 1) = Aj)
once again, this ends the proof.

2. Backward implication : we know that M is Markovian, is time-homogeneous, and M (0) and C(0) are deterministic by hypothesis. We want to create i.i.d. in t ∈ N families of random variables

D (t+1) i→j i,j≤A
such that for every t ∈ N,

C(t + 1) = C(t) + D (t+1) M (t)→M (t+1)
If we succeed, then C will be a C-process, whose underlying Markovian process is M and transition payoffs are the (common) distributions of the created families. To prove that these families of transition payoffs are i.i.d., let for every i, j ≤ A, Hi,j ⊆ R be a measurable set, and

H = A i=1 A j=1 Hi,j ⊆ R A 2
We are going to compute, for t ∈ N,

P (H, t) = P (Di→j(t + 1)) i,j≤A ∈ H|F(t)
and prove that this conditional probability is actually a constant that does not depend in t, which will lead to the result. Hence, let us consider the random variables (Di→j(t + 1)) i,j≤A,t∈N defined by, for every i, j ≤ A, t ∈ N,

• If M (t) = Ai and M (t + 1) = Aj, then Di→j(t + 1) = C(t + 1) -C(t)
• If not, then Di→j(t + 1) is a random variable called Zi→j(t + 1) whose distribution is the distribution of C(s + 1) -C(s) conditionally to M (s) = Ai and M (s + 1) = Aj (no matter s, as ensured by hypothesis). All these random variables named with Z are built to be mutually independent, and independent of all other random variables in this study.

We decompose P (H, t) regarding M (t) and M (t + 1) as follows :

P (H, t) = P A i 0 =1 A j 0 =1 M (t) = Ai 0 ∧ M (t + 1) = Aj 0 ∧ ∀i, j ≤ A, Di→j(t + 1) ∈ Hi,j|F(t)
As the events M (t) = Ai 0 ∧ M (t + 1) = Aj 0 are pairwise exclusive, we get

P (H, t) = A i 0 =1 A j 0 =1 P (M (t) = Ai 0 ∧ M (t + 1) = Aj 0 ∧ ∀i, j ≤ A, Di→j(t + 1) ∈ Hi,j|F(t))
By definition of the payoffs, the conditions M (t) = Ai 0 ∧ M (t + 1) = Aj 0 allow us to rewrite the sought probability P (H, t) as

A i 0 =1 A j 0 =1 P M (t) = Ai 0 ∧ M (t + 1) = Aj 0 ∧ ∀(i, j) = (i0, j0), Zi→j(t + 1) ∈ Hi,j ∧ C(t + 1) -C(t) ∈ Hi 0 ,j 0 |F(t)
The variables Zi→j(t + 1) are independent and independent of F(t) as well as C, thus they are factored. Additionally, M (t) is F(t)-measurable, so

P (H, t) = A i 0 =1 A j 0 =1 1 M (t)=A i 0 P (M (t + 1) = Aj 0 ∧ C(t + 1) -C(t) ∈ Hi 0 ,j 0 |F(t)) (i,j) =(i 0 ,j 0 ) P (Zi→j(t + 1) ∈ Hi,j)
However, we know that • Thanks to the Markovian property,

P (M (t + 1) = Aj 0 ∧ C(t + 1) -C(t) ∈ Hi 0 ,j 0 |F(t)) = P (M (t + 1) = Aj 0 ∧ C(t + 1) -C(t) ∈ Hi 0 ,j 0 |M (t))
and then thanks to the time-homogenous property, ∀k ≤ A, s ∈ N,

P (M (t + 1) = Aj 0 ∧ C(t + 1) -C(t) ∈ Hi 0 ,j 0 |M (t) = A k ) = P k→j 0 P (C(s + 1) -C(s) ∈ Hi 0 ,j 0 |M (s) = A k ∧ M (s + 1) = Aj 0 ) so we get for any s ∈ N P (M (t + 1) = Aj 0 ∧ C(t + 1) -C(t) ∈ Hi 0 ,j 0 |F(t)) = A k=1 1 M (t)=A k P k→j 0 P (C(s + 1) -C(s) ∈ Hi 0 ,j 0 |M (s) = A k ∧ M (s + 1) = Aj 0 )
• By definition of Zi→j(t + 1), we have also for any s ∈ N

P (Zi→j(t + 1) ∈ Hi,j) = P (C(s + 1) -C(s) ∈ Hi,j|M (s) = Ai ∧ M (s + 1) = Aj)
This leads to the expression of P (H, t) being

A i 0 =1 A j 0 =1 A k=1   1 M (t)=A k 1 M (t)=A i 0 P k→j 0 P (C(s + 1) -C(s) ∈ Hi 0 ,j 0 |M (s) = A k ∧ M (s + 1) = Aj 0 ) (i,j) =(i 0 ,j 0 ) P (C(s + 1) -C(s) ∈ Hi,j|M (s) = Ai ∧ M (s + 1) = Aj)   and it simplifies to A i 0 =1 A j 0 =1 1 M (t)=A i 0 Pi 0 →j 0 A i=1 A j=1 P (C(s + 1) -C(s) ∈ Hi,j|M (s) = Ai ∧ M (s + 1) = Aj)
The double product does not depend on i0 or j0, and is a constant term thanks to the timehomogenous property, that will be named P1(H) in the end of the proof. All that remains is now

P (H, t) = P1(H) A i 0 =1 1 M (t)=A i 0 A j 0 =1
Pi 0 →j 0 Hovever, the sum over j0 amounts to 1 by definition of a stochastic matrix, and so does the sum over i0, so P (H, t) = P1(H) does not depend on t. Since for every t ∈ N,

D M (t)→M (t+1) (t + 1) = C(t + 1) -C(t)
then C coincides with a C-process whose transition payoffs are given by the bulit families, which ends the proof.

Time concatenations

A running idea in this work consists in time concatenations, using increasing sequences of (possibly random) times (τ (k)) k∈N ; we shall consider the C-process through its value C(τ (k)) and its state M (τ (k)).

Definition 5.1 Time sequence

• A binary determination sequence is a process of Bernouilli random variables, whose name is ρ :

(N → {0, 1}) such that -ρ(0) = 1 a.s ; -For any t ∈ N * , ρ(t) is F(t)-measurable.
• The time sequence associated with ρ is the increasing random process τ recursively defined by

τ = N → N ∪ {∞} t → min ({u ∈ N; (ρ(u) = 1 ∧ ∀s < t, u > τ (s))}) If the set is empty, we set τ (t) = min (∅) = ∞.
The trick of time concatenation allows us to define C-processes on the random subset of N described by a time sequence : as M is Markovian and C's transition payoffs are i.i.d. in time, one may expect the process (C(τ (t))) t∈N to be a C-process. Although problems arise when τ (t) = ∞, as the definition C(τ (t)) does not make sense there, it will be convenient to set for purposes of time concatenation :

• C(∞) = +∞ ; • M (∞) is a new state, named A∞.

Definition 5.2 Concatenated processes

Let C be a C-process whose underlying Markovian process is M , and τ be a time sequence.

• The concatenated Markovian process of M associated with τ is the process Mτ defined by

∀t ∈ N, Mτ (t) = M (τ (t))
If τ (t) is not finite, we define a new state A∞ and set Mτ (t) = M (∞) = A∞ almost surely.

• The concatenated process of C associated with a time sequence τ is the process

Cτ = N → R ∪ {∞} t → C (τ (t)) If τ (t) is not finite, we set Cτ (t) = +∞ almost surely.
Some time sequences allow concatenation of a C-process into another C-process, as stated by the following fact ; they are called canonical time sequences.

Definition 5.3 Canonical time sequences

Let M be a Markovian process and τ be a time sequence. If τ holds the following properties :

1. Markovian concatenations : for every s ∈ N, t ∈ N * , i ≤ A, H ⊆ R measurable,

P (τ (s + 1) -τ (s) = t ∧ M (τ (s + 1)) = Ai ∧ C (τ (s + 1)) -C (τ (s)) ∈ H|F (τ (s))) = P (τ (s + 1) -τ (s) = t ∧ M (τ (s + 1)) = Ai ∧ C (τ (s + 1)) -C (τ (s)) ∈ H|M (τ (s)))
2. Homogeneous concatenations : for every r, s ∈ N, t ∈ N * , i, j ≤ A,

P (τ (r + 1) -τ (r) = t ∧ M (τ (r + 1)) = Aj ∧ C (τ (r + 1)) -C (τ (r)) ∈ H|M (τ (r)) = Ai) = P (τ (s + 1) -τ (s) = t ∧ M (τ (s + 1)) = Aj ∧ C (τ (s + 1)) -C (τ (s)) ∈ H|M (τ (s)) = Ai)
then it is said to be a canonical time sequence to M .

If a time sequence is canonical, then it allows the concatenated process to be a C-process.

Lemma 5.1 Concatenated C-process

Let C be a C-process whose underlying Markovian process is M , and τ a canonical time sequence to M . Then Cτ is also a C-process, whose underlying Markovian process is Mτ the concatenated Markovian process of M associated with τ .

N.B. : it may happen that concatenation drives some transition probabilities for Mτ to 0. If this happens for transition (i → j), then Cτ 's transition payoff from Ai to Aj may be defined arbitrarily, as it will have no incidence on the sequel.

Time concatenations (with respect to canonical time sequences) will be useful in the sequel of this study. If one chooses an appropriate binary determination sequence, they lead to different forms of C-processes : setting ρ(t) = 1 iff

• M (t) = M (0), we may build a Lévy process from C (the restricted Lévy process), and use it to simplify the computations.

• t ∈ nN, we "merge" multiple transitions at once (the n-periodically concatenated processes), thus considering C's global trend rather than its individual increments.

• C(t) is an all-time low, we may analyze C's successive all-time lows (the descending process), forming a useful tool to get C's default time.

All these will be used later in the proofs.

Concatenated processes

We are going to prove that the concatenated process defined by lemma 5.1 is a C-process. Hence, let ρ be a binary determination sequence, and τ its time sequence, supposed canonical to C. We already know by hypothesis that

• ρ(0) = 1 by definition, so τ (0) = 0 so Mτ (0) = M (0) and Cτ (0) are deterministic ;

• Mτ is Markovian and time homogeneous, because τ is canonical to M ;

We aim at proving that (Cτ , Mτ ) holds 1. The Markovian property ;

2. The time-homogeneous property.

If we succeed, thanks to proposition 1.1, C will be a C-process.

1. Let s ∈ N, H be a measurable subset of R ∪ {+∞}, i ≤ A, and G(s) be the natural filtration associated with Cτ , Mτ up to time s. We want to compute

P (s, H, i, τ ) = P (Cτ (s + 1) -Cτ (s) ∈ H ∧ Mτ (s + 1) = Ai|G(s))
We decompose this probability considering the transition waiting time τ (s + 1) -τ (s).

P (s, H, i, τ ) = ∞ t=1 P (τ (s + 1) -τ (s) = t ∧ Cτ (s + 1) -Cτ (s) ∈ H ∧ Mτ (s + 1) = Ai|G(s))
N.B. : we included t = ∞ in the previous sum. However,

• If t = ∞, then Mτ (s + 1) = A∞ automatically, so the sought probability is zero and has no effect on the sum whenever i < ∞, and t = ∞ may be removed. • When considering i = ∞, we have Cτ (s + 1) -Cτ (s) = ∞ automatically, so P (s, H, i, τ ) is -Either 0 if ∞ / ∈ H ; -Or the complement of the probabilities P (s, H, j, τ ) for j < ∞, so solving i < ∞ suffices to get this case too.

Let Z(s, H, i, τ, t) be the event

τ (s + 1) -τ (s) = t ∧ Cτ (s + 1) -Cτ (s) ∈ H ∧ Mτ (s + 1) = Ai
The σ-algebra G(s) is given by the random variables M (u) and τ (u) for which there is r ≤ s such that u = τ (r), i.e. ρ(u) = 1 ∧ u ≤ τ (s). It follows that

• G(s) ⊆ F (τ (s))
, as the latter is given by all random variables M (u) and τ (u) for u ≤ τ (s), and the event ρ(u

) = 1 is F(u)-measurable ; • G(s) ⊇ σ (M (τ (s))), as the former contains Mτ (s) = M (τ (s)).
Now, let us write (after the first inclusion)

P (Z(s, H, i, τ, t)|G(s)) = E (P (Z(s, H, i, τ, t)|F (τ (s))) |G(s))
Thanks to the property of Markovian concatenations, we know that

P (Z(s, H, i, τ, t)|F (τ (s))) = P (Z(s, H, i, τ, t)|M (τ (s))) So we have P (Z(s, H, i, τ, t)|G(s)) = E (P (Z(s, H, i, τ, t)|M (τ (s))) |G(s))
and the other inclusion yields

P (Z(s, H, i, τ, t)|G(s)) = P (Z(s, H, i, τ, t)|M (τ (s)))
which leads to the sought Markovian property once we sum these equalities for t ∈ N.

2. The second proof is similar, relying on the fact that canonical time sequences hold homogeneous transitions : the main idea is that Z(s, H, i, τ, t) does not depend on s.

Proof of the restricted Lévy process

Hence, we may now prove that the restricted Lévy process from the definition 1.4 is actually a Lévy process. To get it, we set

• The binary determination sequence ρ ∀t ∈ N, ρ(t) = 1 M (t)=A i
• The associated time sequence τ , with finite terms almost surely by hypothesis ;

• The concatenated C-process Cτ .

Then Cτ is C's restricted Lévy process by construction. As a C-process whose state space is trivial is a Lévy process by definition, we are going to prove that

1. τ is canonical to C ; 2. For every t ∈ N, Mτ (t) = Ai almost surely.
This means that Cτ is a C-process, thanks to the proposition 5.1 ; as its underlying Markovian process is constant almost surely, it is a Lévy process.

1. First, τ is a time sequence by construction, indeed M (0) = Ai so τ (0) = 0. The concatenated transitions are Markovian, because ρ derives from M 's present state only and (C, M ) is a Markovian process ; they are time-homogenous, because M itself is time-homogenous. So τ is canonical to C.

2. By construction, Mτ (t) cannot be any Aj for any j = i ≤ A. As M is positive recurrent, it will not be A∞ almost surely either, so it will be Ai.

This proves that Cτ is a Lévy process.

Periodical concatenations

Likewise, this other form of concatenation also yields a C-process. • Its underlying Markovian process is MT n such that ∀t ∈ N, MT n (t) = M (nt). We name its transition probabilities as P i n -→j . • We name its transition payoffs D i n -→j (t), called C's cumulative transition payoffs through n time periods. For every i, j ≤ A, they have the same distributions as random variables called D i n -→j no matter t ∈ N.

• Its starting point is C(0).

N.B. : if P i n

-→j = 0, then D i n -→j (t) and D i n -→j may be defined arbitrarily. The idea here is to "dilute" the state specificities, in the hope that for a large n, C will spend an amount of time on each transition (i → j) proportional to µ [i] Pi→j no matter M (t) at present time : discarding M will simplify the study, and allow for a Lévy-like analysis. Finally, as Tn is deterministic, the value of the concatenated C-process at time t ∈ N may be viewed equivalently as CT n (t) or C(nt).

Analysis of paths

Following paths

First, we show how to use paths, ensuring that every path for C can be followed arbitrarily closely. The non-zero probability of being close to paths is essential to the proofs of the main theorems. Lemma 5.2 Probability of paths 1. "Direct" property : let a path of length T ∈ N, determined by

• Its occupied state numbers at ≤ A for t ∈ [|0, T |] ; • Its payoffs values xt ∈ R for t ∈ [|1, T |].
For every > 0, the probability of following the path with precision is positive :

P ∀t ∈ [|1, T |] , M (t) = Aa t ∧ C(t) ∈ C(0) + t u=1 xu ± |M (0) = Aa 0 > 0 2. "
Inverse" property : let T ∈ N, and families of

• Occupied state numbers at ≤ A for t ∈ [|0, T |], • Payoffs values xt ∈ R for t ∈ [|1, T |]
If for every > 0, the probability of following the path with precision is positive, then this determination of occupied state numbers and payoffs values defines a path of length T .

We begin by the direct implication, as the other one comes from the definition of paths.

1. Let C be a C-process whose underlying Markovian process is M , with transition probabilities given by the matrix by Pi→j for i, j ≤ A. We consider such a path : by definition, the payoffs values must be possible at any precision, e.g. /T :

∀t ∈ [|1, T |] , P M (t) = Aa t ∧ Da t-1 →a t (t) ∈ [xt ± /T ] |M (t -1) = Aa t-1 = Pa t-1 →a t P Da t-1 →a t (t) ∈ [xt ± /T ]
because transition payoffs are independent of M , however by hypothesis both terms of the product on the right-hand side are positive. We decompose a subset of the event of following the path at precision by

P ∀t ∈ [|1, T |] , M (t) = Aa t ∧ Da t-1 →a t (t) ∈ [xt ± /T ] = T t=1 P M (t) = Aa t ∧ Da t-1 →a t (t) ∈ [xt ± /T ] | ∀s ∈ [|1, t -1|] , M (s) = Aa s ∧ Da s-1 →as (s) ∈ [xs ± /T ]
Since M is Markovian and transition payoffs are i.i.d. and independent of M , then this product simplifies as

T t=1 P M (t) = Aa t |M (t -1) = Aa t-1 P Da t-1 →a t (t) ∈ [xt ± /T ]
which is a finite product of positive terms, so is positive. Finally, as the condition

∀t ∈ [|1, T |] , Da t-1 →a t (t) ∈ [xt ± /T ] ∧ M (t -1) = Aa t-1 ∧ M (t) = Aa t implies the condition ∀t ∈ [|1, T |] , C(t) -C(0) ∈ t u=1 xt ±
then the sought probability is also positive.

2. Considering any > 0 and t ≤ T , the given condition at /2 implies

P   M (t) = Aa t ∧ M (t -1) = Aa t-1 ∧ C(t) ∈ C(0) + t u=1 xu ± /2 ∧ C(t -1) ∈ C(0) + t-1 u=1 xu ± /2   > 0
The triangle inequality yields the implication

C(t) ∈ C(0) + t u=1 xu ± /2 ∧ C(t -1) ∈ C(0) + t-1 u=1 xu ± /2 ⇒ C(t) ∈ [C(t -1) + xt ± ]
and leads to

P M (t) = Aa t ∧ M (t -1) = Aa t-1 ∧ C(t) -C(t -1) ∈ [xt ± ] > 0
Under these hypotheses for the states that M hits, we know that C(t) -C(t -1) is Da t-1 →a t (t), so we have both requirements for a path :

• All transitions are possible, because

P M (t) = Aa t ∧ M (t -1) = Aa t-1 = Pa t-1 →a t
is positive ; • All payoffs values are possible, because

P Da t-1 →a t (t) ∈ [xt ± ] > 0
This being for every positive , then xt belongs to Da t-1 →a t 's support.

Existence of paths

When M may eventually hit Aj from Ai with positive probability, we know that it is possible to find a finite sequence of states Aa t for t ∈ [|0, T |] (with T the length of this sequence) such that for any s ∈ N,

P (∀t ∈ [|0, T |] , M (s + t) = Aa t |M (s) = Aa 0 ) = T t=1 Pa t-1 →a t > 0
We aim at extending this notion to paths of C-processes, consisting both in successive states occupied by M and successive values taken by C.

Lemma 5.3 Existence of paths of a given value

Let C be a C-process, whose underlying Markovian process is M , and x ∈ R ∪ {∞}. We consider i, j ≤ A such that for any s ∈ N,

∀ > 0, P (∃t ∈ N * ; M (s + t) = Aj ∧ C(s + t) -C(s) ∈ [x ± ] |M (s) = Ai) > 0
Then for every > 0, there are

• T ∈ N * ; • For every t ∈ [|0, T |], intermediate state numbers a t≤T ≤ A such that a0 = i, aT = j, and ∀t ∈ [|1, T |] , Pa t-1 →a t > 0 • For every t ∈ [|1, T |], intermediate payoffs x t≤T ∈ supp Da t-1 →a t
forming a non-trivial path whose starting state is Ai and finishing state is Aj, of value

T t=1 xt ∈ [x ± ]
Let us take > 0. Decomposing the possibilities for t, then for M (s + t) for t ∈ N, there must be some T ∈ N and state numbers at ≤ A such that a0 = i, aT = j, and

P (∀t ∈ [|1, T |] , M (s + t) = Aa t ∧ C(s + T ) -C(s) ∈ [x ± /2] |M (s) = Ai) > 0 Hence, as [x ± /2] is a compact set, there is some value y ∈ [x ± /2] in the support of C(s + T ) -C(s)'s conditional distribution to ∀t ∈ [|0, T |] , M (s + t) = Aa t .
Since this condition determines the active transition payoffs, we get

C(s + T ) -C(s) = T i=1 Da t-1 →a t (s + t)
Now, we shall use the fact that for any independent random variables X t≤T valued on R ∪ {∞}, the support of X = T t=1 Xt is the topological closure of the sum of supports supp(Xt) for t ≤ T . Taking Xt = Da t-1 →a t (s + t), then there is some value

z ∈ [y ± /2] ∩ T t=1 supp Da t-1 →a t (s + t)
So there are values xt ∈ supp Da t-1 →a t that sum to z; as z is at distance at most /2 + /2 = of x, this ends the proof.

When there are no requirements on the value for the path, one can drop the assumption on C(s + t) -C(s).

Lemma 5.4 Existence of paths of any value

Let C be a C-process, whose underlying Markovian process is M . We consider i, j ≤ A such that for any s ∈ N, P (∃t

∈ N * ; M (s + t) = Aj|M (s) = Ai) > 0 Then there are • T ∈ N * ; • For every t ∈ [|0, T |], intermediate state numbers a t≤T ≤ A such that a0 = i, aT = j, and ∀t ∈ [|1, T |] , Pa t-1 →a t > 0 • For every t ∈ [|1, T |], intermediate payoffs x t≤T ∈ supp Da t-1 →a t
forming a non-trivial path whose starting state is Ai and finishing state is Aj.

To prove this lemma, one only needs to find x that suits lemma 5.3. As M may access to Aj from Ai through some intermediate state numbers a t≤T ≤ A, taking ∀t ≤ T, xt ∈ supp Da t-1 →a t and x their sum, then

P (∃t ∈ N * ; M (s + t) = Aj ∧ C(s + t) -C(s) ∈ [x ± ] |M (s) = Ai) ≥ P (∀t ∈ [|0, T |] , M (s + t) = Aa t ∧ ∀t ∈ [|1, T |] , C(s + t) -C(s + t -1) ∈ [xt ± /T ] |M (s) = Ai) Since C(s + t) -C(s + t -1
) is Da t-1 →a t (s + t) under these conditions, and transition payoffs are independent of M , then the latter term is

P (∀t ∈ [|0, T |] , M (s + t) = Aa t |M (s) = Ai) P ∀t ∈ [|1, T |] , Da t-1 →a t (s + t) ∈ [xt ± /T ]
Thanks to the time-homogeneous property, this is

T i=1 Pa t-1 →a t T i=1 P Da t-1 →a t ∈ [xt ± /T ]
All terms are positive by construction of the transitions and the choice of xt in the support of transition payoffs, so this term is positive. This being for every > 0, x suits lemma 5.3.

Moreover, we notice that if C is positive recurrent, one may choose the path such that x < ∞ by definition of positive recurrence, so we may enforce the requirement that x < ∞. This idea will be useful when considering paths of finite values, in particular to the next lemma.

Paths and recurrence

The existence of paths provides a tool to characterize positive recurrence of C-processes, in a similar fashion to paths for a Markovian process.

Lemma 5.5 Existence of paths and positive recurrence

Let C be a C-process, whose underlying Markovian process is M .

1. M is positive recurrent iff for every state numbers i, j ≤ A, there is a path from Ai to Aj.

2.

C is positive recurrent iff for every state numbers i, j ≤ A, there is a path of finite value from Ai to Aj.

As the statement of M is a direct consequence of the lemma 5.4 and generic properties of positive recurrent Markovian processes, we will only focus on the statement for C.

• The direct implication is a consequence of the definition 1.3 and lemma 5.4 for paths of finite values.

• The reverse implication comes from lemma 5.2 that leads to a suitable lower bound of probability in the definition 1.3. This lemma will be useful when translating the property of positive recurrence into terms of paths.

Concatenation of paths

Another useful idea when following paths of C-processes happens when considering successive paths : if C runs on a path, then proceeds with another path, it is natural that the concatenation of both paths forms a longer (possible) path. If aT a = b0, then we may define a path as

• Occupied state numbers ct ≤ A for any t ∈ [|0, Ta + T b |], such that ∀t ∈ [|0, Ta|] , ct = at ∧ ∀t ∈ [|Ta + 1, Ta + T b |] , ct = bt-T a • Payoffs values zt ∈ R for any t ∈ [|0, Ta + T b |], such that ∀t ∈ [|1, Ta|] , zt = xt ∧ ∀t ∈ [|Ta + 1, Ta + T b |] , zt = yt-T a It is a path from Aa 0 to A b T b , whose value is x + y.
This property is the consequence of the definition of paths. Let > 0, and consider two such paths. Let us consider the events H1 and H2 of following respectively the first and the second path at precision /2 : thanks to lemma 5.2, we know that P(H1|M (0) = Aa 0 ) > 0 and P(H2|M (0) = A b 0 ) > 0. However, we know that

P (H1 ∧ H2|M (0) = Aa 0 ) = P (H1|M (0) = Aa 0 ) P (H2|H1 ∧ M (0) = Aa 0 )
The rightmost probability is P H2|M (Ta) = Aa Ta = A b 0 by definition thanks to C's Markovian property (proposition 1.1). Hence, P (H1 ∧ H2|M (0) = Aa 0 ) > 0, and the triangle inequality states that H1 ∧ H2 implies following the concatenation at precision , so the previous lemma 5.2 implies that it forms a path. The fact that its value is x + y comes from the definiton of values for a path.

Rotation of cycles

As cycles begin and end on the same state, we may be able to start them anytime during their executions, looping to the initial state when reaching the final one as they are the same. This leads to the definition of a rotated cycle, as given below. For every s ≤ T , there is a cycle of the same length whose

• Occupied state numbers are bt = a (t+s) for t ∈ [|0, T -s|], and bt = a (t+s-T ) for t ∈ [|T -s + 1, T |] ; • Payoffs values are yt = x (t+s) for t ∈ [|1, T -s|] and yt = x (t+s-T ) for t ∈ [|T -s + 1, T |].
It is named a Rotated cycle (of the initial cycle) to the state number b0 = as, or to the state A b 0 = Aa s , and has the same value as the initial cycle.

This lemma holds because

• Transitions are still possible, as the loop when hitting Aa T = Aa 0 allows starting the initial cycle over ;

• Addition is a commutative operation, so the value of a cycle is preserved by rotation.

Universal cycles

Among the cycles for C, when M is positive recurrent, some occupy every A i≤A through their executions.

They are called universal cycles, and allow some simplifications of the proofs. • For any n ∈ N * , the set of all possible values for all universal cycles of C is called Un(C) ⊆ R∪{∞}.

• The universal cycle support of C is the set

ucs(C) = ∞ n=1 Un(C)
Universal cycles may be rotated to any state, since they run through every state.

Lemma 5.6 Rotations of a universal cycle For every i ≤ A, every universal cycle of C has a rotated cycle to Ai, that is still universal.

This property came from the definition of a universal cycle, as one may choose s so that As is any arbitrarily chosen state. It leads to the following lemma, linked to the positive recurrence of C's underlying Markovian process : the statements given by the lemma 5.5 have a counterpart when considering ucs(C) instead of supp(C).

Lemma 5.7 Properties of the universal cycle support

Let C be a C-process.

1. ucs(C) = ∅ iff M is positive recurrent. 2. ucs(C) contains a finite value iff C is positive recurrent. 3. If C is positive recurrent, then ucs(C) + supp(C) ⊆ ucs(C).
The proof relies on the previous lemma 5.5 and the lemma 5.5.

1. If M is positive recurrent, then there are paths joining every state Ai to every state Aj. Hence, starting from the state A i≤A-1 there is a path Pi going to the state Ai+1, then there is a path PA going from AA to A1. The lemma 5.5 states that the concatenation of these paths yields a path ; it runs through every state, and is a cycle by construction, so it has a value v ∈ ucs(C). Let i, j ≤ A. If C has a universal cycle, then we rotate it to Ai ; as it runs through Aj for some time t ∈ N because it is universal, the path given by its restriction for time up to t links Ai to Aj. This being for every i, j ≤ A, C is positive recurrent.

2. If C is positive recurrent, we use the same method, this time involving paths of finite values as given by the lemma 5.5.

Let

, x2 ∈ supp(C) such that x1 -x2 = > 0. Then there is X -∈ R -such that for every x ≤ X -, [x ± /2] ∩ ucs(C) = ∅.
First, as ucs(C) is stable by addition (lemma 5.7) and contains a negative value z < 0 (else C would be globally increasing), then by some number n of additions of z to x1 and x2, we may find z1 = x1 + nz and z2 = x2 + nz in ucs(C) ∩ R * -with discrepancy . Now, let x ≤ z1.

For every i ∈ [|0, x/z1 |], we consider yi = z1 ( x/z1 -i) + z2i
As ucs(C) is stable by addition and z1, z2 ∈ ucs(C), then yi ∈ ucs(C). We know that the discrepancy between two successive values for yi is less than , and that y0 ≥ x. Hence, if the last yi (for i = x/z1 ) obeys yi ≤ x, then it follows that one among the values y k will be at distance |y k -x| ≤ /2. This will be the case as soon as

z2 x z1 ≤ x ⇐ z2 x z1 -1 ≤ x
As z1 and z2 have been taken negative, then this will hold whenever

x ≤ -z1z2 |z2 -z1|
As this bound is lower than z1, then X -= -z 1 z 2 |z 2 -z 1 | suits the property.

Density alternative

To use the lemma 5.8, one aims at finding two "close" values z1 and z2 in ucs(C) being less than apart. This is where C's periodicity is an issue :

• If C is aperiodic, the additivity of ucs(C) will allow us to find these values no matter , so we eventually get a "strong" result on ucs(C)'s remote density.

• If C's fundamental period is p ∈ R * + , one cannot beat = p, and the result will be substantially weaker.

What we find is the following lemma.

Lemma 5.9 Density alternative

Let C be a positive recurrent C-process, deemed not globally increasing, whose universal cycle support is ucs(C).

1. If C is aperiodic, then for every > 0 there is X -( ) such that for every x ≤ X -( ), we have

ucs(C) ∩ [x ± /2] = ∅ 2. If C'
s fundamental period is p, then there is X -∈ R such that, for every x ∈ pZ no higher than X -, x belongs to ucs(C).

In this proof, we shall use the set

G = {x2 -x1; x1, x2 ∈ ucs(C)\ {∞}} ⊆ R
To verify that it is an additive subgroup of R, one only verifies that

• 0 ∈ G, because C is positive recurrent by the lemma 5.7, so ucs(C) has a non-trivial value ;

• As ucs(C) is additive, then G is additive ;

• G is symmetrical.

So, we know that G is either

• {0}, which means that C is globally increasing and is excluded ;

• A discrete group, whose form is G = qZ for some q ∈ R * + ; • A dense subgroup of R.
However, we know that

• If G = qZ, then let x ∈ ucs(C)\ {∞} be any value of an universal cycle. By definition of G, it follows that ucs(C) ⊆ (x + qZ) ∪ {∞}
Now, as ucs(C) is additive, then writing x = qn + z with n ∈ Z and z ∈ [0, q) yields x + x = 2qn + 2z ∈ qZ + z and this implies that z = 0, so C is q-periodic.

• If C is q-periodic, then G will be included in qZ. As a consequence, if G is dense, then C must be aperiodic.

We proved that C is aperiodic iff G is dense, and this leads to the lemma the following way :

1. If C is aperiodic, then G is dense, which means by definition that we may find arbitrarily close values in ucs(C). The lemma 5.8 concludes.

2. If C's fundamental period is p ∈ R * + , then G ⊆ pZ is an additive subgroup of R, so G cannot be dense and there is q ∈ R * + such that G = qZ, so (a) There is n ∈ N * such that q = pn ; (b) Like before, C is q-periodic.

However, as p is C's fundamental period, then q ≤ p, which is possible only if p = q. Hence, there must by x1 and x2 in ucs(C) such that x2 -x1 = p. Finally, applying the lemma 5.8 to x1 and x2 yields that

∃X -∈ R -; ∀x ≤ X -, [x ± p/2] ∩ ucs(C) = ∅ However, as ucs(C) ⊆ pZ, we get [x ± p/2] ∩ ucs(C) ∩ pZ = ∅
Taking every x ∈ pZ not above X -yields {x} ∩ ucs(C) = ∅, thus such an x must belong to ucs(C), which ends the proof.

Monotonicity properties

Thanks to the lemmata about paths and concatenations, we are now able to prove the propositions involving globally monotone C-processes.

Globally increasing C-processes

This paragraph aims at proving the proposition 1.2. First we note that if C is positive recurrent, the lemma 5.3 may be applied to C, considering any starting and finishing states, with some finite given value x because C is positive recurrent.

• (1 implies 2) Let us assume that C is globally increasing. We consider its restricted Lévy process Cτ , and we want to prove that ∀t ∈ N, Cτ (t + 1) ≥ Cτ (t).

-If Cτ (t) = ∞, we know by construction of Cτ that Cτ (t + 1) = ∞ automatically, be it because

τ (t) = ∞ or C (τ (t)) = ∞. Hence this case is solved. -If Cτ (t + 1) = ∞, the case is immediately solved.
-So we will deem that Cτ (t) < ∞ and Cτ (t + 1) < ∞ in the rest of this proof. In particular, this implies τ (t) < ∞ and τ (t + 1) < ∞.

To compute the probability

P (Cτ (t + 1) < Cτ (t)|τ (t) < ∞ ∧ τ (t + 1) < ∞)
we make use of the canonical time sequence τ of Cτ . As Ai is accessible from itself using a path of finite value (because C is positive recurrent), there is x ∈ R such that

∀ > 0, P (Cτ (t + 1) -Cτ (t) ∈ [x ± ] |Cτ (t) < ∞ ∧ Cτ (t + 1) < ∞) > 0
So, for every > 0, there are s1 (a finite value for τ (t)) and s2 (a finite value for τ (t + 1)) such that

∀ > 0, P (τ (t + 1) = s2 ∧ τ (t) = s1 ∧ C(s2) -C(s1) ∈ [x ± ] |Cτ (t) < ∞ ∧ Cτ (t + 1) < ∞) > 0
The events describing τ indicate that M (s1) = M (s2) = Ai, hence

P (M (s1) = Ai ∧ M (s2) = Ai ∧ C(s2) -C(s1) ∈ [x ± ] |Cτ (t) < ∞ ∧ Cτ (t + 1) < ∞) > 0
As the event Cτ (t) < ∞ ∧ Cτ (t + 1) < ∞ has a non-zero probability because C is positive recurrent, we get

P (M (s1) = Ai ∧ M (s2) = Ai ∧ C(s2) -C(s1) ∈ [x ± ]) > 0
Thanks to the lemma 5.3, for every > 0, this means that there is a non-trivial path from Ai to Ai whose value belongs to [x ± ]. As a cycle of a globally increasing C-process must have a non-negative value, then x cannot be negative (else take = -x/2). So we proved that if

∀ > 0, P (Cτ (t + 1) -Cτ (t) ∈ [x ± ]) > 0 i.e.
x belongs to the support of one of Cτ 's increments, then it is non-negative : this means that Cτ 's increments are non-negative almost surely, so Cτ is increasing almost surely.

• (2 implies 1) Let us assume that C has a cycle P0 of negative value v0 ∈ R * -, whose starting and finishing states are both A j≤A . As C is positive recurrent, lemma 5.5 builds a path P1 of finite value v1 from M (0) to Aj, and another path P2 of finite value v2 from Aj to M (0). Concatenating P1, then n ∈ N times P0, then P2, yields a cycle whose value is v1 + nv0 + v2 ; since v1 + v2 < ∞ and v0 < 0, there is n ∈ N large enough such that this cycle has a negative value v < 0. Following it thanks to lemma 5.2 at precision -v/2 yields that we eventually hit Ai at a time t such that C(t) ≤ C(0) -v/2 with positive probability, so there is s ∈ N such that P (Cτ (s) < Cτ (0)) > 0.

• (1 implies 4) For this proof, we consider the starting state M (0) to be our reference state, and we define the height hi ∈ R of any state Ai to be h M (0) = 0 and

∀i ≤ A, hi = sup ({x ∈ R; ∀t ∈ N, P (C(t) ≤ x ∧ M (t) = Ai) = 0})
Hence, stating that Ai has height hi means that paths going from M (0) to Ai have values no less than hi, this being ensured by lemma 5.2.

-No hi may be -∞, because then lemma 5.3 would create paths of arbitrarily low values from M (0) to Ai ; as lemma 5.5 creates a path of finite value from Ai to M (0), then their concatenation may have a negative value, which is excluded. -No hi may be +∞ either, because then C would not be positive recurrent because of the state Ai, that would be inaccessible using finite transition payoffs.

So, we may consider the C-process C = defined by C = (0) = C(0) and transitions from Ai to Aj being hj -hi ∈ R almost surely, and

C + defined by ∀t ∈ N, C + (t) = C(t) -C = (t).
-For every t ∈ N, we have

C + (t + 1) -C + (t) = (C(t + 1) -C(t)) -(C = (t + 1) -C = (t)) = D M (t)→M (t+1) (t + 1) -h M (t+1) -h M (t)
However, by definition of heights, * For every > 0, i ≤ A, there is a path P from M (0) to Ai whose value belongs to

[hi, hi + ]
thanks to lemma 5.3 ; * There is no path from M (0) to Aj whose value is less than hj thanks to lemma 5.2 (no matter j ≤ A). Thus, if there is x > 0 such that

P D M (t)→M (t+1) (t + 1) ≤ h M (t+1) -h M (t) -x > 0
then we get some states Ai (a possibility for M (t)) and Aj (a possibility for M (t + 1)) such that

P (M (t) = Ai ∧ M (t + 1) = Aj ∧ Di→j(t + 1) ≤ (hj -hi) -x) > 0
It follows that there is a one-step path from Ai to Aj whose value is at most (hj -hi) -x, and concatenation of it after P yields a path from M (0) to Aj whose value is at most hj + -x. Taking = x/2 makes this contradictory to hj's value. So, for every x > 0,

P D M (t)→M (t+1) (t + 1) ≤ h M (t+1) -h M (t) -x = 0
It follows from this that almost surely

D M (t)→M (t+1) (t + 1) ≥ h M (t+1) -h M (t)
and then C + (t + 1) -C + (t) ≥ 0 almost surely, so that C + is almost surely non-decreasing. -C = is globally constant, because for every s < t ∈ N, C = (t) -C = (s) is the telescopic sum of height differences between successive states hit by M , i.e. h M (t) -h M (s) . If M (t) = M (s), then C = (t) -C = (s) = 0, so the only value in C = 's cycle support is 0, which is the definition for a globally constant C-process.

As C = is a C-process by construction and is finite almost surely, then C -C = is a C-process, so this ends the proof.

• (4 implies 3) Considering that C = C + + C = , let us deem that for every Q ∈ R, there are s < t ∈ N such that P (C(t) < C(s) -Q) > 0. By the lemma 5.3, C must have paths of arbitrarily low values ; as M 's state space is finite, there are two states Ai and Aj such that there are paths of arbitrarily low values from Ai to Aj. The lemma 5.5 now creates a path of finite value from Aj to Ai, so their concatenation yields cycles of arbitrarily low values for C. However, C = C + + C = by hypothesis, C + is non-decreasing so must have a non-negative cycle support, and C = is globally constant so has {0} as a cycle support : by addition, it is impossible to get cycles of arbitrarily low values. This ends the proof.

• (3 implies 1) If C has a cycle P0 of length t ∈ N and negative value v0 ∈ R * -, whose starting and finishing states are both A j≤A , then as C is positive recurrent, lemma 5.5 builds a path P1 of length u ∈ N and finite value v1 from M (0) to Aj, which leads by concatenation of P1 with n ∈ N times P0 to a path of length u + nv and value v1 + nv0 from M (0) to Aj. As v1 + nv0 can be made arbitrarily low, then with positive probability, C(u + nv) -C(0) may be arbitrarily low, so no Q of statement 3 may exist.

To see why positive recurrence of C is required, let us take the C-process defined by

• A = 2 states, A1 being the starting state ;

• ∀i, j ≤ A, Pi→j = 1/2 ;

• The transition payoffs are D1→1 = 1, D1→2 = D2→2 = -1, and D2→1 = +∞ almost surely.

Then the return to A1 starting from A1 means either a cycle going only through A1 (so it has a positive value), or going through the transition (A2 → A1) if having gone through A2 (so it has a value of +∞) ; hence C's Lévy process is non-decreasing. However, as C allows for the cycle going (A2 → A2) of value -1, it is not globally increasing.

Globally constant C-processes

This paragraph aims at proving the proposition 1.3.

• (1 implies 2) We consider the heights hi of M 's states like in the step (1 implies 4) of the previous proof, for the proposition 1.2 : they still cannot be ±∞, as C globally increasing implies in both cases that there is a transition payoff whose value is +∞ almost surely like before, and so supp(C) ∞, which is impossible since C is globally constant. For the same reasons as before, there are

C = such that ∀t ∈ N, C = (t) = C(0) + h M (t)
, and C + non-decreasing, with C = C = + C + . It follows that C and C = are globally constant, so C + = C -C = is globally constant, non-decreasing, and starts from C(0) -C = (0) = 0. As C + is non-decreasing, then every path of C + must have a non-negative value, but as C + is globally constant, concatenations of paths into cycles must have zero values. This is possible only if every path of C + has zero value, and so C + ≡ 0 almost surely, so C = C = which satisfies the statement 2 with ∀i ≤ A, ci = C(0) + hi.

• (2 implies 3) This is because Q = maxi (ci) -mini (ci) works.

• (3 implies 1) As C is a bounded process (bounded by 2Q), then -C is also a C-process. We use the implication (3 implies 1) of the previous lemma 1.2 to get that both C and -C are globally increasing, so C is both globally increasing and globally decreasing.

Finally, if C is positive recurrent and globally constant, then it is bounded by 2Q thanks to statement 3.

Properties of the Laplace matrix function

Before working on C's martingale parameter, we focus on its Laplace matrix function, the extension of the Laplace transform of a single random variable. Indeed, the proposition 2.1 involves a matrix LC (α), the "Laplace transform" of the C-process C.

Characterization of a C-process

We prove the proposition 1.4. To do this, we will build a third C-process C, whose underlying Markovian process has a state space being {Ai; i ≤ A} ∪ {A∞}

where A∞ is a new state, reserved to transition payoffs amounting to +∞. We shall 1. Build C and M using only L1(= L2) ;

2. Verify that C has the same distribution as C1 (and thus C2 because we used only L1, which ends the proof of the first part) ;

3. Verify that when no transitions amount to +∞, A∞ is useless so M has the same distribution as M1 (and thus M2).

Hence, let us start from C1, whose transition probabilities P1,i→j and transition payoffs D1,i→j are used in its Laplace matrix function L1.

1. By definition of the Laplace transform, for every i, j ≤ A we have

(L1(0)) i,j = P1,i→jP (D1,i→j < ∞)
We define P (M 's transition matrix) to be the (A + 1)-dimensional matrix, whose entries are named after the states Ai and A∞ and defined as follows :

• Over the "old" states, ∀i, j ≤ A, Pi,j = (L1(0)) i,j

• We go from any old state A i≤A to A∞ with the remaining probability

Pi,∞ = 1 - A j=1 (L1(0)) i,j = 1 -L1(0) 1 [i]
• A∞ is absorbing, i.e. P∞,∞ = 1 and ∀j ≤ A, P∞,j = 0.

We verify that P is a stochastic matrix, because ∀i, j ≤ A, 0 ≤ Pi,j ≤ P1,i→j, so Pi,∞ is not an issue. We define the random transition payoffs of C as follows :

• For i, j ≤ A with P1,i→j > 0, we divide the entry number (i, j) of L1 by P1,i→j, yielding the expression E e -αD 1,i→j for every α ∈ I (where I is the convergence domain of L1). As one knows, this is sufficient to recover D1,i→j's distribution, after which we create a transition payoff Di→j of the same distribution between Ai and Aj. Thanks to

E e -αD 1,i→j = P (D1,i→j < ∞) E e -αD 1,i→j |D1,i→j < ∞
this is tantamount to computing the distribution of D1,i→j conditionally to being finite. • For i, j ≤ A with P1,i→j = 0, we set arbitrarily Di→j = 0.

• For i = ∞ or j = ∞, we set Di→j = +∞.

Finally, we set C's starting point and state as given for C1 and C2. This definition has been done only with items common to C1 and C2, so if C and C1 share the same distribution, the first proposition is solved.

2. Let n ∈ N * and xt ∈ R ∪ {∞} for t ≤ n, defining events X = {∀t ≤ n, C(t) ≤ xt} ∧ X1 = {∀t ≤ n, C1(t) ≤ xt}
We want to compare q = P (X) with q1 = P (X1), which is done by decomposition over the states M may hit over the first n time periods. Hence, let us take a family of state numbers

a = (ai) i≤A ∈ ([|1, A|] ∪ {∞}) n
For every such a, we name Za the event of M following the states described by a successively, i.e.

Za = {∀t ≤ n, M (t) = at}
Its probability pa is obtained thanks to the Markovian property pa = n t=1 Pa t-1 →a t whereas the probability of event X conditionally to event Za is

n t=1 P Da t-1 →a t + C(t -1) ≤ xt
Now, let us distinguish the cases for infinite values of xt.

• If there are t1 < t2 ≤ n with xt 1 = ∞ and xt 2 < ∞, then q = 0 as C gets stuck on +∞ after t1, while q1 = 0 as well.

• If there is t0 ≤ n -1 with ∀t ≤ t0, xt < ∞ and ∀t > t0, xt = ∞, then q is by definition

       A a 1 =1 . . . A a t 0 =1 t 0 t 0 t=1 P1,a t-1 →a t P D1,a t-1 →a t < ∞ P Da t-1 →a t + C(t -1) ≤ xt        Pa t 0 →a∞
However, Da t-1 →a t is D1,a t-1 →a t conditionally to being finite whenever P1,a t-1 →a t > 0, and the terms with P1,a t-1 →a t = 0 have no influence whatsoever. As it is independent of C(t -1), we recognize

       A a 1 =1 . . . A a t 0 =1 t 0 t 0 t=1 P1,a t-1 →a t P D1,a t-1 →a t + C(t -1) ≤ xt        P (C1(t0 + 1) = +∞)
which is q1 by construction. • If ∀t ≤ n, xt < ∞, the same idea works without the final term P (C1(t0 + 1) = +∞).

In all cases, the distributions of C and C1 coincide, which solves this part.

3. Under the given condition, the probabilities Pi→∞ are all zero. It follows that the sub-matrix of rows and columns 1 to A of P defines a stochastic matrix, thus L1(0) is a stochastic matrix, which must be M1's transition matrix. As a consequence, we recovered the whole C-process.

As we recovered the initial terms using only L1, this ends the proof.

Effects of periodic concatenation

We present the proof of the propostion 1.5, which is a recursion on n.

• If n = 1, then T1 is the identity (∀t ∈ N, T1(t) = t), so CT 1 = C and the equality is a tautology ;

• Admitting equality for n, then for every i, j ≤ A, the entry number (i, j) of (LC (α)) n+1 is

(LC (α)) n+1 i,j = A k=1 ((LC (α)) n ) i,k (LC (α)) k,j
By recursion hypothesis, this comes down to

(LC (α)) n+1 i,j = A k=1 P i n -→k E e -αD i n -→k
P k→j E e -αD k→j However, the above random variables D i n -→k and D k→j are independent, and by definition of

D i n+1 -→j
, we recognize the expression of

A k=1 P i n -→k E e -αD i n -→k P k→j E e -αD k→j = P i n+1 -→j E e -αD i n+1 -→ j
when conditionning on the n th step, which ends the proof.

Positive recurrent matrices

Throughout the proofs, we shall use Perron-Frobenius' theorem as a basis tool granting properties of LC (α)'s dominant eigenvalue. Interestingly, theorem 2.1 breaks down when C is not positive recurrent, which is related to LC (α)'s failure to hold the hypothesis of "positive recurrence" defined below that voids Perron-Frobenius' result.

Definition 5.8 Positive recurrent matrix

Let n ∈ N * , and L ∈ Mn R + be a non-negative matrix. We say that L is positive recurrent iff ∀i, j ≤ n, ∃k ∈ N * ; L k i,j > 0

In particular,

• For M a time-homogeneous Markovian process, its transition matrix L is positive recurrent iff M itself is positive recurrent.

• If M is positive recurrent and aperiodic, it is possible to select the same k large enough for all couples (i, j), and L k will be a positive matrix.

Since Perron-Frobenius' theorem works for positive recurrent matrices, and we shall apply it to C's Laplace matrix function, we look for the Laplace matrix functions holding positive recurrence.

Lemma 5.10 Positive recurrent Laplace matrix functions Let C be a sEI C-process. C is positive recurrent iff for every α ∈ R + , LC (α) is a positive recurrent matrix.

To prove this lemma, we take C a C-process. Thanks to this path choice, the Laplace transforms of the involved transition payoffs are positive and the transition probabilities are positive. For every u ≤ k, we note

LD a u-1 →au (α) = E e -αDa u-1 →au = lu(α) > 0
The k-periodically concatenated transition matrix (whose entries are

P i k -→j
) is the k th power of M 's transition matrix, and the entry number (i, j) of (LC (α)) k is no less than

k u=1 PA a u-1 →Aa u lu(α)
by construction of the matrix product, which is positive because all terms in this product are positive. This being for every i, j ≤ A, LC (α) is positive recurrent.

2. If C is not positive recurrent, there are states i, j ≤ A such that for every k ∈ N * ,

P (M (t + k) = Aj ∧ C(t + k) < ∞|M (t) = Ai ∧ C(t) < ∞) = 0
so this rewrites as

P (M (t + k) = Aj|M (t) = Ai ∧ C(t) < ∞) P (C(t + k) < ∞|M (t) = Ai ∧ M (t + k) = Aj ∧ C(t) < ∞) = 0
However, M is Markovian, and one recognizes the conditional probability of

D i k -→j
being finite in the rightmost term, so

P i k -→j P D i k -→j < ∞ = 0
Hence, for every k ∈ N * , the entry (i, j) of (LC (α)) k must be 0 by definition, so LC (α) is not positive recurrent.

This ends the proof.

Martingale parameter

This section aims at proving all statements about the martingale parameter of a C-process C : existence, unicity, behaviour, properties. By definition, its value at point a ∈ R * + should allow for the following equation, hereafter called the "eigenvector equation" in vector w (a) ∈ R A , to have a non-zero solution :

LC (α(a))w (a) = w (a) e a
This section is divided in multiple steps :

1. Introduce some preliminary notions ; 2. Introduce the trick of "reduced processes" ; 3. Prove that these reduced processes allow for the dominant eigenvalue to stand through reductions ; 4. Find the correct value for α(a), leading to the proposition 2.1 for a ∈ R * + ; 5. Prove the other properties of C's martingale parameter and extend them to a ∈ R + .

Preliminary explanations

Let us start with C's Laplace matrix function LC and its eigenvector equation. We are going to ensure that there really is a solution to the eigenvector equation for every value of a ∈ R * + when C is sEI, positive recurrent, and not globally increasing. We also explain why there is no solution when C is globally increasing.

Existence of the martingale parameter

First, we prove that for every a ∈ R * + , a martingale parameter α(a) exists. Lemma 6.1 Existence of a martingale parameter Let C be a positive recurrent, sEI, not globally increasing C-process. For every a ∈ R * + , there is α(a) ∈ R * + such that e a is the dominant eigenvalue of LC (α(a)). To ensure that, we start with Perron-Frobenius' theorem, stating that every (square) non-negative and positive recurrent matrix L has a single non-negative dominant eigenvalue f (L) such that

• Every eigenvalue λ of L holds |λ| ≤ f (L) ;
• The associated eigenspace has dimension one (f (L) is a single root of L's characteristic polynomial), directed by a vector w(L) whose coordinates are positive.

Let us consider this function f . Tyrtyshnikov's result about eigenvalues indicates that f is continuous, so let us consider the function g defined by

g = R + → R + α → f (LC (α))
As LC is continuous over R + by construction because C is sEI, then g is a continuous function. We find its limits on the boundaries :

• At point α = 0, g(0)
is the dominant eigenvalue of the matrix whose entry number (i, j) is Pi→jP (Di→j < ∞). This matrix is thus dominated by M 's transition matrix, so g(0) ≤ 1.

• To analyze when α goes to infinity, we use the hypothesis under which C is not globally increasing. It follows from this assumption and lemma 5.2 that there are i ≤ A and n ∈ N * such that there is a cycle going from Ai to Ai whose value is negative, so P i n -→i > 0 and there is x > 0 such that P D i n -→i ≤ -x = p > 0. Hence, noting CT n as C's n-periodically concatenated C-process, the coefficient (i, i) of LC Tn (α) is at least pe αx . Since lemma 1.5 indicates that LC Tn (α) = (LC (α)) n and LC (α) is a non-negative matrix, its dominant eigenvalue is at least p (1/n) e αx/n , so g must go to infinity when α goes to infinity.

It follows from these facts and the intermediate value theorem that there is (at least) one real solution to g(α) = e a in R * + , no matter a ∈ R * + . • In the sequel of this proof, we define α(a) as any such solution to this equation.

• We might also define a "negative martingale parameter" when C is not globally decreasing, as a negative solution to g(α) = e a . Even if this definition comes useful when computing C's differentiated process, we shall not discuss it as it is symmetrical to the study of C's natural martingale parameter.

Case of a globally increasing C-process

To explain why the study fails when C is globally increasing, let us assume for now that it is a Lévy process : it means that its increments D are non-negative almost surely. It follows that the Laplace transform of D (over R * + ) will not hit any value in (1, ∞), so the eigenvector equation will have no positive solution in α. The same phenomenon happens in the general case for C globally increasing, although some of its increments may be negative.

Lemma 6.2 Global increase and dominant eigenvalue

Let C be a positive recurrent C-process, whose Laplace matrix function is LC . The following assertions are equivalent :

1. C is globally increasing.

2. LC is well-defined over R * + , and the dominant eigenvalue of LC (α) is non-increasing and bounded by 1 when α goes to infinity.

3. LC is well-defined over R * + , and the dominant eigenvalue of LC (α) is bounded when α goes to infinity.

1. If C is globally increasing, we know after the proposition 1.2 that it rewrites as C + + C = with C + non-decreasing almost surely and C = globally constant. Thanks to the proposition 1.3, we rewrite

C = 's increments from Ai to Aj as cj -ci ∈ R, so C's Laplace matrix function at point α ∈ R * + is LC (α) = Pi→jE e -αD + i→j e -α(c j -c i)
This expression is non-increasing and converges because cj -ci is finite and C + 's increments D + i→j are non-negative almost surely by hypothesis. It follows from it that LC 's dominant eigenvalue, being a non-decreasing function of its coefficients, is no more than the one of LC= (α) ; however, setting D(α) as the diagonal matrix whose coefficents are e -c i α and P as C's transition matrix, we have

LC= (α) = (D(α)) -1 P D(α)
As a consequence, LC= (α)'s eigenvalues are the same as P 's ones, so are dominated by 1, which ends the proof.

2. If C is not globally increasing, then thanks to the lemma 6.1, for every a ∈ R * + , there is a martingale parameter α(a) that can make LC (α(a))'s dominant eigenvalue arbitrarily large. Since LC is continuous, divergence may only happen when α(a) goes to infinity, which ends the proof.

When C is, so to speak, "strictly" globally increasing, a similar property holds.

Lemma 6.3 Stict global increase and dominant eigenvalue

Let C be a positive recurrent C-process, whose Laplace matrix function is LC . The following assertions are equivalent :

1. C is globally increasing and not globally constant.

∀α ∈ R * + , LC (α) = (∆(α)) -1 L C + (α)∆(α)
so LC (α) and L C + (α) have the same eigenvalues, so we shall look at the dominant eigenvalue of L C + (α). Hence, let us take α and look at the dominant eigenvalue λ ∈ R + of L C + (α) and its dominant eigenvector w : thanks to the concatenation property, we have after T time periods

(L C + (α)) T w = λ T w
However, this matrix is the Laplace matrix function of C + 's T -periodically concatenated C-process thanks to the proposition 1.5, so its entry number (i, j) is

P i T -→j E e -αD i T -→j
Thanks to the cycle of positive value, we know that its entry number (k, k) is strictly less than

P k T -→k
. As the other entries hold the large inequality because transition payoffs are non-negative, there is a non-negative matrix X, whose entry number (k, k) is positive, such that

(L C + (α)) T = P T -X
where P is M 's transition matrix. It follows that

λ T w = (L C + (α)) T w = P T -X w so P T w = λ T Id + X w.
Multiplying by µ being M 's invariant measure (P 's row eigenvector), this leads to µw = λ T µw + µXw ; as µXw > 0, this is possible only if λ < 1, which ends the proof.

2. We already know after the lemma 6.2 that this condition implies C globally increasing. However, if

C is a globally constant process, rewriting it as C + + C = yields that LC (α)'s dominant eigenvalue is L C + (α)'s one ; as C + ≡ 0, L C + (α) = P and thus the dominant eigenvalue must be 1.

Drifted C-processes

In the incoming work, we shall use a trick consisting in "drifting" C-processes by a fixed trend. This consists in adding a constant value d ∈ R to each of its transitions, so that the value of C's drifted process at time t ∈ N will be C(t) + td.

i→j = Di→j + d • Starting point is C(0). It follows from this that ∀t ∈ N, C [d] (t) = C(t) + dt
This definition is useful when looking at C [d] 's Laplace matrix function, given by

∀α ∈ R, L C [d] (α) = LC (α) e -αd
This property will allow us several simplifications during the incoming work.

Reductions

The next step in our study is to enable the method of "reduction" of a C-process. The main idea is to turn the C-process into its restricted Lévy process by elimination of all its states but one, getting "reduced C-processes" in the work.

Notions of reduction

The main idea is to eliminate one of M 's states A k = M (0), considering that the time steps when M hits A k are skipped by the C-process : this is tantamount to looking at the concatenated process Mτ where τ comes from the binary determination sequence ρ given by

∀t ∈ N, ρ(t) = 1 M (t) =A k
In other words, eliminating a state A k≤A is skipping all times t ∈ N for which M (t) = A k , concatenating the "previous" and the "next" transitions

(M (t -1) → A k ) , (A k → M (t + 1))
to get a single transition (M (t -1) → M (t + 1))

However, for computations purposes, we allow the reductions to take into account some drifts for C : indeed, the reduced C-process will be obtained as follows.

1. Drift all transition payoffs by d ∈ R, yielding C [d] ;

2. Concatenate C [d] with the above time sequence ;

3. "Un-drift" the resuling transition payoffs, drifting them by -d.

Concatenated transition probabilities, transition payoffs and C-processes then arise like told below.

Definition 6.2 Reduced C-process

Let C be a positive recurrent, sEI C-process and A k = M (0) a state to eliminate.

• The time sequence for reduction with respect to state A k is τ k , whose binary determination sequence is

ρ k such that ∀t ∈ N, ρ k (t) = 1 M (t) =A k
We define M -k the reduced Markovian process with respect to the time sequence τ k .

• For every d ∈ R, the reduced C-process with respect to state A k and drift d is obtained by

∀t ∈ N, C (-k,d) (t) = C [d] (τ k (t)) -td
This definition yields a C-process because τ k is a canonical time sequence to C [d] , which follows from the definition of canonical time sequences because ρ k relies only on present time. We may now compute the transition probabilities and transition payoffs of C -k .

k→k (i.i.d. copies of D k→k ) and D k→j , • X1 a Bernouilli random variable indicating "direct" transition (Ai → Aj), with

P(X1 = 1) = Pi→j P -k,i→j
• X2 a geometric random variable indicating the number of loops in state A k , of parameter P k→k , such that the transition payoff from Ai to Aj is k,d) is also a positive recurrent C-process over the state space

D (-k,d),i→j = 1X 1 =1Di→j + 1X 1 =0 D i→k + X 2 n=1 D (i) k→k + D k→j + (X2 + 1)d Moreover, C (-
{Ai; i ≤ A ∧ i = k}
We note that • P k→k < 1 because M is positive recurrent and A ≥ 2 (the states M (0) and A k being distinct) ;

• If P -k,i→j = 0, then D (-k,d),i→j may be defined arbitrarily, as it will have almost surely no subsequent effects on C.

As the proof of these statements are similar, we will only explain the construction of the transition probabilities of M -k , the reduced Markovian process with respect to A k≤A .

• Previous transition probabilities going from Ai to A j =k are not modified, so transfer to M -k . This case corresponds to X1 = 1.

• With probability P i→k , one gets in A k , this case corresponding to X1 = 0. Then the probability of going out of A k by state Aj after exactly n loops (corresponding to X2 = i) of the transition

(A k → A k ) is P n k→k P k→j So, the probability of going out of A k by state Aj is ∞ n=0 P n k→k P k→j = P k→j 1 -P k→k

It follows that

• The transition probabilities of process M -k are increased to

P -k,i→j = Pi→j + P i→k P k→j 1 -P k→k • C (-k,d
) is positive recurrent, as for every states Ai, Aj = M (0), if there is a path

(Ai → Aa 1 → . . . Aj)
of finite value going through A k , then the sequence given by skipping steps in A k still forms a path of finite value. Indeed, i, j = k ⇒ P -k,i→j ≥ Pi→j > 0 and "skipped" times correspond to some positive term P i→k (P k→k ) n P k→j that appears in P -k,i→j too.

For transition payoffs, the same method and the fact that C is a C-process lead to the result a similar way, after disjunction with respect to values taken by X1 and X2. The term (X2 + 1)d itself is the consequence of the drift ; for some value of X2, the concatenated path takes a time of 1 + X2 + 1, leading to the remaining drift after multiplication.

This gets us a sequence (tu) u≤A such that ∀u ≤ A -1, tu+1 = τu (tu). In particular, we get t = tA = υA(t1), so s = t1 solves this part.

Hence, there is s ∈ N such that υA(s) = t iff M (t) = M (0). As υA is increasing (N → N) (composition of increasing functions τu), then it must coincide with τ * . Now, to get the value of C1(t), we make use of

∀u ≤ A -1, Cu(t) = (Cu+1) (-k u+1 ,d) (t) = C [d] u+1 (τu(t)) -td = (Cu+1) (τu(t)) + (τu(t) -t) d
This leads by induction to

C1(t) = CA (υA(t)) + A-1 u=1 (υu+1(t) -υu(t)) d which simplifies to C1(t) = CA (τ * (t)) + (τ * (t) -t) d
This ends the proof.

Mean expectancy and reductions

When C is positive recurrent and integrable, we aim at evaluating the mean expectancy of its reduced C-process by virtue of the following lemma. In particular, we shall be interested only in the case d = 0, so we shall abbreviate

• C (-k,0) by C -k ; • D (-k,0),i→j by D -k,i→j .
in the next paragraphs.

Lemma 6.6 Scaling of mean expectancies

Let C be a positive recurrent, integrable C-process ; let A k = M (0), and C -k be C's reduced process with respect to A k and drift 0.

1. C -k is positive recurrent and integrable.

C

-k 's invariant distribution µ -k is given by ∀i = a, µ -k,[i] = µ[i] 1 -µ[k] 3. C -k 's mean expectancy is E (C -k ) = E(C) 1 -µ [k]
4. In particular, the mean expectancy of C's restricted Lévy process Cτ is given by

E (Cτ ) = E(C) µ [M (0)]
Noting by µ the row-vector of M 's invariant probabilities, E may be expressed by definition as

E = A i=1 A j=1 µ [i] Pi→jE (Di→j) = µRC (0) 1
Considering the matrix RC -k (0), the statements to prove are dealt with as follows :

1. Its entries converge, because C -k 's transition payoffs are sub-geometrical. Indeed, let us take i, j ≤ A : using D -k,i→j as written in lemma 6.4,

E (|D -k,i→j |) ≤ P (X1 = 1) E (|Di→j|) + P (X1 = 0) E (|D i→k |) + ∞ n=0 P (X2 = n) nE (|D k→k |) + E (|D k→j |)

Now, we know that

The definition of C -k 's transition payoffs leads to

E -k = i =k j =k µ [i] 1 -µ [k]     Pi→jE (Di→j) + P i→k P k→j ∞ n=0 P n k→k   E (D i→k ) + nE (D k→k ) + E (D k→j )      
We focus on the rightmost sum

x = i =k j =k ∞ n=0 µ [i] P i→k P k→j P n k→k (E (D i→k ) + nE (D k→k ) + E (D k→j ))
We split it into the three relevant terms

x = i =k µ [i] P i→k E (D i→k ) j =k P k→j ∞ n=0 P n k→k + j =k P k→j E (D k→j ) i =k µ [i] P i→k ∞ n=0 P n k→k + ∞ n=0 P n k→k nE (D k→k ) i =k µ [i] P i→k j =k P k→j
The terms now simplify as follows :

• For the first term, we know that

j =k P k→j = 1 -P k→k = 1 ∞ n=0 P n k→k It means that the first term of x is i =k µ [i] P i→k E (D i→k )
• For the second term, since µ is the invariant distribution

i =k µ [i] P i→k = µ [k] -µ [k] P k→k
It means that the second term of x is

j =k P k→j E (D k→j ) µ [k]
• For the third term, we know that

∞ n=0 P n k→k n = P k→k (1 -P k→k ) 2
It means that the third term of x is

P k→k (1 -P k→k ) 2 E (D k→k ) (1 -P k→k ) (1 -P k→k ) = P k→k E (D k→k )
Hence,

E -k = 1 1 -µ [k]     i =k j =k µ [i] Pi→jE (Di→j) + i =k µ [i] P i→k E (D i→k ) + j =k P k→j E (D k→j ) µ [k] + P k→k E (D k→k )    
This finally simplifies to

E -k = 1 1 -µ [k] A i=1 A j=1 µ [i] Pi→jE (Di→j) = E 1 -µ [k]
3. The entries of L C (d) u (α) are rational fractions of e αd and entries of LC (α), being well-defined over Su.

4. For every (α1, d1) ∈ g -1 (Su), if α2 ∈ (0, α1] and d2 ≥ d1, then (α2, d2) ∈ g -1 (Su).

We do a proof by induction on u from A down to 1. First, we verify the properties for u = A :

• By definition, C (d) A = C is sEI, so SA = R * + × R is an opened set.
• The entries of LC (α) constitute the sought rational fractions themselves. Now, we assume the properties to be true at step u + 1 and prove them for step u. ; we compute the Laplace matrix function thanks to the lemma 6.4, by

L C (d) u (α) i,j = Pu+1,i→jE e -αD u+1,d,i→j +      P u+1,i→k u+1 E e -αD u+1,d,i→k u+1 ∞ n=0 P n u+1,k u+1 →k u+1 E e -αD u+1,d,k u+1 →k u+1 n e -α(n+1)d P u+1,k u+1 →j E e -αD u+1,d,k u+1 →j      If (α, d) / ∈ Su+1, there are i, j ∈ {kx; x ≤ u + 1} such that L C (d) u+1 (α) i,j = ∞. • If i, j = ku+1, then the term Pu+1,i→jE e -αD u+1,d,i→j = L C (d) u+1 (α) i,j = ∞ appears in L C (d) u (α) i,j
, so (α, d) / ∈ Su.

• If i = ku+1 and j = ku+1, we use the fact C is positive recurrent, so there is i ∈ {kx; x ≤ u} such that P u+1,i →k u+1 E e -αD u+1,d,i→k u+1

is bounded from below by some v > 0. This time, it is the term

P u+1,i →k u+1 P u+1,k u+1 →j E e -αD u+1,d,i →k u+1 E e -αD u+1,d,k u+1 →j that amounts to ∞ and appears in L C (d) u (α) i ,j
, so (α, d) / ∈ Su.

• If i = ku+1 and j = ku+1, we find j ∈ {kx; x ≤ u} such that

P u+1,k u+1 →j E e -αD u+1,-d,k u+1 →j
is bounded from below by v > 0 the same way, so a convenient term appears in

L C (d) u (α) i,j
.

• If i = ku+1 and j = ku+1, we find both i and j the same way.

Hence, we proved that if (α, d) / ∈ Su+1, then (α, d) / ∈ Su, from which follows that Su ⊆ Su+1. Now, let us take (α, d) ∈ Su+1. Since all entries of L C (d) u+1 (α) are finite by hypothesis, the only possibility of divergence comes from the sum over n. Hence, let us define the function hu as

hu = Su+1 → R + (α, d) → P u+1,k u+1 →k u+1 E e -αD u+1,d,k u+1 →k u+1 e -αd
As hu is non-negative over Su+1, determining whether or not (α, d) ∈ Su is equivalent to comparing hu(α, d) with 1. Here, we recall that

L C (d) u+1 (α) k u+1 ,k u+1
is a rational fraction of e αd and LC (α)'s entries, so the set

S u+1 = h -1 u ((-∞, 1)
) is opened, which means that the domain of integrability Su is Su+1 ∩ S u+1 , which is opened.

2. It suffices to prove that the property holds for S u+1 . We recall that over Su+1, hu(α, d) is a rational fraction of e αd and LC (α)'s entries by definition, and thanks to the induction property there is α such that

k(α) = E e -αD u+1,d,k u+1 →k u+1 < ∞
k being a Laplace transform, it is continuous over (0, α) and has a limit no higher than 1 at point 0. It follows that there is α ∈ (0, α) such that hu(α , d) < 1, because since C is positive recurrent we have P u+1,k u+1 →k u+1 < 1 and this α solves the statement.

3. In the set Su, the sum in the expression of L C (d) u (α) converges by definition of S u+1 . Hence, the entry number (i, j) converges to that rewrites to

L C (d) u (α) i,j = Pu+1,i→jE e -αD
L C (d) u (α) i,j = L C (d) u+1 (α) i,j + L C (d) u+1 (α) i,k u+1 L C (d) u+1 (α) k u+1 ,j e a -L C (d) u+1 (α) k u+1 ,k u+1
so it is a rational fraction of the desired form, well-defined by construction of Su.

4. Let us take (α1, d1) ∈ g -1 (Su), α2 ∈ (0, α1] and d2 ≥ d1. By construction, the matrix

L C (d 1 /α 1 ) u (α1)
converges. As it is the Laplace matrix function of a concatenated process with a drift d1/α1, its entries rewrite as ∀i, j ∈ {kx; x ≤ u} , L

C (d 1 /α 1 ) u (α1) i,j = Pu,i→jE e -α 1 (Du,i→j+(d1/α1)Tu,i→j)
where Tu,i→j is a random non-negative variable indicating the number of concatenated steps between hitting Ai and Aj. Hence,

∀i, j ∈ {kx; x ≤ u} , L C (d 1 /α 1 ) u (α1) i,j = Pu,i→jE e -α 1 D u,i→j e -d 1 T u,i→j
As Tu,i→j is always non-negative, and Laplace transforms are convex, we have ∀α2 ∈ (0, α1], d2 ∈ [d1, ∞),

L C (d 2 /α 2 ) u (α2) i,j ≤ Pu,i→jE 1 + e -α 1 D u,i→j e -d 1 T u,i→j ≤ 1 + L C (d 1 /α 1 ) u (α1) i,j < ∞ so g (α2, d2) ∈ Su.
This induction scheme allows us to state the properties for u = 1 :

1. The subset S1 of (R * + × R) defined by

S1 = A u=2 S u = (α, d) ∈ R * + × R; L C (d) 1 (α) < ∞ is opened ; 2.
For every d ∈ R, there is α > 0 such that (α, d) ∈ S1 ;

3. The only entry of L C (d) 1

(α) is a rational fraction of e αd and entries of LC (α), being well-defined over S1.

4. If (α1, d1) ∈ g -1 (S1), α2 ∈ (0, α1] and d2 ≥ d1, then (α2, d2) ∈ g -1 (S1).

However, we know thanks to the lemma 6.

∀(α, d) ∈ S1, f (α, d) = L C (d/α) 1 (α) = E e -αDτ e -τ d
We are now able to prove the given statements.

1. S is given by g -1 (S1) ; g being continuous over its domain, S is an opened set.

2. This is a direct consequence of S = g -1 (S1) and the property obtained at step 1.

3. As S is opened, for every (α, d) ∈ S, we can select (α , d ) ∈ S with α > α and d < d. The use of

∂ n 1 +n 2 f ∂1 n 1 ∂2 n 2 (α , d ) = E (-Dτ ) n 1 e -α Dτ (-τ ) n 2 e -τ d
allows Leibniz's integral rule to work over some opened set containing (α, d). In particular, with n1 + n2 = 2, one gets the Hessian matrix, whose determinant amounts to

H(α, d) = det ∂ 2 f ∂1 2 ∂ 2 f ∂1∂2 ∂ 2 f ∂1∂2 ∂ 2 f ∂2 2 (α, d) = E D 2
τ e -αDτ e -τ d E τ 2 e -αDτ e -τ d -E Dτ τ e -αDτ e -τ d 2

To simplify matters, let us define (D τ , τ ) to be an independent copy of (Dτ , τ ). We get It follows that, as both the trace and the determinant of f 's Hessian matrix are non-negative, f is convex.

H(α, d) = 1 2 E D 2 τ e -

For every u and d, let us consider

Ju(d) = {α ∈ R * + ; g (α, d) ∈ Su}
We verify that

• These sets Ju(d) are opened : indeed, if α ∈ Ju(d), then (α, d/α) ∈ Su which is opened, so one may find an opened subset of Su containing (α, d/α) ; as g is continuous, its inverse image by g yields an opened subset of Ju(d) containing α.

• Let us take α1 > α2 ∈ R * + . If α1 ∈ Ju(d), thenwe get that (α1, d) ∈ g -1 (Su), so thanks to the above property (α2, d) ∈ g -1 (Su) and α2 ∈ Ju(d). Recalling that C is not globally increasing, C has a cycle of negative value and thus there are n ∈ N, x > 0 and p > 0 such that

P (Dτ ≤ -x ∧ τ = n) = p > 0 so we get E e -αDτ e -τ d ≥ pe αx e -nd
When α goes to ∞ at a fixed d, x > 0 solves this case.

This ends the proof.

In particular, as f is continuous over S, and for every d ∈ R and x > 1, there is α > 0 such that (α, d) ∈ S and f (α, d) < x, the use of the intermediate value theorem proves that for every d ∈ R and a ∈ R * + , there is α ∈ R * + such that (α, a) ∈ S and

f (α, d) = E e -αDτ e -τ d = e a
This property shall be used later, especially when making d = a.

Solution

When d = a, the latter property exhibits a value α that solves

f (α, a) = e a
We recall that f is convex and goes to a limit no larger than 1 when α goes to 0. Hence, there may not be any other solution α to this equation, so we define the function α that maps a ∈ R * + to the single corresponding solution. Our next step is to ensure that 1. If β ∈ R * + solves the proposition 2.1, then β = α(a) ; 2. This value of α(a) solves the proposition 2.1.

So, in this paragraph, we shall set a ∈ R * + and assume that β ∈ R * + solves the proposition 2.1. Our idea is to reduce the matrix LC (β) until it becomes a single element, and then identify β with α(a), because the solution to f (β, a) = e a is single.

Reduced Laplace matrix function

Now that issues of integrability have been taken care of, we may define the Laplace matrix function of a reduced process.

Definition 6.3 Reduced matrix

Let L ∈ MA R + be a positive recurrent matrix. We define its reduced matrix with respect to a dimension number k ≤ A and a parameter x > ln (L k,k ) as the matrix L (-k,x) , whose rows and columns are indexed by [|1, A|] \ {k} in the natural order, and whose entry number (i, j) for i, j = k is

L (-k,x) i,j = Li,j + L i,k L k,j e x -L k,k
For the sake of simplicity, we henceforth note a matrix L ∈ Mn (C) whose row number r ≤ n and column number c ≤ n are removed as 2. Noting by H the following row-vector and V the following column-vector :

H = (LC (β)) k,j j =k ∧ V = (LC (β)) i,k i =k
the previous sentence rewrites as

LC (-k,d) (β) = LC (β) k,k + V H e βd -(LC (β)) k,k
This is a consequence of the formula

LC (-k,d) (β) i,j = Pi→jE e -βD i→j + (P i→k P k→j ) ∞ n=0 P n k→k E e -βD i→k E e -wD k→k n E e -wD k→j e -β(n+1)d
and the fact that by definition of

L C [d] (β), ∞ n=0 P k→k E e -βD k→k e -βd n = 1 e βd -e -βd (L C [d] (β)) k,k
The idea is to state that as e a is the dominant eigenvalue of C's Laplace matrix function at point β ∈ R * + by hypothesis, it is still the dominant eigenvalue of C (-k,d) 's Laplace matrix function at the same point if one chooses d = a/β.

Lemma 6.10 Conservation of the dominant eigenvalue

Let C be a positive recurrent C-process, and a ∈ R * + , β ∈ R * + such that LC (β) is well-defined and its dominant eigenvalue is e a . Let A k = M (0) be a state of C's underlying Markovian process M and C (-k,a/β) be C's reduced C-process with respect to A k and drift a/β.

1. Its Laplace matrix function at point β is well-defined.

2. e a is also the dominant eigenvalue of LC (-k,a/β) (β).

To prove this lemma, we will separately prove that 1. LC (-k,a/β) (β) is well-defined ;

2. e a remains a eigenvalue for it ;

3. If LC (-k,a/β) (β) has an eigenvalue greater than e a , so does LC (β).

If all properties are true, the lemma 6.10 will follow.

Conservation of the dominant eigenvalue

First we verify that LC (-k,a/β) (β) is well-defined. We know by hypothesis that LC (β)'s dominant eigenvalue is e a , so as LC (β) is positive recurrent, its diagonal entries must be less than e a . Hence, the corresponding term hu(β, a/β) from the proof is lower than 1 ; as it was the only condition for integrability, the reduced matrix is well-defined. To prove that every β ∈ R such that e a is the dominant eigenvalue of LC (β) also gives e a as an eigenvalue of LC (-k,a/β) (β), we are going to state this intermediate result.

Lemma 6.11 Determinant of a reduced matrix

Let L ∈ MA R + be a positive recurrent matrix. Let k ≤ A be a state number, with A ≥ 2. For every x > ln (L k,k ), det (e x Id -L) = det e x Id -L (-k,x) (e x -L k,k )

We write down L (-k,x) using the vectors H and V denoting L's removed row and column, like in lemma 6.9. We develop the determinant with respect to row number k, which leads to

det (e x Id -L) = (e x -L k,k ) det e x Id -LC (β) k,k + u =k (-1) u-k Hu det e x Id k,u -LC (β) k,u
As the determinant is multilinear and anti-symmetric, swapping the column k with the skipped column u in the rightmost matrix has a signature (-1) k-u+1 . Let us note by Xu the matrix L k,k modified as follows :

• Its column number u is replaced by V ;

• After this, we add e x to its entry number (u, u), to "cancel" the entry (u, u) of the incoming identity matrix.

This gives

det (e x Id -L) = (e x -L k,k ) det e x Id -L k,k - u =k
Hu det (e x Id -Xu)

Now, we deal with L (-k,x) .

• First, L k,k < e x by hypothesis.

• The determinant is multilinear and anti-symmetric, and columns proportionnal to V are added to L k,k to get L (-k,x) .

So, we get

det e x Id -L k,k - V H e x -L k,k = det e x Id -L k,k - u =k Hu e x -L k,k
det (e x Id -Xu)

The lemma 6.11 follows. The idea is now to apply this lemma to L = LC (β). Since e a is its dominant eigenvalue, its entry number (k, k) is lower than e a because C is positive recurrent, thus det (e a Id -

LC (β)) = det e a Id -(LC (β)) (-k,a) (e a -L k,k )
Since the determinant is 0 by hypothesis and e a > L k,k , so det e a Id -(LC (β)) (-k,a) = 0 However, we recall after the lemma 6.9 that

LC (-k,d) (β) = (LC (β)) (-k,βd) so setting d = a/β yields det e a Id -LC (-k,a/β) (β) = 0
which means that e a is an eigenvalue for LC (-k,a/β) (β).

Dominant eigenvalue

We want to ensure that the eigenvalue e a is still dominant for LC (-k,a/β) (β). As Perron-Frobenius' theorem ensures that its dominant eigenvalue is non-negative, we only need to prove that LC (-k,a/β) (β) has no eigenvalue λ > e a . By contradiction, we shall deem that w is an eigenvector of LC (-k,a/β) (β) with an associated eigenvalue λ > e a , and use it to build a vector w ∈ R A such that the sequence e -a LC (β) n w n∈N geometrically diverges, which will indicate that LC (β) has an eigenvalue greater than e a . First, as C (-k,a/β) is still positive recurrent by construction (lemma 6.4), then LC (-k,a/β) (β) is a positive recurrent matrix whenever defined, thanks to the lemma 5.10, and Perron-Frobenius' theorem states that we may take w with all coordinates being positive. We start with the eigenvector equation LC (-k,a/β) (β)w = λw . By definition of the reduced matrix, we get

LC (β) k,k w + 1 e a -(LC (β)) k,k V Hw = λw
Let w be the vector w with the additionnal coordinate

w [k] = z = Hw e a -(LC (β)) k,k
Then we rewrite LC (β)w decomposing the product between terms issued from column number k and other columns :

• At row number k, we get (LC (β)w) [k] = Hw + (LC (β)) k,k z = ze a
• At other rows, considered as a whole as a vector,

(LC (β)w) [i] i =k = LC (β) k,k w + V z
However, we know by construction that

LC (β) k,k w = LC (-k,a/β) (β)w - 1 e a -(LC (β)) k,k V Hw = LC (-k,a/β) (β)w -V z
So, the eigenvalue equation leads to

(LC (β)w) [i] i =a = LC (-k,a/β) (β)w = λw
Since λ > e a , we get that LC (β)w is we a plus a non-zero vector y ∈ R + A , which rewrites as LC (β)w = we a + y Now, let n ∈ N, so the previous equation leads to

(LC (β)) n w = we na + n-1 k=0 (LC (β)) k y
Since y = 0 is non-negative, it has a q ≤ A such that y [q] > 0. However, we recall that C is positive recurrent, so thanks to lemma 5.10, for any i, j ≤ A, there is ni,j ∈ N * such that ((LC (β)) n i,j ) i,j > 0 Hence, set any i ≤ A and j = q : the product ((LC (β)) n i,j ) i,q y will yield a nonnegative vector whose coordinate number i is positive. Hence, setting n = 1 + max ({ni,q; i ≤ A})

the above sum will evaluate to a vector y ∈ (R * + ) A , which leads in turn to c > 0 and a nonnegative vector y such that (LC (β)) n w = (e na + c)w + y Finally, we find out that the sequence ((LC (β)) nu w) u∈N grows at least at speed n √ e na + c u , so the dominant eigenvalue of LC (β) is greater than e a . By contradiction, we proved that the spectral radius of LC (-k,a/β) (β) cannot be greater than e a , which means that e a is still the its dominant eigenvalue.

Result of successive reductions

Assuming that e a was LC (β)'s dominant eigenvalue, we found out that LC (-k,a/β) (β) is well-defined and e a is its dominant eigenvalue. It follows that we may start over with C (-k,a/β) instead of C, as it is still positive recurrent thanks to the lemma 6.4. We eliminate successively all states of M other than M (0) : starting from C, whose Laplace matrix function has e a as a dominant eigenvalue at point β, we 1. Select any state A k of its underlying Markovian process other than M (0) ; 2. Build its reduced C-process C (-k,a/β) with respect to A k : thanks to the above remark, it is still positive recurrent and its Laplace matrix function has e β as a dominant eigenvalue ;

3. Start again, until only the state M (0) remains.

The final reduced C-process is thus C's (a/β)-restricted Lévy process C (a/β) thanks to the lemma 6.5, whose increments are written as Dτ + (a/β)τ like above. Thanks to the lemma 6.10, the term E e -βDτ e -aτ is well-defined and amounts to e β , so β solves f (β, a) = e a ; as α(a) is the single solution to this equation, then β = α(a). Hence, we proved that the eigenvector equation has at most one solution α(a) ; as the lemma 6.1 assures the existence, we proved that α(a) is the one and only solution, which ends the proof of the first statement of the proposition 2.1 for a ∈ R * + .

Properties of the martingale parameter

Now that the martingale parameter of a C-process has been defined, we focus on its properties as given through the study.

1. Eigenspaces : to prove the proposition 2.1, we focus on the eigenspaces spanned by the dominant eigenvalue e a of the Laplace matrix function LC (α(a)) at point a ∈ R * + . • Dimension of the column eigenspace : since C is positive recurrent, the lemma 5.10 ensures that LC (α) is a positive recurrent matrix no matter α ∈ R * + . It follows from Perron-Frobenius' theorem that the dominant eigenspace is one-dimensional and directed by a positive vector.

• The row eigenspace holds the same properties a similar way.

• The scaling may be chosen as desired provided that the dot products of vectors in the considered eigenspaces are not 0, which is true because these eigenvectors may be chosen positive.

Finally, as the eigenspaces are one-dimensional and the affine equations of scaling are not collinear with them as proved above, the solutions w (a) and µ (a) are unique.

2. Martingale process : to prove that the martingale process X By definition of a C-process, we get (conditionning over M (t + 1))

E X (a) C (t + 1)|F(t) = A k=1 P M (t)→k w (a) [k] E e -α(a)(C(t)+D M (t)→k ) e -at |F(t) e -a that simplifies to E X (a) C (t + 1)|F(t) = e -at e -α(a)C(t) A k=1 P M (t)→k E e -α(a)D M (t)→k w (a) [k] e -a
We recognize the sum as the entry number M (t) of the vector LC (α(a))w (a) . However, since w (a) is an eigenvector of LC (α(a)) associated with the eigenvalue e a by definition of α(a), this equation simplifies to E X Hence, we proved the whole proposition 2.1, as well as the correctness of the definition 2.1, for a ∈ R * + .

Regularity of the dominant eigenvectors

In this paragraph, we prove that the martingale parameter α and the dominant eigenvectors of a C-process are C ∞ functions over R * + . The method we shall use is 1. Verify that α is continuous, as an inverse function of LC (α)'s dominant eigenvalue ; 2. Use α's continuity to apply the implicit functions theorem, expressing it through the equation governing C's restricted Lévy processes, so α is C ∞ ;

3. Verify that µ (a) and w (a) , viewed as functions of a, are locally bounded ;

4. Use this fact to get continuity over R * + ; 5. Like previously, apply the implicit functions theorem and get that they are C ∞ .

In particular, we shall use α's second derivative to verify that is is concave.

Dominant eigenvalue

For a non-negative matrix L ∈ Mn R + , let us consider L's dominant eigenvalue : it is a continuous expression of L's entries (in R + ), and it is positive as soon as L is positive recurrent. For every β ∈ R * + , LC (β) is a positive recurrent matrix thanks to the lemma 5.10, so we may define λ(β) to be the logarithm of LC (β)'s dominant eigenvalue : λ is a continuous function (R * + → R). Let us look at the set

X = λ -1 (R * + ) ⊆ R * + .
We know that λ is bijective over X thanks to the proposition 2.1, which indicates that • X is an opened set, being the inverse image of R * + by the continuous function λ ; • λ is not bounded from above (this is the lemma 6.1), so X must contain arbitrarily large values and is a non-empty interval.

• X is a convex set : indeed, if a < b ∈ X,
It follows that there is α0 ∈ R + such that X = (α0, ∞), so λ is bijective over this set onto R * + ; being continuous, and not decreasing (else we would get λ (α0) = ∞, which is incompatible with X's definition and C sEI), it must be increasing. Therefore, the martingale parameter α, being λ's inverse function (R * + → X), is also continuous and increasing.

Implicit function

We recall that α(a) is the only solution to the equation h(a, α(a)) = 0, where f is the function defined by

f = S → R (a, β) → E e -βDτ e -τ a
h is the function defined by

h = S → R (a, β) → f (a, β) -e a
and S is the opened set coming from the lemma 6.8. We also know that h is C ∞ over S, which allows us to apply the implicit function theorem as follows.

• Let us start from a ∈ R * + . We know by the proposition 2.1 that there is a single α(a) ∈ R * + , with (a, α(a)) ∈ S, such that h (a, α(a)) = 0.

• We compute ∂h/∂2 at this point (a, α(a)).

∀a ∈ R * + , ∂h ∂2 (a, α(a)) = ∂f ∂2 (a, α(a)) = E -Dτ e -α(a)Dτ e -τ a
However, at a fixed a ∈ R * + , we know that f is a convex function so ∂f /∂2 must be non-decreasing of α ; f goes to a limit no higher than 1 when β goes to 0, and f (a, α(a)) = e a is larger than 1.

The mean value theorem indicates that ∂f /∂2 must be positive somewhere in (0, α(a)), so ∂h ∂2 (a, α(a)) > 0

Hence, the implicit function theorem states that there are

• Opened sets Ua a, Va α(a) ;

• A single function ha ∈ C ∞ (Ua → Va),
such that the following subsets of Ua × Va coincide :

{(x, y) ∈ Ua × Va; h(x, y) = 0} = {(x, ha(x)) ; x ∈ Ua}
We recall that α is continuous and solves h(a, α(a)) = 0, so there is an opened set U a a such that α(a) ⊆ Va. Hence, (x, α(x)) ; x ∈ U a ⊆ {(x, ha(x)) ; x ∈ Ua} so α(x) and ha(x) must coincide for x ∈ U a . As ha is C ∞ , it follows that α is C ∞ around a ; this being for every a ∈ R * + , α is C ∞ over R * + .

Spread of w (a)

We want to prove that µ (a) and w (a) are locally bounded. We know that µ (a) 's coordinates are bounded by 1 thanks to the first equation of scaling ; to get a similar property for w (a) , we make use of its spread introduced by the definition 2.2.

Lemma 6.12 Spread of C's dominant eigenvector Let C be a positive recurrent, sEI, not globally ingreasing C-process. For every a ∈ R + , we define C's dominant eigenvector w In particular, we get that

∀i, j ≤ A, n ∈ N, w (a) [i] w (a) [j] ≥ e -na P i n -→j E e -α(a)D i n -→j
Since C is not globally increasing, we know that for every i, j ≤ A, there is a path from Ai to Aj whose value is -v < 0 and length is ni,j ∈ N ; noting by p the probability of following it at precision v, we have p > 0 and

∀a ∈ R + , P i n i,j -→ j E e -α(a)D i n i,j -→ j ≥ P i n i,j -→ j E e -α(a)D i n i,j -→ j 1 D i n i,j -→ j ≤0 ≥ p Thus we have ∀i, j ≤ A, n ∈ N, w (a) [i] w (a) [j]
≥ pe -an i,j

and this rewrites as

∀i, j ≤ A, n ∈ N, ln w (a) [j] w (a) [i]
≤ -ln(p) + ani,j

Taking c = -ln(p) and n = max i,j≤A

(ni,j) solves this lemma. Now, to bound w (a) , we know that there is always i ≤ A such that w (a)

[i] ≤ 1, as else

µ (a) [i] w (a) [i] > µ (a) [i] 1 = 1
By definition of the spread, since we have

∀j ≤ A, ln w (a) [j] w (a) [i]
≤ c + an then we get the desired inequality for every w (a)

[j] , which ends the proof.

Continuity

Let a, b ∈ R * + . We start with the equality

µ (b) -µ (a) (LC (α(a)) -e a Id) = µ (b) e b -e a -µ (b) (LC (α(a)) -LC (α(b)))
that one may verify, using

Fa = R A → R A × R x → x (LC (α(a)) -e a Id) , x 1 It is injective, because if x ∈ R A is such that x (LC (α(a)) -e a Id) = 0 then ∃k ∈ R; x = kµ (a)
, so the second part of Fa(x) amounts to kµ (a) 1 = k by the first equation of scaling : if it is also 0, then x = 0. However, we proved that = 0

We also know that

µ (a) w (b) -w (a) = µ (a) -µ (b) w (b)
so it goes to 0 when b goes to a, and this time we use the function

Ga = R A → R A × R x → (LC (α(a)
) -e a Id) x, µ (a) x that is injective thanks to the second equation of scaling, on x = w (b) -w (a) , with a similar ending.

Differentiability

We introduce the function

f = R * + × R A × R A → R A × R A (a, x, y) → xLC (α(a)) -x 1 xe a , LC (α(a))y -yxye a N.B.:
x is a row vector and y is a column vector in this definition. As α is C ∞ as we proved before, f itself is C ∞ over its domain, so we consider f 's Jacobian matrix : we shall note by ∂fx/∂x, ∂fx/∂y, ∂fy/∂x and ∂fy/∂y its sub-matrices (A × A) related to

• For fx, the first A coordinates of f , and fy are the other ones ;

• For ∂x, differentiation with respect to the A coordinates of x, and ∂y for y's ones. then we have

• Either v 1 = 0, thus vLC (α(a)) = e a v so v belongs to LC (α(a))'s dominant row eigenspace (which is µ (a) R), but then v 1 = 0 implies v = 0 because µ (a) 1 = 0.

• Or there is v ∈ R A such that µ (a) = v (LC (α(a)) -e a Id)
and by µ (a) 's definition of an eigenvector, we get by right multiplication that v (LC (α(a)) -e a Id) 2 = 0

As e a 's is an eigenvalue of order 1, this is possible only if already v (LC (α(a)) -e a Id) = 0 so µ (a) = 0, which is impossible.

Likewise, the case of the matrix LC (α(a)) -e a Id -w (a) µ (a) e a is solved when looking for a right kernel, through the use of the other equation of scaling. Hence, the restriction of f 's Jacobian matrix related to derivatives with respect to (x, y) at point µ (a) , w (a) is invertible : as f a, µ (a) , w (a) = 0 by definition of µ (a) and w (a) , one may apply the implicit functions theorem : there are a) , and Wa ⊆ R A containing w (a) ;

• Opened sets Ua ⊆ R * + containing a, Va ⊆ R A containing µ (
• A single function ga ∈ C ∞ (Ua → Va) and a single function ha ∈ C 1 (Ua → Wa) such that {(z, ga(z), ha(z)) ; z ∈ Ua} = {(z, x, y) ∈ Ua × Va × Wa; f (z, x, y) = 0}
Once again, as a, µ (a) , w (a) cancels f , and µ (a) and w (a) are continuous of a (as we proved above), there is an opened set U a such that µ (a) (U a ) ⊆ Va and w (a) (U a ) ⊆ Wa. It follows that z, µ (z) , w (z) ; z ∈ U a ⊆ {(z, ga(z), ha(z)) ; z ∈ Ua} so µ (z) and ga(z), as well as w (z) and ha(z), coincide over U a that contains a ; as ga and ha are C ∞ over U a , it follows that µ and w are C 1 around a. This being for every a ∈ R * + , this ends the proof.

Concavity

Finally, we prove that α is concave over R * + , which requires no more than defining

f = S → R (a, β) → E e -βDτ e -a(τ +1)
and computing the second derivative of the equation f (a, α(a)) = 1.

• The first derivative yields and we recall that this term is well-defined and non-negative, thus positive because C is positive recurrent.

E - dα (
• The second derivative yields The rightmost term is non-negative, and as dα(a)/da > 0 and C is positive recurrent, α's second derivative is non-positive, which ends the proof.

E - d 2 α(

Limit at point zero

We aim at proving the propositions 2.1 for a = 0, leading later to properties about C's default time.

Throughout this paragraph, we shall name fC the function defined by

fC = R + × R + → R ∪ {∞} (a, β) → E 1D τ <∞e -βDτ e -a(τ +1)
so that fC (a, α(a)) = 1 no matter a ∈ R * + : we are going to control this function fC to get the desired results when a goes to 0.

Integrability

We aim at finding β0 ∈ R * + such that fC (0, β0) < ∞. If we succeed, then as we know that for every a ∈ R + and β ∈ [0, β0], fC (a, β) will be bounded by fC (a, β0), the dominated convergence theorem indicates that

• fC is continuous over the whole set R + × [0, β0] ;

• fC is C ∞ over R * + × (0, β0) ; • In particular, fC (0, 0) = P (Dτ < ∞).

We remember that the expression of L C (d) (β), for d ∈ R, was obtained as a rational fraction of LC (β)'s entries and terms e βd . When d = 0, the successive constraints we find on the successive reductions come to ∀u ∈ [|2, A|] , P u,ku→ku E e -βD u,ku →ku < 1 where D u,ku→ku is exponentially integrable over some opened set (0, βu) with βu > 0 thanks to the induction hypothesis. As P u,ku→ku < 1 because u ≥ 2, the induction loop follows. Finally, for u = 1, we get some β1 > 0, and taking any β0 ∈ (0, β1) works.

Limit of eigenvectors

We prove that α, µ and w are continuous at point a = 0.

• Let us start with α. As it is increasing and bounded from below by 0, it converges to a non-negative limit l at point 0. Hence, we may extend α's domain to 0 by α(0) = l ; as α was already continuous over R * + , it is now continuous over R + . • The Laplace matrix function LC may also be continued by a limit L at point 0, being the matrix whose entry (i, j) is given by Li,j = Pi→jP (Di→j < ∞)

It follows that LC (α(0)) is a well-defined limit, whether α(0) = 0 or not, and is still a positive recurrent matrix whenever C is positive recurrent. As the function

R + → R a → det (LC (α(a)) -e a Id)
is then well-defined and continuous over R + , and identically zero over R * + , then it is also zero at point 0, so 1 is an eigenvalue for LC (α(0)). This unit eigenvalue must be dominant, because (complex) eigenvalues of LC (α(a)) are continuous functions of a and dominated by e a over R * + by construction of α. Hence, we may note by -µ0 ∈ (R * + ) A a row eigenvector of LC (α(0)) such that µ0 1 = 1 ; -w0 ∈ (R * + ) A a column eigenvector of LC (α(0)) such that µ0w = 1. • We rewrite the eigenvector equation as

   µ (a) (LC (α(a)) -LC (α(0))) + µ (a) -µ0 LC (α(0)) + µ0LC (α(0))    =    µ (a) (e a -1) + µ (a) -µ0 + µ0   
After simplifications, this leads to

µ (a) -µ0 (LC (α(0)) -Id) = µ (a) (LC (α(a)) -LC (α(0)) -(e a -1)Id)
However, we know that µ (a) 's coordinates are bounded by 1 because they are non-negative and µ (a) 1 = 1 ;

-LC (α(a)) -LC (α(0)) converges to 0 when a goes to 0 because α is continuous at point 0 and LC is continuous at point α(0).

We note by F the linear function defined by

F = R A → R A+1 x → x (LC (α(0)) -Id) , x 1
F is injective : indeed, if xLC (α(0)) = x, then there is k ∈ R such that x = kµ0 (thanks to
Perron-Frobenius' theorem), and then k = 0 thanks to the first equation of scaling. As we proved that F µ (a) -µ0 goes to zero when a goes to zero, this implies that we may continue the function µ by µ (0) = µ0, and µ is now continuous over R + .

• A similar property holds for w : we use the same line of thought, requiring only (when using the scaling properties) that µ is continuous at point 0, which was just proved ; w is locally bounded around 0, which is true because µ being continuous around 0, its lowest coordinates hold ∀i ≤ A, ∃x ∈ [0, 1] ; inf

y∈[0,1] µ (y) [i] = µ (x) [i] > 0 so there is a positive real r > 0 such that ∀i ≤ A, ∀x ∈ [0, 1] , µ (x)
[i] ≥ r, which implies that w's coordinates are bounded bt 1/r thanks to the second equation of scaling.

It follows that we may define α(0), µ (0) and w (0) . Henceforth, we shall rename α, µ and w their respective continuations at point a = 0.

Case E(C) = ∞

This paragraph and the following one aim at proving that α(0) = 0 iff E(C) ≤ 0 ; we begin here by the case E(C) = ∞. There is a transition (i → j) such that

µ [i] Pi→jE (Di→j) = ∞
As C is positive recurrent, we use it to build a cycle going from M (0) to M (0) of infinite value, thus P (Dτ = ∞) > 0. Let us assume that α(0) = 0. As α is continuous and increasing, there is a0 ∈ R * + such that ∀a ∈ (0, a0), α(a) ∈ (0, β0) where β0 was defined earlier. However, as fC is continuous over

R + × [0, β0], then fC (0, α(0)) = lim a→0 (fC (a, α(a))) = 1
whereas fC (0, 0) = P (Dτ < ∞) < 1. It follows that α(0) cannot be 0 ; as it is non-negative, it is positive.

Case E(C) > 0 and finite

As C is positive recurrent and sEI, the negative part of its increments is integrable, so E(C) < ∞ indicates that C's increments are all integrable. Let us assume that α(0) = 0. Once again, there is a0 ∈ R * + such that ∀a ∈ (0, a0), α(a) ∈ (0, β 0 ) where β 0 < β0 that was defined earlier. Differentiation of the equation fC (a, α(a)) = 1 at such a point yields dα(a) da = E (τ + 1)e -α(a)Dτ e -a(τ +1)

E (-Dτ e -α(a)Dτ e -a(τ +1) )

However, when a goes to 0, we analyze the numerator :

• We know that E(τ ) < ∞, because τ + 1 is M 's return time to M (0), so holds a sub-geometric distribution ;

• As a consequence, the dominated convergence theorem indicates that the numerator converges to a positive value (because Dτ is not ∞ almost surely, as C is positive recurrent) named x1 ∈ R * + . Now we analyze the denominator :

• The positive part of the denominator is controlled by

E -Dτ e -β 0 Dτ 1D τ <0
However, this is bounded by 1 β0 -β 0 E e -β 0 Dτ < ∞ so the positive part is bounded ;

• The negative part of the denominator is controlled by

E (Dτ 1 Dτ ≥0 )
This value is finite, because thanks to the lemma 6.6, as C is integrable, Dτ is integrable.

• The theorem of dominated convergence thus yields

x2 = lim a→0 E -Dτ e -α(a)Dτ e -a(τ +1) = -E (Dτ )
and thanks to the lemma 6.6, this is

x2 = - E(C) µ [M (0)] ∈ R * - Hence, we get that lim a→0 dα(a) da = x1 x2 < 0
This is impossible because α is increasing over R * + , so once again α(0) > 0.

Case E(C) ≤ 0

This time we assume that α(0) > 0. First, as by definition 1 is the dominant eigenvalue of LC (α(0)) and C is positive recurrent, the same construction as in the lemma 6.8 enables successive reductions of C up to its 0-restricted Lévy process, i.e. its restricted Lévy process. It follows that fC (α(0), 0) = 1 ; noting by g the function defined by

g = [0, α(0)] → R β → fC (β, 0)
then g is continuous over [0, α(0)] and C 2 over (0, α(0)). Moreover, we know that

∀β ∈ (0, α(0)) , d 2 g(β) dβ 2 ≥ 0
However, when β goes to 0, we find out that

g(0) = P (Dτ < ∞) = 1
because the lemma 6.6 indicates that C is integrable. So, g is a convex function that evaluates to 1 at both points 0 and α(0), but we also know that

lim β→0 dg(β) dβ = E (-Dτ 1D τ <∞)
and this term is non-negative thanks to the lemma 6.6. Hence, g must be constant over [0, α(0)], which is possible only if Dτ is either 0 or ∞ almost surely, which is excluded as C is not globally increasing. So, we proved that α(0) = 0 ; to find the dominant eigenvectors, we shall verify that ∀i, j ≤ A, (LC (0

)) i,j = Pi→j indeed if Di→j = ∞, then Pi→j = 0 else one would get E(C) = ∞.
As the dominant eigenspaces have unit dimension, we only need to exhibit any non-zero eigenvector of LC (0), then scale it to find µ (0) and w (0) ; by definition of M 's transition matrix, µ (0) = µ and w (0) = 1 work, which ends the proof for the limit terms at a = 0.

Special cases

In this part, we take C as a C-process whose underlying Markovian process is M . We aim at finding what happens when C does not fall in the scope of the main method, mainly because it is periodic : the Laplace transforms end up not being of the desired form. Indeed, when C is globally increasing, it will never default provided that its starting point is high enough (when C0 amounts at least to the Q from the proposition 1.2). As a consequence, we shall hereby assume that C is aperiodic, and has p ∈ R * + as a fundamental period. We ultimately aim at proving the proposition 2.4.

Values for the offsets

First, we introduce the offsets as given along with the regular process from the definition 2.3. Let i, j ≤ A ; as C is positive recurrent, we create some paths of finite value, thanks to the lemma 5.5 and the requirement for C to be positive recurrent.

• From Ai to Aj, we define Si→j ⊆ R the set of finite values for all such paths. Si→j is not empty thanks to the lemma 5.5, so we may choose any vi,j ∈ Si→j ;

• From Aj to Ai, we choose a path of finite value named x ∈ R (this is possible thanks to lemma 5.5 again).

The concatenation of the path (Ai → Aj) of value vi,j and the path (Aj → Ai) of value x yields a cycle of value vi,j + x that must belong to pZ by hypothesis. It follows that ∃x ∈ R; ∀vi,j ∈ Si→j, vi,j + x ∈ pZ So, for every i, j ≤ A, there is pi,j (it may be thought of as -x modulo p) such that Si→j ⊆ pi,j + pZ C being positive recurrent, all states are accessible using paths of finite values, and concatenation of paths leads to ∀i, j, k ≤ A, Si→j + S j→k ⊆ S i→k Since the sets are not empty, then pi,j + p j,k -p i,k ∈ pZ. We set e.g. for i = M (0) and some x ∈ R the values pj = p M (0),j + x, which leads to the condition

∀j, k ≤ A, p j,k ∈ p k -pj + pZ
For any fixed value of x leading to p M (0) = x, this condition leads to a single solution vector (pi) i≤A ∈ ([0, p)) A , which creates offsets once choosing x ≡ C(0) modulo p.

Regular process

To prove that the definition 2.3 is correct, we take C a C-process as given, and prove by induction on t that ∀t ∈ N, C(t) ∈ Z ∪ {∞} almost surely.

• For t = 0,

C(0) = C(0) -p M (0)
p is an integer by hypothesis on p M (0) .

• For every t ∈ N, let us assume that C(t) ∈ Z ∪ {∞} almost surely and write

C(t) = p C(t) + p M (t)
In the equation C(t + 1) = C(t) + D M (t)→M (t+1) , we know that D M (t)→M (t+1) belongs to p M (t+1)p M (t) + pZ almost surely thanks to the definition of the offsets, so C(t + 1) may be written with a random integer increment Z (maybe Z = ∞) as

C(t + 1) = C(t) + p M (t+1) -p M (t) + pZ So, this leads to C(t + 1) = p C(t) + Z + p M (t+1)
As C(t) is either an integer or ∞ by induction hypothesis, then C(t + 1) is of the right form, so C(t + 1) is either an integer or ∞.

Finally, since C's fundamental period is p ∈ R * + , then after scaling by 1/p, C's one will be 1.

Transfer of properties

We are going to express the martingale parameter for C, along with its dominant eigenvectors. xt -pa t + pa t-1 p = x -pa T + pa 0 p

However, a0 = aT by definition of a cycle, so y = x/p, and it follows that

R ∩ supp (C) ⊆ R ∩ supp C p
As this works also the other way around, we get the desired property. In particular, if C is not globally increasing, then C cannot be either.

(c) We prove that C and C have the same default time. Let t ∈ N. If C(t) < 0, then C(t) < 0 because the offsets are nonnegative. Conversely, if C(t) < 0, then C(t) ≤ -1 because C(t) ∈ Z almost surely, so C(t) ≤ p M (t) -p < 0 because the offsets are lower than p. It follows that C(t) < 0 iff C(t) < 0, which ends the proof. (d) Finally, if C is sEI (or bounded), then C is still sEI (or bounded), because

• The "relevant" transitions (i.e. (i, j) ∈ Γ) are the same for both processes because M is still C's Markovian process (in particular, if C is positive recurrent, then C is also positive recurrent) ; • Adding a bounded (by p) constant to the relevant transition payoffs does not change their integrability properties.

Hence, if the conditions to the existence of a martingale parameter are satified for C, they are for C.

2. We compare the Laplace matrix functions of C and C. By definition of C's transition payoffs, we have ∀i, j ≤ A, Di→j = Di→j -pj + pi p In particular, for every α ∈ R, using the diagonal change-of-basis matrix ∆(α) defined by ∀i ≤ A, (∆(α)) i,i = e -αp i then by definition of Laplace matrix functions,

∀α ∈ R, LC (α) = (∆(α)) -1 L C (pα)∆(α)
As we recall that the martingale parameter α(a) and the dominant eigenspaces w (a) were defined by the proposition 2.1, we get that (a) The eigenvalues of LC (α) and L C (pα) being identical, the martingale parameter α(a) must be given by pα(a) ; (b) If w ∈ R A is a dominant eigenvector for LC (α), then using the change-of-basis matrix ∆(α) and the above equation, v = ∆(α)w is a dominant eigenvector for L C (pα). Likewise, if ν ∈ R A is a dominant row eigenvector for LC (α), then ν (∆(α)) -1 is a dominant row eigenvector for L C (pα).

Prerequisites to the proof for periodic C-processes

We aim at proving the proposition 2.4 for a C-process C responding to the hypotheses. Since its default time is identical to the one of its regular process C thanks to the above work, we are going to prove a similar property for C. Let us note respectively

• α is C's martingale parameter (so ∀a ∈ R + , α(a) = pα(a)) ;

• C0 is C's starting point, i.e. C0 = C0 p ∈ Z
by definition of the regular process ;

• w(a) is C's dominant eigenvector, defined previously.

We want to prove this lemma : Lemma 6.13 Convergence for the regular process Let C be a C-process whose underlying Markovian process is M . We assume that

• C is positive recurrent, bounded and not globally increasing ;

• For every t ∈ N, C(t) ∈ Z almost surely ;

• C's fundamental period is 1.

Such a C-process will be called a regular C-process. Let a ∈ R + and s ≤ A. We set

• Its random default time as T ∈ N ∪ {∞} ;

• T 's log-Laplace at point a starting from M (0) = As and C(0) = x ∈ N as

Λ (a) s = N → R x → ln E e -aT 1T <∞|M (0) = As ∧ C(0) = x
• C's martingale parameter at point a as α(a) ; • C's dominant eigenvector at point a as w (a) .

For every a ∈ R + , there is

K (a) ∈ R such that ∀x ∈ N, -Λ (a) s (x) ∈ α(a)x -ln w (a) [s] + K (a) ± e (x, a)
where e is an error function

(N × R * + ) → R + , uniformly exponentially convergent to 0 • Over any subset of the form a ∈ [0, b] with b ∈ R + ;
• When x goes to +∞ (keeping integer values). Moreover,

1. K (a) is a continuous expression of a ; 2. If C is a Lévy process, then ∀a ∈ R + , K (a) ≥ α(a).
The proof for this lemma itself is postponed until the next part, as it relies on the definitions of descending processes.

Proof of the theorem for periodic C-processes

In this paragraph, we assume the previous lemma 6.13 to be granted, and we aim at proving the proposition 2.4. More specifically, we want to build the functions K and e, using the error functions given by this lemma 6.13. Henceforth, we shall note

• For every x ∈ R + , C[s,x] is C's regular process where C starts from C(0) = x and M (0) = As ; • For every i ≤ A, p [s]
i is the natural offset of the state Ai when C starts from M (0) = As and any C(0) ∈ pN ;

• For x ∈ R and y ∈ R * + , y

x is the only element of x + yZ ∩ [0, y) (so-called "x modulo y") ;

For example, it follows from these notations that the natural offset of the state Ai when C starts from M (0) = As and C(0

) = x ∈ R + is p p [s]
i + x

Breaking points

Our basis is to remark that Λ (a) i is piecewise constant over R + in a p-periodic fashion.

Definition 6.4 Breaking points

Let C be a positive recurrent, periodic C-process, whose fundamental period is p ∈ R * + and random default time is T . Let a ∈ R + and s ≤ A, so that we set Λ (a) s as in lemma 6.13. There is a family

q [s] i i∈[|0,A-1|] ∈ [0, p) A
sorted by ascending order with q

[s] 0 = 0, such that for every n ∈ N, • Λ (a) s is constant over [np + q [s] A-1 , (n + 1)p) ; • For every i ∈ [|1, A -1|], Λ (a) s is constant over [np + q [s] i-1 , np + q [s] i ).
We shall call such points q We prove the correctness of this definition making use of C's regular process C[s,x] for several starting points C(0) = x ∈ R + , with the same starting state As. The main idea is to remark that C[s,x] is the same C-process for "close" values of x, as long as no offset of C loops back from p to 0.

• Let us start with x = 0. We get C's natural offsets p

[s]

i , and as they are in a finite number, some value

p [s] max = max i≤A p [s] i < p • As long as x < p -p [s] max called q [s]
1 , then the family of offsets p

[s]

i + x keeps defining offsets of C that satisfy the condition for natural offsets (because p

[s] i + x < 1). As natural offsets are unique, they are the natural offsets of C starting from this x ; by definition of C[s,x] , x cancels out in the computations of the values of the regular process, so C[s,x] is the same process for every x ∈ [0, q

[s] 1 ).

• Then we start again with x = q

[s] 1 , getting some value p

[s] max (the second maximum among the offsets p

[s] i ) and q [s] 2 = p -p [s] max , such that C[s,x] is the same process for every x ∈ [q [s] 1 , q [s]
2 ), and so on.

• We proceed until A -1 states have been reached. As the last maximum is the minimum p

[s] s = 0 by definition of C[s,0] , then C[s,x] is the same process for every x ∈ [q [s] A-1 , p).
• After that, the whole thing restarts with the same successive values q

It follows by induction that the family q

[s] i i

(with q

[s] 0 = 0) forms breaking points for Λ (a) s .

Identification of the terms for breaking points

Let C be a C-process, deemed positive recurrent, bounded and not globally increasing, starting from M (0) = As and C(0) = x ∈ R + , whose fundamental period is p ∈ R * + . We aim at finding suitable terms K and e for the proposition 2.4, for now only when x ∈ q

[s] i + pN where q [s] i is some breaking point for Λ (a)
s . To do that, we set any a ∈ R + , as the choice of a is irrelevant to the breaking points. The main idea is that C[s,x] 's transition payoffs do not change when x stays in q

[s] i + pN, so only the starting point changes for C[s,x] , which means that we may apply the lemma 6.13 and get, for every i ≤ A, suitable values of K and errors e that do not depend on s ; we note them respectively K i and ei. However, we recall that since x ∈ q

[s] i + pN, • C[s,x]
martingale parameter is pα(a), and its starting point is

z = x -q [s] i p • For every j ≤ A, let p [s] j (x) = p p [s]
j + x be Aj's natural offset of C starting from x and M (0) = As, viewed as a p-periodic function of x.

When x ∈ q [s] i + pN, it keeps a single value, noted p [s] j,[i] = p p [s] j + q [s] i
Let ∆ [s] (i, α) be the diagonal matrix whose non-zero entries are these offsets

∀j ≤ A, ∆ [s] (i, α) j,j = e -αp [s] j,[i]
We know that there is k

[s,i] (a) ∈ R * + such that C[s,x] 's dominant eigenvector is w[s,i],(a) = k [s,i] (a)∆ [s] (i, α(a))w (a)
The statement from lemma 6.13 applied to C[s,x] yields (for the Laplace transform Λ(a)

s of C[s,x] 's default time) ∀z ∈ N, -Λ(a) s (z) ∈ pα(a)z -ln k [s,i] (a) ∆ [s] (i, α (a)) s,s w (a) [s]
+ K i (a) ± ei (z, a)

Hence, when x = q

[s]

i + pz, we know that : • By construction of the regular process,

Λ(a) s (z) = Λ (a) s (x) • For every α ∈ R + , ∆ [s] (i, α) s,s = e -αp [s] s,[i] = e -α p p [s] s + q [s] i • As p [s]
s = 0 by definition of the starting state,

p p [s] s + q [s] i = q [s] i
it follows that the expression in the above control simplifies to

-Λ (a) s (x) ∈ α(a)x -ln w (a) [s] + K i (a) -ln k [s,i] (a) ± ei x -q [s] i p , a
Hence, one may set

∀x ∈ q [s] i + pN, ∀a ∈ R + , K(x, a) = K i (a) -ln k [s,i] (a)
and ∀x ∈ q

[s] i + pN, ∀a ∈ R + , e(x, a) = ej x p , a N.B. : if q [s] i-1 = q [s] i
for some i, these definitions not are contradictory because the regular processes C[s,x] would coincide : so, K i-1 = K i , ki-1 = ki and ei-1 = ei, and the definitions for x ∈ q

[s] i-1 + pN and x ∈ q

[s] i + pN coincide.

Extension to R

+ × R + Now that the cases x ∈ q [s]
i + pN are settled, we use the breaking points given by definition 6.4 to extend K and e to R + × R + . For conveniency, we shall note

• The "boundary" breaking points of Λ (a) s are q [s] 0 = 0 and q [s] A = p ; • The intervals where Λ (a) s is constant are, for every n ∈ N and i ≤ A, I [s] n,i = [q [s] i-1 + pn, q [s] i + pn)
We should begin with the remark

∞ n=0 A i=1 I [s] n,i = R +
where the sets

I [s]
n,i are pairwise disjoint, so with every x ∈ R + one may associate a single (n, i) such that x ∈ I

[s] n,i . Now, let n ∈ N and i ≤ A. Λ (a) s is constant over I [s]
n,i , and we already know that for x ∈ qi-1 + pN, we have

∀a ∈ R + , i ≤ A, -Λ (a) s (x) ∈ α(a)x -ln w (a) [s] + K i-1 (a) -ln k [s,i-1] (a) ± ei-1 x p , a
Hence, setting for x ∈ I

[s]

n,i the function K as

K(x, a) = K q [s] i-1 + pn, a -x -q [s] i-1 + pn α(a)
and the function e as e(x, a) = ei-1 (n, a)

then the control

∀a ∈ R + , s ≤ A, -Λ (a) s (x) ∈ α(a)x -ln w (a) [s]
+ K(x, a) ± e(x, a)

will still hold over x ∈ I [s]
n,i : doing this for every i ≤ A and n ∈ N thus yields the desired control over the whole R + . Finally, we verify that this function K is p-periodic of its first variable. Let us take x ∈ I n+1,i , so

K(x + p, a) = K q [s] i-1 + (n + 1)p, a -(x + p) -q [s] i-1 + (n + 1)p α(a)
p cancels out in the rightmost term, and we recall that

K q [s] i-1 + (n + 1)p, a = K i (a) -ln k [s,i] (a)
that does not depend on n, which ends the proof. As a consequence, we will • Redefine the function K as being [0, p) × R + → R , as given by its periodicity ;

• Keep the function e, in the subsequent paragraphs.

Continuity of K

We verify that K is piecewise continuous as required by the theorem 2.4. When x ∈ I [s] n,i , the terms n and i are constant, and so is y = q

[s] i-1 + pn. We recall that

∀x ∈ I [s] n,i , K(x, a) = K (y, a) -(x -y) α(a)
However, α is continuous and K(y, a) is defined by

∀a ∈ R + , K(y, a) = K i (a) -ln k [s,i] (a)
K i is continuous thanks to the lemma 6.13. To get this property for k [s,i] (a), we use the fact that it is the normalization constant of w[s,i],(a) : noting μ[s,i],(a) the dominant row eigenvector of C[s,i] , one has

1 = μ[s,i],(a) w[s,i],(a) = k [s,i] (a)μ [s,i],(a) ∆ [s] (i, α(a))w (a)
however α(a), μ[s,i],(a) , and w[s,i],(a) are continuous of a, with the vectors being positive and the diagonal matrix having positive entries, so k [s,i] is positive and continuous, which ends the proof of K's continuity in a. It follows that

• K is continuous in a no matter x (fixed) ;
• This being for every i ≤ A, K is continuous over every I [s] 0,i × R + , thus piecewise continuous over its domain.

Sign of K

We want to ensure that when C is a Lévy process, K(x, a) > 0 for every x ∈ R + and a ∈ R * + . As Lévy means A = 1, let us take n ∈ N such that x ∈ I

[s]

n,1 ; according to the previous notations, we have

K(x, a) = K0(a) -ln k [1,0] (a) -x -q [1] 0 + pn α(a)
We know that K0(a) ≥ pα(a) thanks to the lemma 6.13 and the fact that the regular process has a martingale parameter pα(a) at point a. To evaluate ln k [1,0] (a) , we use the equations of scaling : the dominant row eigenvector μ(a) of C becomes the unit 1 * , and it follows from this that k [1,0] (a) = ∆ [1] (0, α(a))

-1
We recall that, by definition of ∆ [s] (i -1, α(a)), its only term amounts to ∆ [1] (0, α(a))

1,1 = e -α(a) p p [1] 1 + q [1] 0 However, p [1]
1 = 0 by construction and q [START_REF] Asmussen | Risk theory in a markovian environment[END_REF] 0 is also 0 modulo p, so everything simplifies and finally

k [1,0] (a) = 1 As a consequence, for x ∈ I [1] n,1 = [np, (n + 1)p), we have K(x, a) = K0(a) -(x -np) α(a) > pα(a) -((n + 1)p -np) α(a) = 0
which ends the proof.

Exponential convergence

We may now verify that e converges uniformly exponentially to 0. To do that, let us set amax ∈ R * + and look at the functions ei for i < A. By hypothesis on them given by lemma 6.13, for each i there are Bi ∈ R * + and βi > 0 such that ∀n ∈ N, ∀a ≤ amax, ei(n, a) < Bie -nβ i When looking for n ∈ N and i such that x ∈ I which ends the proof of the theorem proposition 2.4.

Convolution processes

In this paragraph, we shall deem that C is a bounded and non-globally increasing C-process, and we eventually aim at proving the main theorem 1, for now only when a ∈ R * + . We shall note the Laplace transform of its default time starting from the state M (0) = A i≤A and the point

C(0) = x ∈ R + at point a ∈ R * + by L (a) i (x) = E e -aT 0 |M (0) = Ai ∧ C(0) = x
The core of the proof for ΛT 0 's asymptotical behaviour for large values of C0 lies in the properties governing the Laplace transform. Indeed, by definition of the default time and the Markovian, timehomogeneous behaviour of C, we have

∀i ≤ A, x ∈ R+, a ∈ R * + , L (a)
i (x) = A k=1 P i→k E L (a) k (x + D i→k )
To simplify the future study, we want to assume the transition payoffs to be negative almost surely, for a reason that will appear below. The aim of this part is to build a new C-process, named "Convolution process", whose ΛT 0 is preserved but whose increments are negative almost surely and easier to deal with.

Goal : the convolution equation

For the sake of simplicity, let us consider that C is a Lévy process for the time being, and that we seek its default probability when starting from the point x ∈ R + , defined by

∀x ∈ R + , P (x) = P (T0 < ∞|C(0) = x) = L (0) (x)
The naive analysis of C's default time will yield, by the Markovian property, an equation like

∀x ∈ R + , P (x) = ∞ y=-∞ P (x + y)σ(y)dy
where σ is the distribution of C's transition payoff D. Unfortunately, as σ takes both positive and negative support values, trying to use the Laplace transform on this equation does not work ; this is somewhat equivalent to saying that the equation is not in "solved" form, with P (x) expressed as a combination of previous values P (y) for y < x. For instance, when C is periodic, its regular process is C, that expresses P (x) for x ∈ N through a recurrence equation that is not in solved form.

Convolution equation

For this reason, the next idea is transforming the previous equation in another equation where D's support is contained in R * -.

Lemma 7.1 Convolution equation

Let C be a bounded (by Q ∈ R * + ), non-globally increasing C-process and a ∈ R * + , allowing us to define • C's martingale parameter at point a as α(a) ∈ R + ;

• C's dominant eigenvector at point a as w (a) ∈ (R * + ) A . Then for every i, j ≤ A, there are An alternate formulation of the descending equation may be written as follows.

• Random variables G (a) i→j ∈ (0, Q] ; • Constants named P (a) i→j ∈ [0, 1] ; • Measurable functions K (a) i : R → R + such that 1. The sought Laplace transforms hold ∀i ≤ A, x ∈ R + , L (a) i (x) = e -α(a)x w (a) [i] K (a) i (x)

Lemma 7.2 Convolution process

Let C be a bounded (by Q ∈ R * + ), non-globally increasing C-process and a ∈ R * + as above (lemma 7.1). We define the convolution random variables G Then the process Y (a) defined by

Y (a) = N → R + t → K (a) M (a) (t) Φ (a) (t) is a martingale. Admitting lemma 7.1, we have indeed E Y (a) (t + 1)|F(t) = A j=1 P M (a) (t)→j E K (a) j Y (a) (t) -G (a) Ma(t)→j |F(t) This is K (a)
M (a) (t) Y (a) (t) by lemma 7.1, which proves the statement.

Introduction to the descending process

To prove the lemma 7.1, we will use another concatenation trick, skipping times t ∈ N * for which C(t) is not less than its previous minimum min

u<t ({C(u)})
For some value of C(t), we wait until C hits an inferior value, which is equivalent to considering min u ∈ N;

u k=1 D M (t+k-1)→M (t+k) < 0
If there is no such u, then we set u = min (∅) = ∞ and we also know that (provided that C(t) ≥ 0) C will never default, so T0 = ∞. The descending process of C is defined as the process of successive all-time low values and associated wating times.

Let us note

ν (a) i→j = ∞ n=1 Q i n -→j e -an ν i n -→j
These are nonnegative measures of finite masses, that solve

∀i ≤ A, a ∈ R * + , x ∈ R + , L (a)
i (x) = A j=1 L (a) j * ν (a) i→j (x)
Noting by Qi→j the sum of probabilities Q i n -→j for n ∈ N, we notice that • These measures have finite masses because a > 0, Qi→j ≤ 1, and the measures ν i n -→j have unit masses ;

• When a goes to 0, we get

ν (0) i→j = ∞ n=1 Q i n -→j ν i n -→j
whose mass is Qi→j ; when it is non-zero, dividing ν

i→j by its mass yields the distribution of -C(t + 1) -C(t) conditionally to M (t) = Ai and M (t) = Aj. Now we prove this lemma 7.3. To get LT 0 using C, we use the fact that C's default time must be an all-time low for C, so must be recorded by C : we get C's default time, reading C's default time T C ∈ N ∪ {∞} and computing the total time

T0 = τ (T C ) = T C t=1 (τ (t) -τ (t -1))
As C is a C-process, then the values L (a) i (x) decomposes following the possibilities of immediate future for C as follows.

1. A random determination of both a transition (starting from Ai) and a descent time is drawn, with respect to C's transition probabilities

Q i n -→j i,j≤A;n∈N *
This has a multiplicative effect of Q i n -→j on the following. 2. If j = ∞, after n time periods, M goes to Aj and C goes to x -F i n -→j : waiting n time periods has a multiplicative effect of e -na on the conditionnal expected value of the Laplace transform we get there

E e -aT 0 |C(0) = x -F i n -→j ∧ M (0) = Aj = L (a) j x -F i n -→j
If j = ∞, then T0 = ∞ almost surely, so the value we get there is 0.

So we finally find out that

∀i ≤ A, a ∈ R * + , x ∈ R + , L (a)
i (x) = A j=1 ∞ n=1 Q i n -→j e -an E L (a) j x -F i n -→j
Rewriting this equation as a convolution equation using the distributions ν i n -→j , one gets

∀i ≤ A, a ∈ R * + , x ∈ R + , L (a)
i (x) = A j=1 L (a) j * ∞ n=1 Q i n -→j e -an ν i n -→j (x)
By definition of ν (a) i→j , this ends the proof.

Masses

This lemma 7.3 will be especially useful when the distributions ν (a) i→j , having a nonnegative support, have a unit mass, meaning that the values L (a) i (x) will be convex combinations of previous values L (a) i (y) for y ≤ x. Our next step is now to twist these distributions to get the convex combinations. Hence, let β ∈ R, and set

∀i ≤ A, a ∈ R * + , x ∈ R + , dν (a)
m (a) i→j,[β] = ∞ n=1 Q i n -→j e -an E e βF i n -→j
Recalling that F i n -→j is upper bounded by Q by construction and positive almost surely, then for a ∈ R * + and β ∈ R, this mass is finite. Moreover, it is

• Non-zero whenever Qi→j > 0 ;

• 0 when Qi→j = 0, but this is no issue as it will have no effect on further computations.

After properties of the convolution product, and defining the twisted Laplace transforms

∀i ≤ A, a ∈ R * + , x ∈ R + , L (a) i,[β] (x) = L (a)
i (x)e βx then lemma 7.3 rewrites as

∀i ≤ A, a ∈ R * + , x ∈ R + , L (a) i,[β] (x) = A j=1 L (a) j,[β] * ν (a) i→j,[β] (x)
If the measures ν (a) i→j,[β] have non-zero masses, we may divide them by their finite masses, getting probability measures that will be noted φ

∀i ≤ A, a ∈ R * + , x ∈ R + , L (a)
i,[β] (x) = A j=1 m (a) i→j,[β] L (a) j,[β] * φ (a) i→j,[β] (x)
Since the measures φ

∀i ≤ A, a ∈ R * + , x ∈ R + , L (a)
i,[β] (x) = A j=1 m (a) i→j,[β] E L (a) j,[β] x -G (a) i→j,[β]
We shall use this equation in the next paragraph.

Scaling

The previous paragraph ensured the measures φ

E L (a) j,[β] x -G (a) i→j,[β]
To do that, we allow the functions L (a)

i, [β] to be scaled by multiplicative positive constants r

[i] i≤A : ∀i ≤ A, L (a) i,[β] = r [i] K (a) i,[β]
After this scaling, the convolution equation becomes

∀i ≤ A, a ∈ R * + , x ∈ R + , r [i] K (a) i,[β] (x) = A j=1 m (a) i→j,[β] r [j] E K (a) j,[β] x -G (a) i→j,[β]
Hence, we shall get the desired result whenever

∀i ≤ A, r [i] = A j=1 m (a) i→j,[β] r [j]
which happens when r is an eigenvector associated to eigenvalue 1 of the "mass" matrix m (a), [β] , whose entries are ∀i, j ≤ A, m

(a),[β] i,j = m (a) i→j,[β]
The goal is now to adjust β to a, so that m (a), [β] will have 1 as an eigenvalue.

Martingale equation

We recall, after paragraph 7.1.4, that

∀i, j ≤ A, m (a),[β] i,j = m (a) i→j,[β]
= ∞ n=1 Q i n -→j e -an E e βF i n -→j
Let τ be the waiting time for C's first descent below its starting point, so that C(1) = C(τ ). For i, j ≤ A, a ∈ R + , and β ∈ R + , we define the function

gi,j = R + × R + → R + ∪ {∞} (a, β) → E e -β(C(τ )-C(0)) e -aτ 1 τ <∞∧M (τ )=A j |M (0) = Ai
Incidentally, we notice how close this definition comes to the function fC , where the concatenation time τ is now given by the descent time instead of the return time. The computation of gi,j is obtained by disjunction on C's future :

• C's first descent happens at time n ∈ N * , with M (n) = Aj : this happens with probability Q i n -→j , and yields an expectancy of

E e βF i n
-→j e -an |M (0) = Ai = e -an E e βF i n -→j

• C's first descent happens at time n ∈ N * , but with M (n) = A k = Aj : this happens with probability Q i n -→k , and yields an expectancy of 0. • C does not descend, which happens with probability Qi→∞ and yields an expectancy of 0 by construction.

It follows that gi,j(a, β) = ∞ n=1 Q i n -→j e -an E e βF i n -→j = m (a),[β] i,j
Hence, looking for the vector r like above, one gets the equation

E r [M (τ )] e -β(C(τ )-C(0)) e -aτ 1τ<∞|M (0) = Ai = A j=1 m (a),[β] i,j r [j]
so the question is to find β and r such that

∀i ≤ A, r [i] = E r [M (τ )] e -β(C(τ )-C(0)) e -aτ 1τ<∞|M (0) = Ai 96 7.1.

Choice of β

We want to prove that β = α(a) and r = w (a) work for this equation. Thanks to the proposition 2.1, we know that the process X

C is a martingale, so we build a stopping time for it, using parameters • T1, T2 ∈ N * (deterministic times) ; • X ∈ R + a barrier for C ; defining stopping times through • τ0 the descent time of C, as before ; • τ1(X, T1) such that τ1(X, T1) = min ({t > T1; C(t) -C(0) > X}) which is a stopping time ;

• The stopping time that we shall use is τ (X, T1, T2) = min (τ0, τ1(X, T1), T2) so it is bounded by T2.

Thanks to the martingale property for X (a)

C , we have E w (a) [M (τ (X,T 1 ,T 2 ))] e -α(a)(C(τ (X,T 1 ,T 2 ))-C(0)) e -aτ (X,T 1 ,T 2 ) = w (a) M (0)
and this decomposes to

w (a) M (0) = E w (a) [M (τ 0 )] e -α(a)(C(τ 0 )-C(0)) e -aτ 0 1 τ (X,T 1 ,T 2 )=τ 0 + E w (a) [M (τ 1 (X,T 1 ))] e -α(a)(C(τ 1 (X,T 1 ))-C(0)) e -aτ 1 (X,T 1 ) 1 τ (X,T 1 ,T 2 )=τ 1 (X,T 1 )<τ 0 + E w (a) [M (T 2 )] e -α(a)(C(T 2 )-C(0)) e -aT 2 1 τ (X,T 1 ,T 2 )=T 2 <τ 1 (X,T 1 ),τ 0
We deal with these terms separately.

1. When T1 and T2 go to ∞ (no matter X), the monotone convergence theorem indicates that the first term converges to E w

[M (τ 0 )] e -α(a)(C(τ 0 )-C(0)) e -aτ 0 1τ 0 <∞ which is the desired term in the equality to prove ; 2. Recalling that w (a) 's coordinates are bounded through its spread, because it must have a coordinate no larger than 1 thanks to the second equation of scaling, the second term is bounded by e δ(w (a) ) e -α(a)X P (τ (X, T1, T2) = τ1(X, T1) < τ0)

3. Finally, the third term is bounded by e δ(w (a) ) P (τ (X, T1, T2) = T2 < τ1(X, T1), τ0)

Now, we control the second and the third term, looking for values of X, T1, T2 such that they become arbitrarily small. As α(a) > 0 because a > 0, one chooses X such that e -α(a)X is as small as wanted, and for this value of X, C -C(0) has a zero probability of staying eternally in the interval [0, X] (because it is true for its restricted Lévy process once C is not globally constant), so one may choose T2 such that the probability involved in the third term is as small as wanted. Hence, we proved that

E w (a) [M (τ 0 )] e -α(a)(C(τ 0 )-C(0)) e -aτ 0 1τ 0 <∞ = w (a) [M (0)]
So, the values of functions K

Properties of the convolution process

In this paragraph, we take C as a positive recurrent, not globally increasing, bounded C-process. For a ∈ R + , let us define

• Φ (a)
C its convolution process ; • M (a) the underlying Markovian process of Φ C . As a consequence, we want to ensure that C's useful properties (positive recurrence, periodicity) are not broken by this transformation.

Regularity of the convolution processes

We want to state some properties for the regularity of Φ (a) C 's parameters, especially when a goes to 0 so that we will be able to extend the analysis to a = 0. As the dominant eigenvectors are known to be continuous, we only have to verify that the mass matrix is continuous, which is the consequence of α's continuity and

∀a ∈ R + , β ∈ R + , m (a) i→j,[β] = ∞ n=1 Q i n -→j e -na E e βF i n -→j
The random variables F i n -→j are bounded by Q and the probabilities Q i n -→j sum to Qi→j ≤ 1, so the continuity follows by monotonous convergence.

2. When Qi→j > 0, let us recall that the measures defined by

ν (a) i→j = ∞ i=1 Q i n
-→j e -an ν i n Thus, it is a continuous expression of a, which ends the proof. 98

Irrelevance of a to the existence of paths

We mention here that Φ (a)

C has a path of value z ∈ R for any a ∈ R + , then this path exists for all values of a simultaneously.

Lemma 7.5 Simultaneous existence of paths

Let a ∈ R + . We assume that P is a path for Φ (a)

C , and consider for every > 0 the probability p (a) ( ) of following P at precision , as in lemma 5.2. Then for every b ∈ R + ,

• P is a also a path for Φ The probability of Φ (a)

C following P is T t=1 P (a) a t-1 →a t P ∀t ≤ T, T t=1 G (a),t a t-1 →a t ∈ [vt ± ]
The notation G (a),t a t-1 →a t rather than G (a)

a t-1 →a t highlights that these random variables are independent. Now,

• Since the distribution of the partial sums are given through convolution equations and convolution preserves continuity with respect to probabilities, then this probability is continuous of a.

• By construction, P

a t-1 →a t will be non-zero iff the descending process M yields a non-zero transition from Aa t-1 to Aa t ; since P is a path, we get Qa t-1 →a t > 0, so for every b ∈ R + we still have

P (b) a t-1 →a t > 0.
• As all random variables G (b) a t-1 →a t have the same support, it follows that P is still a path for Φ (b) C . This ends the proof.

Closed communicating classes

As M (a) was defined thanks to a time concatenation through C, we may have lost M 's positive recurrence during the concatenation. However, as we shall see here, this is no issue.

Definition 7.3 Descending class

There is A ⊆ [|1, A|] such that, for every a ∈ R + , M (a) has A as only closed communicating class. We name it C's descending class.

To ensure that this definition is correct, we start by proving that a plays no part. By construction the increments G (a) i→j have the same support no matter a, and both zero and non-zero transition probabilities are conserved when a changes by construction of P (a) i→j (like above, when paths were conservated). Now, let i, j ≤ A be state numbers indicating states Ai and Aj, belonging to closed communicating classes A and A of M (a) . As C is positive recurrent and not globally increasing, there is a universal cycle from Ai to itself whose value is v < 0.

• We concatenate this cycle an arbitrarily large number of times, and consider the successive all-time lows hit by this concatenation, at successive times t k ∈ N, whose values are named v (t k ) : the sequence (t k ) k∈N is infinite because v < 0.

• We do the same thing but with analysis starting at the first hitting time of Aj (that exists because the cycle is universal), and get a sequence (u k ) k∈N describing all-time lows named v (u k ).

Once v (t k ) and v (t k ) are both negative, which happens because v < 0, then the times u k and v k will coincide (maybe with a shift in k), because they will describe the same all-time lows. Hence, let t ∈ N that is simultaneously a t k and a u k . As A is closed, then any all-time low v (t k ) hit by a process starting from Ai must hold M (t k ) ∈ A . The same idea for A leads to M (u k ) ∈ A , so M (t) ∈ A ∩ A . However, as closed communicating classes are either disjoint or identical, this is possible only if A = A , which proves that there is at most one closed communicating class.

Periodicity issues

The property of periodicity is not impaired when changing C into Φ C to A is periodic, and its fundamental period is the same p.

1. First, we shall deal with the case where C is aperiodic. We need to recall Φ -→j > 0 and x ∈ supp ν i n -→j . Hence, let us assume that there is a path from Ai to Aj, of length n ∈ N * such that, for some > 0,

• Its final value v is lower than -(then negative) ; • All of its partial values are higher than (then positive).

Then with positive probability, a C-process starting from Ai will follow this path, taking positive partial values higher than /2 and a negative final value lower than -/2, which means that it hits an all-time low at time n : hence, v must be in supp σ i n -→j . Now, let Aa i∈[|0,T |] be a cycle of length T ∈ N and value v < 0. We "turn around" this cycle, as in the definition 5.6, so that Aa T is an all-time low. We rename by Aa i∈[|0,T |] this cycle with z(T ) = v is its (single) all-time low. To prove that v belongs to Φa's cycle support, it suffices to prove that it is the value of a cycle a) for some U ∈ N * . Let us consider Aa i∈[|0,T |] as a cycle for M , whose successive all-time lows are hit at times t k ≤ T , for k ≤ U . We consider the restriction of this path over [|t k-1 , t k |] (with t0 = 0). For the sake of simplicity, we will deem that none of its partial values z k (t) for t ∈ [|t k-1 + 1, t k |] are exactly zero, a case that will be discussed later. Then its value x k holds

A b i∈[|0,U |] of M (
x k ∈ supp σ A t k-1 n -→A t k
thanks to the previous analysis, so x k belongs to -G (a) A t k-1 →A t k 's support. By concatenation, it follows that the sum of such values x k for k ≤ U (which amounts to v) defines a cycle for Φa whose value is v. The case with partial sums (at t ∈ [|t k-1 + 1, t k -1|]) equal to zero is a bit more tedious, as C(t) may or may not be lesser than C(t k-1 ). However, this is no issue, as one may proceed to the following disjunction :

• If P (C(t) -C(t k-1 ) < 0) = 0, then we just "skip" t, as it cannot be an all-time low for paths of C. • If not, then we can consider in the above some > 0 such that P (C(t) -C(t k-1 ) < -) > 0, and only paths for which C(t) -C(t k-1 ) < -, as this does not drive the sought probability to 0 ; then we introduce t as an intermediate time like t k-1/2 between t k-1 and t k .

It follows that any cycle value v < 0 of C is a cycle value for Φ

(a) C , so if Φ (a)
C has a period p, then we get ucs(C) ∩ R * -⊆ -pN * is non-empty because C is positive recurrent, and as ucs(C) + supp(C) ⊆ ucs(C), then supp(C) (and also ucs(C)) must be a subset of pZ ∪ {∞}, so C will be periodic, which is contradictory.

2. Now, we deal with the case where C is periodic, of fundamental period p ∈ R + , thus has a regular process C of period 1. C's convolution process is governed by integer increments, so it must be periodic with an integer period e at least 1. As all-time lows for C are governed only by its "integer" part, then C's descending process C, thus C's convolution process Φ (a)

C , is periodic of period pe , so we want to prove that e is at most 1. In the rest of the proof, we shall note :

• V = supp( C) is C's cycle support ; • U = ucs( C) is C's universal cycle support.
As C has fundamental period 1, we may find coprime integers u k≤q ∈ -V , denoting (negated) cycle values of C for each cycle number k, for some q ∈ N * . After Bézout's identity, there are integers m k≤q ∈ Z such that

q k=1 u k m k = 1
Hence, noting m = 2 + max k (-m k ) ∈ N * for example, then for each k ≤ q, (m + m k )u k is the value of (m + m k ) > 0 concatenations of cycle number k, thus belongs to -V . Now, let us take any universal cycle for C of value x ∈ Z. As U + V ⊆ U , then we concatenate it to the following cycles to obtain new universal cycles :

• Considering m concatenations of each cycle, we get a cycle of negated value

y1 = -x + q k=1 u k m ∈ -U
• Considering m + m k concatenations of cycle number k and m of the others, we get a cycle of negated value

y2 = -x + q k=1 u k (m + m k ) = y1 + 1 ∈ -U
Once again, we rotate the cycles so that their final values y1 and y2 are all-time lows, so belong to Φ

C 's cycle support. We apply lemma 5.8 to -y1 and -y2, which proves that for some X -∈ R, then for every x ≤ X -, [x ± 1/2] ∩ U = ∅. However, as U ⊆ Z, then [x ± 1/2] ∩ U ∩ Z = ∅ : taking integer values for x, this is only possible if for any x ∈ Z and lower than X -we have x ∈ U . This implies that

• There is a universal cycle of any value lower than X -(and this will be used later) ; • U has period at most 1.

So U has 1 as a fundamental period. This ends the proof.

Splitting between periodic and aperiodic cases

In this paragraph, we explain how to use the previous convolution process to prove the proposition 2.4 or the theorem 1. We remember that periodic C-processes were explicitly excluded from the theorem 1 because, as said above, "the references to previous values do not merge", and we are now going to explain this assertion in here.

Explanation

To clarify matters, we assume for now that C is a Lévy process, so ∀a, w (a) = 1. We consider a bounded stopping time τ for C's convolution process Φ (a) C ; the martingale property from lemma 7.2 thus yields

∀a ∈ R + , K (a) (C(0)) = E K (a) (C(τ ))
We seek the local extrema for K (a) ; for example, let us imagine that we have sequences of reals x + n (maxima) and x - n (minima) such that

lim n→∞ K (a) (x + n ) = l + ∧ lim n→∞ K (a) (x - n ) = l -
where l -≤ l + . To prove that l -= l + , our idea is to use Φ

C starting from some x + n , and halt it at a τ featuring with a positive probability some value C(τ ) = x - m (for n, m ∈ N). The line of thought is roughly :

1. l -keeps "pulling" values of K (a) at points x + n to itself through a convex combination (given by the martingale property for Y (a) ) involving this positive coefficient ; 2. As a consequence, the discrepancy between K (a) x - m and K (a) x + n must vanish over time ; 3. So, as K (a) is bounded, it must converge. For now, we are not interested in the value of this limit l -= l + , which will be the subject of a further part of the study.

Unfortunately, this approach fails when C(τ ) cannot hit any x - m . Indeed, the lemma 5.9 ensures that remote density allows "closing to" remote enough values x - m only if C is aperiodic, so the main theorem 1 will not hold for periodic C-processes : one gets the proposition 2.4 instead. The behaviour of a C-process may be described by either one of incoming lemmata, depending on whether C is periodic or not.

• When C is periodic, we recall that it suffices to prove the lemma 6.13 to get the theorem 2.4. To get it, we target a weaker form given below, the lemma 7.10.

• When it is not, we directly tackle the theorem 1, using the values of Φ (a)

C hit by τ (a) , chosen to be "close" to the values x - m . We target the weaker lemma 7.11. Before splitting the study, we start by creating τ .

Stopping time

As explained above, we aim at finding a suitable stopping time τ , allowing to express the value

Y (a) (0) = K (a) M (a) (0) Φ (a) C (0) = K (a) M (0) (C(0))
as a convex combination of well-chosen values K C be C's convolution process at some point a ∈ R * + , and x ∈ R + . We set the random process σ (a) as, for every t ∈ N,

• If Φ (a) C (t) ≥ x, then σ (a) (t) = 1 ; • If Φ (a) C (t) ∈ [x -Q, x), then σ (a) (t) = 1/2 ; • Else σ (a) (t) = 0.
We define random variables Z(t) to be i.i.d. of uniform distribution over (0, 1) and independent from anything else, and set ρ (a) to be the binary sequence defined by

∀t ∈ N, ρ (a) (t) = 1 Z(t)>σ (a) (t)
Finally, we set τ (a) to be the stopping time defined by

τ (a) = min t ∈ N; ρ (a) (t) = 1
We call τ (a) the x-halting time of Φ (a)

C . Before studying the convolution process itself through the martingale property, we ensure that τ (a) is eligible to this martingale property. As the functions K Let C be a positive recurrent, bounded, not globally increasing C-process, a ∈ R + , and x ∈ R + . Let τ (a) be the x-halting time of C's convolution process at point a :

P τ (a) < ∞ = 1
Let us set a ∈ R + , and for every η > 0 the value p(η) = max We know that p is a non-increasing function of η and that lim η→0 (p(η)) = 0 so there is η > 0 such that p(η) < 1/2 (so η ≤ Q), and we keep this value of η in the sequel of the proof. By construction, if Φ (a)

C goes below x -Q at some time t ∈ N, then τ (a) ≤ t < ∞, so one only needs to prove that Φ (a)

C will eventually lose C(0) + Q -x. Let us note

n = 1 + C(0) + Q -x η If Φ (a)
C loses at least η at least n times, then Φ

C lost more than C(0) + Q -x and thus stops τ (a) . However, we know at at each step t ∈ N, no matter the present situation Φ M (t)→M (t+1) > η will happen infinitely often almost surely, so this ends the proof.

Case of periodic C-processes

In this paragraph, we assume that C is periodic, and look at the hypotheses to the lemma 6.13 : hence, we may deem that C lies in Z and has 1 as a fundamental period. Later on, this condition will be called "C is regular". Hence, rather than controlling K (a)

i 's values over whole x-intervals, we shall only look at their values over intervals like (when x ∈ N)

[x -Q, x) ∩ Z = [|x -Q, x -1|]
because the convolution equation will only involve negative integer increments of Φ be the functions defined by the lemma 7.1. We define their local discrete extrema as follows.

• The local discrete maxima of positive recurrent functions K (a) i over the x-interval is a function of x, given by

K (a) + = N → R x → max y∈[|x-Q,x-1|],i∈A K (a) i (x)
• The local discrete maxima of all functions K (a) i over the x-interval is a function of x given by

K (a) ++ = N → R x → max y∈[|x-Q,x-1|],i≤A K (a) i (x)
• The local discrete minima of positive recurrent, and all functions K (a) i over the x-interval, are defined likewise and named repectively K To simplify the incoming work, we want to prove that they are monotone.

Lemma 7.8 Monotonicity of local extrema

The functions

K (a) + , K (a)
++ , K

-, K

--given in the definition 7.5 hold the following properties. because the coefficients form a convex combination. This being for every i ∈ A , we have that

• All values K (a) j (x -d) for j ∈ A , d ∈ [|1, Q -1|] are involved in the maximum K (a) + (x) ; • The values K (a) j (x) for j ∈ A are no higher than K (a) + (x). It follows that K (a) + (x + 1) = max max j∈A ,d∈[|1,Q-1|] K (a) j (x -d) , max j∈A K (a) j (x) ≤ K (a) + (x)
The case of K (a)

++ being similar (removing the assumption on A ) and the cases of K (a)

-and K (a)

--being symmetrical, this ends the proof.

Case of aperiodic C-processes

When C is aperiodic, we need to control K

K (a) + = R + → R x → sup y∈[x-Q,x),i∈A K (a) i (x)
• The local maxima of all functions K (a) i over the x-interval as a function of x given by

K (a) ++ = R + → R x → sup y∈[x-Q,x),i≤A K (a) i (x)
• The local minima of positive recurrent, and all functions K (a) i over the x-interval, likewise and named repectively K

++ , K

-, K

--given in the definition 7.6 hold the following properties. 1. They are sorted by ascending order as in

∀x ∈ R + , K (a) --(x) ≤ K (a) -(x) ≤ K (a) + ≤ K
--. This proof uses the η and p(η) from the proof of the lemma 7.7. Sorting these functions in ascending order comes from their definitions and A ⊆ [|1, A|], so we shall focus on the monotonicity property ; since the proofs are almost similar, we shall only present the case of K (a) + . We want to prove by induction on n ∈ N that for every n ∈ N,

sup i≤A,z∈[x-η,x+nη) (fi(z)) ≤ K (a) + (x)
For n = 0, this is true because [x -η, x) ⊆ [x -Q, x) ; so we focus on the transmission of the property. Now, let us assume by contradiction that for some

x ∈ R + , ∃i0 ∈ A , z0 ∈ [x + nη, x + (n + 1)η); (fi 0 (z0)) -K (a) + (x) = y0 > 0
We write fi 0 (z0) as given by the convolution equation. Since p(η) < 1/2, then there must be i1 ∈ A and z1 ∈ [x + nη, z0) such that (fi 1 (z1)) -K

• xu ∈ [x + nη, x + (n + 1)η) ; • iu ∈ A (in ≤ A for K (a) ++ ) ; • yu ∈ R * + , such that (fi u (zu)) -s(x) = yu ≥ 2 u y0
but this is impossible if y0 > 0 since all points xn belong to the finite interval [x + nη, x + (n + 1)η) and the functions fi are locally bounded. As the cases of K (a)

-and K (a)

--are symmetrical, we proved the monotonicity property, which in turn implies that the functions converge thanks to their natural inequalities.

Separate lemmata

We present the goal of the next step : intermediate lemmata, indicating that the twisted Laplace transforms converge (disregarding their limits). The rest of the work is now to : 1. Prove them ; 2. Find the value of the limits ; 3. Get back to the sought statements : proposition 2.4 and theorem 1, using them.

When C is periodic, we aim at proving this lemma : Lemma 7.10 Weak theorem for periodic C-processes Let C be a C-process such that

• C is positive recurrent, bounded and not globally increasing ;

• For every t ∈ N, C(t) ∈ Z almost surely ;

• C's fundamental period is 1.

For every a ∈ R + , we set • C's martingale parameter at point a as α(a) ;

• C's dominant eigenvector at point a as w (a) .

We define the functions K

such that ∀i ≤ A, x ∈ R + , a ∈ R + , K (a)
i (x) ∈ [K∞(a) ± e(a, x)] In both cases, by definition of K (a) i , the remaining work will be to focus on the function K∞ to end the proofs.

Proof of the weak theorem for periodic C-processes

We want to prove the lemma 7.10. To do this, we will set some x ∈ R + defining the x-halting time τ (a) , and follow these steps :

1. Find a suitable lower bound p (a) to the probabilities of τ (a) halting Φ 4. Find out that the parameters controlling the latter discrepancy may be chosen as continuous functions of a, so they lead to the lemma 7.10.

Lower bound

As Φ (a)

C stays in Z, we only need to prove that for some x ∈ N, the x-halting time τ (a) may halt Φ C be C's convolution process at any point a ∈ R * + , whose starting point is C0 ∈ N. We name A its descending class.

1. There is a minimal x0 ∈ N * such that, for every a, the (C0 -x0)-halting time τ (a) of Φ (a)

C holds ∀y ∈ [|C0 -x0 -Q, C0 -x0 -1|] , i ∈ A , P Φ (a) C τ (a) = y ∧ M (a) τ (a) = Ai > 0
We name :

• This minimal x0 is Φ (a)
C 's ergodic pace, noted e(C) ; • Any value h (a) (C) no higher than both 1/2 and all these probabilities is a reducing constant of

Φ (a) C . 2. When a varies into R + ,
• e(C) does not change ;

• We may choose reducing constants so that h (a) (C) varies continuously.

To prove the lemma 7.12, we take Φ C has a positive probability of following this path and then being stopped by τ (a) at the end. The lemma 7.12 will follow, when considering how e(C) and h (a) (C) are built.

1. We begin by building a path from As to Ai.

(a) First, as A is the only closed communicating class of Φ (a) C , if s / ∈ A , there is a way to A (as else [|1, A|] \A would be closed, thus would have a closed communicating sub-class). We get (through the lemma 5.3) a path P1 of minimal length from As to some state Ar with r ∈ A , of finite value v1 ∈ -N.

• This path P1 works for all values of a ∈ R * + simultaneously thanks to the lemma 7.5.

• As Φ (a)

C 's values are integers, following this path at precision e.g. = 1/3 in the terms of lemma 5.2 means following it exactly, so P1 has a positive probability ps(a) > 0, that is continuous of a ∈ R + (and depends on As).

• As P1's length is minimal, it is at most A ; since its payoffs are bounded by Q, then

-v1 ≤ AQ. (b) Since A is a closed communicating class of Φ (a)
C , there is a path P2 of minimal length from Ar to Ai. For the same reasons, its value is noted v2 ∈ [| -AQ, 0|], and its probability is noted qi(a) > 0.

(c) We also know after lemma 5.9 that Φ (a)

C has universal (in A ) cycles of any low enough integer value. In particular, there is a universal cycle starting from Ai of every integer value k ∈ [| -X -Q, -X + 2AQ -1|] for some X ∈ N. For each k, we note by r k (a) > 0 the probability of such a cycle.

We concatenate the paths with every cycle. It follows that there are paths from As to Ai for every value in

[| -X -Q + v1 + v2, -X + 2AQ -1 + v1 + v2|] As v1 + v2 ∈ [| -2AQ, 0|], there are paths from As to Ai of every value in 2AQ v=0 [| -X -Q -v, -X + 2AQ -1 -v|] = [| -X -Q, -X -1|]
whose probabilities are at least min

(s≤A,i∈A ,k∈[|-X-Q,-X+2AQ-1|])
(ps(a)qi(a)r k (a)) = q(a) > 0 So, we proved that there are q(a) > 0 and X ∈ N such that for every s

≤ A, i ∈ A and v ∈ [| -X -Q, -X -1|],
there is a path for from As to Ai whose value is v and probability at least q(a).

2. We shall prove that x0 = X works. Let us take

• A "target" point y ∈ [|C0 -X -Q, C0 -X -1|] • A "target" state A i∈A .
We start the convolution process Φ

C from C0 ∈ N and A s≤A : as proved above, it has a probability at least q(a) of following the path leading to the target point y and the target state Ai. If it follows this path, then the successive values ρ(t) amount to

• 1 as long as Φ (a) C remains at least C0 -X ; • 1/2 when Φ (a)
C crosses C0 -X -1, staying there until the end of the path so for at most Q time periods, since it decreases by at least 1 per step and becomes 0 when Φ (a)

C (t) goes below C0 -X -Q.
Hence, by construction of τ , the probability of τ stopping precisely at the end of the path is at least 1/2 Q no matter the path taken. Since the path probability is at least q(a), then ∀y

∈ [|C0 -X -Q, C0 -X -1|] , i ∈ A , P (B(y, i)) ≥ q(a) 2 Q
3. We move on to the properties of e(C) and h (a) (C) when a changes.

• We remark that, no matter a, we have

∀i ∈ A , y ∈ [|C0 -x0 -Q, C0 -x0 -1|] , P (B(y, i)) > 0 iff there is a path for Φ (a)
C from As to Ai whose value is C0 -y. Since this property does not depend on a thanks to the lemma 7.5, then the minimal x0 does not either.

• As e(C) is fixed, we investigate on q(a) : as the probabilities of paths are non-zero and continuous of a ∈ R + thanks to the lemma 7.5, then by construction q is continuous, which implies that we may choose

h (a) (C) = min 1 2 , q(a) 2 Q
that is continuous of a.

This ends the proof.

Use of anterior values

We aim at proving that all (for i ∈ A ) twisted Laplace transforms K (a) i converge exponentially, to a common limit named l(a). Let us look at the value taken by Φ (a)

C at its (z -e(C))-halting time τ (a) as given by the lemma 7.12, when starting from z = Φ is a martingale, the martingale property leads to

K (a) i (z) = Y (a) C (0) = E Y (a) C τ (a) = x∈Z A j=1 P Φ (a) C τ (a) = x ∧ M (a) τ (a) = Aj K (a) j (x)
However, as A is closed, i ∈ A , and by construction of τ (a) , the sum restricts to

K (a) i (z) = z-e(C)-1 x=z-e(C)-2Q j∈A P Φ (a) C τ (a) = x ∧ M (a) τ (a) = Aj K (a) j (x)
We know that

• All terms K (a)
j (x) in this expression are bounded from above by -K

(a) + (z -e(C)) if they belong to [|z -e(C) -Q, z -e(C) -1|] ; -K (a) + (z -e(C) -Q) if they belong to [|z -e(C) -2Q, z -e(C) -Q -1|]. As J (a) + is non-increasing, we select K (a) + (z -e(C) -Q). • As K (a)
-is non-decreasing, by definition of the minimum we know that there are

j ∈ A , x ∈ [|z -e(C) -Q, z -e(C) -1|] such that (noting l-(a) the limit of K (a)
-as in the definition 7.5)

K (a) j (x) = K (a) -(z -e(C)) ≤ l-(a)
Moreover, we know thanks to the lemma 7.12 that its "weight" given by P (a) i→j is at least h (a) (C). As a consequence,

K (a) i (z) ≤ 1 -h (a) (C) K (a) + (z -e(C) -Q) + h (a) (C)l-(a) Doing this for every i ∈ A and z ∈ [|C0, C0 + Q -1|] yields K (a) i (z) ≤ 1 -h (a) (C) max z∈[|C 0 ,C 0 +Q-1|] K (a) + (z -e(C) -Q) + h (a) (C)l-(a)
However, the maximum appearing here is K

(a) + (C0 -e(C) -Q) because K (a) +
is non-increasing. This being for every i and z, we get

∀C0 ∈ N, K (a) + (C0) ≤ 1 -h (a) (C) K (a) + (C0 -e(C) -Q) + h (a) (C)l-(a)
In particular, the sequence u defined by

u = N → R n → K (a) + ((e(C) + Q) n) -l-(a) holds the geometric property ∀n ∈ N, u(n) ≤ u(0) 1 -h (a) (C) n As K (a)
+ is non-increasing, it follows that ∀x ∈ N, K

+ (x) ≤ K (a) + (0) 1 -h (a) (C) x e(C)+Q + l-(a) (a)
Finally, we simplify this expression :

• K e -α(a)

• h (a) (C) was deemed no higher than 1/2, so we may replace the integer part of x/(e(C) + Q) by

x/(e(C) + Q) -1 and multiply by 2.

Hence, setting

β (a) 1 = -ln 1 -h (a) (C) e(C) + Q ∧ Z (a) 1 = max i∈A 2 w (a) [i] e -α(a)
then by definition of J

Extension to outside the descending class

When the starting state Ai does not belong to A , we use a similar trick, involving a reference to a value in A . Indeed, when waiting for A time periods, M (a) must contain a path from Ai to some state Aj with j ∈ A (as else Ai would belong to a closed communicating class, however the lemma 7.3 forbids it). The proof for all states is quite similar to the one for states in A , so we shall only explain its main steps. When waiting for A time periods, we note by g(a) a (continuous) lower bound for the probability of going from any state of M (a) to a state in A ; hence, we have

∀x ∈ N, K (a)
++ (x + AQ) ≤ (1 -g(a)) K (a) ++ (x) + g(a)K (a) + (x)
Solving this inequation through the auxiliary sequence defined by

v (a) = N → R n → 1 1-g(a) n K (a) ++ (nAQ) -l-(a)
yields eventually, thanks to the previous work,

∀n ∈ N, v (a) (n + 1) -v (a) (n) = Z (a) 1 1 -g(a) e -n β (a)
1 AQ+ln(1-g(a))

Taking g(a) to be positive and low enough so that The symmetrical inequality (for K (a)

∀a ∈ R + , β (a) 1 AQ + ln (1 -g(a)) = β (a) 2 > 0 then one gets ∀n ∈ N, v (a) (n) ≤ v (a) (0) + Z (a) 1 1 -g(a) 1 1 -e -β (a) 2 = Z2(a) So, noting ∀a ∈ R + , β (
--) holds a similar way, eventually leading to functions Z4 and β4 such that ∀i ≤ A, x ∈ N, K we have the solutions to the lemma 7.10, which ends the proof.

Proof for aperiodic C-processes

When C is aperiodic, one cannot force τ (a) to stop Φ

C precisely on a given point x ∈ R + instead of N, e.g. when Φ (a)

C 's increments behave as a continuous distribution. However, the density alternative states that we may stop Φ (a)

C arbitrarily close to x, so our idea is to 1. Control the functions K (a) i locally around any x ∈ R + , to ensure that their values around x may be viewed through K 5. Rewrite the convex combination, finally yielding an arithmetico-geometric recursion scheme, whose "constant" part is driven by . As may be chosen arbitrarily small, we shall get the result for i ∈ A ; 6. Extend to every i ≤ A a similar way as before.

Local control

The functions K (a) i are differentiable over [ -Q, 0) by definition. As a consequence, there is a constant Ξ

∈ R * + such that ∀x < y ∈ R * -, K (a) i (y) -K (a) i (x) y -x ≤ Ξ
We want to prove that this property extends to R + , so that any value K

Ξ = R + → R + a → α(a) min i≤A w (a) [i]
Then Ξ is a continuous function satisfying

∀i ≤ A, ∀x < y ∈ R + , K (a) i (y) -K (a) i (x) y -x ≤ Ξ(a)
We call it the half-Lipschitz function of C.

To prove this lemma, we take x < y ∈ [ -Q, ∞), writing y = x + z with z > 0, and solve the property depending on whether x, y are negative or not. For every i ≤ A, we write

ξi(x, z) = K (a) i (x + z) -K (a)
i (x) z so we want ∀i ≤ A, ∀x ∈ R + , ∀z > 0, ξi(x, z) ≤ Ξ(a).

• When x + z < 0, the statement is given by the mean value inequality, implying that

K (a) i (x + z) -K (a) i (x) z ≤ sup u∈R * - dK (a)
i (u) du

The definition of Ξ comes from the limit value of this derivative at point u = 0.

• When x < 0 and x + z ≥ 0, we know thanks to the definition of Ξ that

∀i ≤ A, u ∈ R * -, K (a) i (u) ≤ K (a) i (x) -xΞ(a)
However, as K

++ is non-increasing, this implies that the same is true for every z ∈ R + , so

ξi(x, z) ≤ K (a) i (x) -xΞ -K (a) i (x) z = Ξ -x z ≤ Ξ(a)
because 0 < -x ≤ z, which solves this case.

• When x ≥ 0, we consider Φ

C starting from the state Ai and the point x, and its random hitting time τ of R * -(it is eligible to the martingale property, thanks to the lemma 7.7). The martingale property thus yields which ends the proof.

K (a) i (x + z) -K (a) i (x) = A j=1 P M (a) (τ ) = Aj E K (a) j z + Φ

Density over intervals

Let η > 0. We set m = Q/η and split any x-interval on sub-intervals of length at most η, as in

[x -Q, x) ⊆ m k=1 [x -kη, x -(k -1)η)
We want to prove, thanks to the density alternative, that τ (a) may halt Φ

C on every such interval. The following lemma "looks like" the lemma 7.12, changing only when considering that Φ (a)

C hits an interval rather than a point : as a consequence, its length η will be involved in the result. C be C's convolution process at any point a ∈ R + , whose starting point is C0 ∈ N. We name A its descending class. Let η > 0 be called a margin, supposed to be an integer fraction of Q, i.e. and m = Q/η ∈ N * . 1. There is some x0(η) ∈ R * + such that, for every a, the (C0 -x0(η))-halting time τ (a) of Φ (a)

C holds, for every ∀k ∈ [|1, m|] and i ∈ A ,

P Φ (a) C τ (a) ∈ [C0 -x0(η) -kη, C0 -x0(η) -(k -1)η) ∧ M (a) τ (a) = Ai > 0
We name :

• Any value x0(η) is an ergodic pace of Φ (a)
C at margin η, noted e(C, η) ; • Any value h (a) (C, η) no higher than both 1/2 and all these probabilities is a reducing constant of Φ

C at margin η. 2. When a varies into R + and η is fixed,

• We may choose e(C, η) so that it does not change ;

• We may choose reducing constants so that h (a) (C, η) varies continuously.

The proof of this lemma follows the same steps as for the lemma 7.12.

1. We begin by building a path from any starting state As to Ai with i ∈ A . ∈ A , there is a way to A . We get (through the lemma 5.3) a path P1 of minimal length from As to some state Ar with r ∈ A , of finite value v1 ∈ -N.

• This path P1 works for all values of a ∈ R * + simultaneously thanks to the lemma 7.5. • We take η/9 as a precision for following this path at precision in the terms of lemma 5.2.

P1 has a positive probability to be followed at this precision, noted ps(a, η) > 0, that is continuous of a ∈ R + thanks to the lemma 7.5 (and depends on As). • As P1's length is minimal, it is at most A ; since its payoffs are bounded by

Q, then v1 ∈ [-AQ, 0]. (b) Since A is a closed communicating class of Φ (a)
C , there is a path P2 of minimal length from Ar to Ai. For the same reasons, its value is noted v2 ∈ [-AQ, 0], and its probability to be followed at precision η/9 is noted qi(a, η) > 0. (c) We also know after lemma 5.9 that setting = η/9 and the associated X -, then for every y ≤ X -, Φ

We know that v1

+ v2 ∈ [-2AQ, 0] ; hence, taking any x ∈ X --(2A + 1)Q, X --2AQ , we know that y = -x + X -+ v1 + v2 ∈ 0, (2A + 1)Q ⊆ [0, b]
The closest integer k to y is thus at a distance at most 1/2 of y itself and belongs to

[|0, b|]. Hence, there is a k ∈ [|0, b|] such that X --kη ∈ x -v1 -v2 ± 2
We concatenate the paths P1 and P2 with the cycle Q k for this k, whose value belongs to

X --kη ± η 18 113
This leads to a path from As to Ai whose value is in

X --kη + v1 + v2 ± η 18 ⊆ x ± η 9
Hence, we proved that for every x ∈ X --(2A + 1)Q, X --2AQ , there is a path from As to Ai whose value is at most η/9 apart from x. Besides, the probability of following it at a precision η/9 + η/9 + η/9 (each η/9 comes respectively from P1, P2 and

Q k ) is at least min (s≤A,i∈A ,k∈[|0,b|])
(ps(a, η)qi(a, η)r k (a, η)) = q(a, η) > 0 2. We shall prove that x0(η) = X -works. Let us take

• A "target" interval, for k ≤ m, [C0 -X --kη, C0 -X --(k -1)η) = [y k ± η/2)
described by its central point

y k = C0 -X --(k - 1 2
)η

• A "target" state A i∈A .

We shall note, for k ≤ m and i ∈ A ,

B(k, i) = P Φ (a) C τ (a) ∈ [y k ± η/2) ∧ M (a) τ (a) = Ai
We start the convolution process Φ (a) C from C0 ∈ R + and A s≤A : as proved above, it has a probability at least q(a, η) of following the path leading to

• At most η/9 apart of the target point y,

• And the target state Ai, at precision η/3. If it follows this path at this precision, then the successive values ρ(t) amount to

• 1 as long as Φ (a) C remains at least C0 -X -; • 1/2 when Φ (a) C crosses C0 -X -; • 0 if Φ (a) C crosses C0 -Q -X -.
However, the latter case is impossible, because Φ (a) C τ (a) is not further than η/9 + η/3 apart (approximation of y + precision) from y k , and

y k = C0 -X --(k - 1 2 )η ≥ C0 -X --Q + η 2
thanks to m = Q/η ∈ N, and η/2 > 4η/9. Now, the given paths are in a finite number, so we note by N (η) ∈ N the maximum of their lengths : it follows that the probability of τ stopping precisely at the end of the path is at least 1/2 N (η) no matter the path taken. Since the path probability is at least q(a, η), then

∀k ∈ [|1, m|] , i ∈ A , P (B(k, i)) ≥ q(a, η) 2 N (η)
which ends this proof.

3. We move on to the properties of e(C, η) and h (a) (C, η) when a changes.

• We remark that, no matter a, we have ∀i ∈ A , k ∈ [|1, m|] , P (B(k, i)) > 0 as soon as there is a path for Φ (a)

C from As to Ai whose value is at most η/9 apart from C0 -y k . Since this property does not depend on a thanks to the lemma 7.5, then there is an x0 that does not either.

• As e(C, η) is fixed, we investigate on q(a, η) : as the probabilities of paths are non-zero and continuous of a ∈ R + thanks to the lemma 7.5, then by construction q is continuous, which implies that we may choose

h (a) (C, η) = min 1 2 , q(a, η) 2 N (η)
that is continuous of a.

This ends the proof.

Use of the local control

Let us take x0 ∈ R + , yielding a value y0 = K

≤ m, j ∈ A , ∃c k,j ≥ h (a) (C, η); K (a) i 2 (x2) = j∈A m k=1 c k,j E K (a) j Φ (a) C τ (a) |B(k, j) + P   ¬ j∈A m k=1 B(k, j)   E   K (a) j Φ (a) C τ (a) |¬
τ (a) ≥ x1 -Q When B(k0, j0) does not happen, then ∀j ∈ A , K (a) j 0 Φ (a) C τ (a) ≤ K (a) + (x1 -Q) because K (a)
+ is non-increasing (provided that x1 ≥ Q). As a consequence, we may bound K (a)

i 2 (x2) from above, by

K (a) i 2 (x2) ≤ (1 -P (B(k0, j0))) K (a) + (x1 -Q) + P (B(k0, j0)) (y0 + + 2ηΞ(a)) As K (a) i 2 (x2) ≤ K (a)
+ (x1 -Q), we have the alternative :

• If y0 + + 2ηΞ(a) is not higher than K (a)
+ (x1 -Q), then we may replace P (B(k0, j0)) by its lower bound h (a) (C, η), and get

K (a) i 2 (x2) ≤ 1 -h (a) (C, η) K (a) + (x1 -Q) + h (a) (C, η) (y0 + + 2ηΞ(a))
• If it is, the latter inequality still holds because

K (a) i 2 (x2) ≤ K (a) + (x1 -Q) by definition of K (a)
+ . So, we proved that for every i2 ∈ A and x2 ∈

[x1 + η + e(C, η), x1 + η + e(C, η) + Q), K (a) i 2 (x2) ≤ 1 -h (a) (C, η) K (a) + (x1 -Q) + h (a) (C, η) (y0 + + 2ηΞ(a))
which leads by definition of K (a) + and y0 to

K (a) + (x1 + η + e(C, η) + Q) ≤ 1 -h (a) (C, η) K (a) + (x1 -Q) + h (a) (C, η) K (a) -(x0) + + 2ηΞ(a)
Finally, as

• x1 ∈ [x0 -Q, x0) by construction ; • K (a) + is non-increasing ; • K (a)
-is non-decreasing and converges to l (a) -, then we get, provided that x1 ≥ Q, so whenever x0 ≥ 2Q,

K (a) + (x0 + η + e(C, η) + Q) ≤ 1 -h (a) (C, η) K (a) + (x0 -2Q) + h (a) (C, η)l (a) -+ h (a) (C, η) ( + 2ηΞ(a))
We shall use this inequation to get an arithmetico-geometric inequality for K (a) + .

Arithmetico-geometric convergence

Let us take > 0 and η an integer fraction of Q. We define the sequence

u ( ,η,a) = N → R n → K (a) + (2Q + (η + e(C, η) + 3Q) n)
The previous inequality leads to the arithmetico-geometric inequality

∀n ∈ N, u ( ,η,a) (n + 1) ≤ 1 -h (a) (C, η) u ( ,η,a) (n) + h (a) (C, η)l (a) -+ h (a) (C, η) ( + 2ηΞ(a))
Noting by Z1( , η, a) = u ( ,η,a) (0), we eventually get + , the common limit being K∞(a). So, we proved the desired property for the states of A .

∀n ∈ N, u ( ,η,a) (n) ≤ l (a) -+ + 2ηΞ(a) + Z1( , η, a) 1 -h (a) (C, η) n Since K (a) + is non-increasing, ∀x ≥ 2Q, K (a) + (x) ≤ l (a) -+ + 2ηΞ(a) + Z1( , η, a) 1 -h (a) (C, η) x-2Q η+e(C,η)+3Q

Extension to outside the descending class

When considering i / ∈ A , the same idea as before still works : when waiting for A time periods, we have g(a) as a lower bound for the probability of hitting A . Let us take a ∈ R + and > 0 : we know that, thanks to the previous work, there is x0 ∈ R + such that ∀b ≤ a, ∀x ≥ x0, K

(x + n ( , a) AQ) = P M (b) (n ( , a) A) ∈ A E K (b) i Φ (a) C (n ( , a) A) |M (b) (n ( , a) A) ∈ A + P M (b) (n ( , a) A) / ∈ A E K (b) i Φ (a) C (n ( , a) A) |M (b) (n ( , a) A) / ∈ A
We know that after n ( , a) A time periods, Φ

End of the main proofs

Now that the lemmata 7.12 and 7.14 are stated, we shall use them to find the term K∞(a), that will lead to

• The lemma 6.13 for periodic C-processes ;

• The theorem 1 for aperiodic ones.

We shall also explain why the convergence may be arbitrarily slow (proposition 2.6).

Affine vector equation

Now that we are ensured of K (a) i 's convergence to K∞(a), our idea is to 1. Apply the Laplace transformation to the convolution equation governing K We notice that, as the convergence is to be taken in a different sense when C is periodic, we shall use the discrete Laplace transform (linked with the Z-transform) instead of the usual Laplace transform in this case.

Regular C-processes

When C is regular, we shall use the discrete Laplace transform.

Definition 8.1 Discrete Laplace transform

Let u : N → R + be a non-negative sequence. Its discrete Laplace transform (DLT) is defined, whenever possible, by

û = R → R + ∪ {∞} w → ∞ n=0 u(n)e -nw
The DLT obeys the usual properties. As these properties are similar to those of the usual Laplace transform and come from computations, we shall admit them, so we can proceed with the study. Let us take the convolution equation from lemma 7.1 : we view the functions K (a) i as sequences, and apply the DLT on them. We get that, for every i ≤ A, ∀w ∈ R, K We notice that, when w = α(a), the rightmost fraction is continuously prolongated to de -α(a)d , so that this point is not an issue. In particular, V V (a) (w) whenever 1 is not an eigenvalue of L Φ (a) C (-w), which is automatic if w > 0.

Aperiodic C-processes

When C is aperiodic, we use the usual Laplace transform, that obeys the usual properties. Once again, we admit these properties, so we proceed immediately with the study. Let us take the convolution equation from lemma 7.1 again, and apply the Laplace transform on it. We get that, for every i ≤ A, ∀w ∈ R, L K V (a) (w) (when C is regular, it is K (a) that plays the part of L K (a) ).

Preliminary formula

A scheme that we will encounter in this paragraph is the computation of lim w→0 + (Id -L(-w)) -1 w when L is a differentiable matrix function R → MA R + , in particular when 1 is an eigenvalue for L(0). We hereby evaluate this term.

Lemma 8.3 Computation of the local inverse

Let C be a bounded, positive recurrent, not globally increasing C-process, and L be the Laplace matrix function of its convolution process (at any point). We deem that 1 is the dominant eigenvalue of L(0), whose characteristic eigenspace is 1-dimensional, spanned by a column vector c ∈ (R * + ) A or a row vector r ∈ (R * + ) A , with rc = 1. Then lim w→0 + (Id -L(-w)) -1 w = cr -r dL(w) dw (0)c

To prove this lemma, we first verify that L(0) has a single dominant eigenvalue : thanks to Perron-Frobenius' theorem, it suffices to verify that it has a single closed communicating class, which is true thanks to the lemma 7.3. We use Jordan's reduction of the matrix L(0), considering P a change-of-basis matrix of L(0) to ∆ through the equation

L(0) = P ∆P -1
where • ∆ is an upper triangular matrix, with ∆1,1 = 1 and the rest of the first row is zero ;

• P 's first column is c (which means that P -1 's first row is r).

Now, let us note by ∀w ∈ R * + , f (w) = (Id -L(-w)) -1 w and rewrite it as f (w) = Id -P ∆P -1 + L(0) -L(-w) -1 w

As L is continuously differentiable, we get a continuous function H such that f (w) = P (Id -∆) P -1 -wH(w) -1 w that is also f (w) = P Id -∆ w -P -1 H(w)P -1

P -1
In particular, -H(0) is the local derivative of L at point 0, which means by construction that

H(0) = RΦ C (0)
As ΦC is decreasing almost surely by construction, H(0)'s entries are non-positive. Now, we look for the matrix P -1 f (w)P . We note by

• T the upper triangular matrix of dimension A -1, being the restriction of Id -∆ to its rows and columns 2 to A (we exclude the dominant eigenspace) ;

• B(w), C(w), D(w), E(w) the block decomposition of F (w) = -P -1 H(w)P relatively to the first or the other dimensions, written in the natural order (so that the block C(w) is 1 × (A -1)).

We are going to find matrices Mn(w), for n ∈ N, decomposed in blocks Un(w), Vn(w), Wn(w), Xn(w) the similar way, such that Id -∆ w + F (w)

∞ n=0
Mn(w)w n = Id where this sum converges to a matrix M (w) for w small enough.

• At order -1 in w, we find the equations T W0(w) = 0 and T X0(w) = 0, that translate to W0(w) = 0 and X0(w) = 0 because T is triangular and its diagonal terms are not zero, since 1 was a single eigenvalue of L(0), thus invertible.

• At order 0, we get four equations (one per block), yielding The similar way, we get successive values for Un(w), Vn(w), then for Wn+1(w), Xn+1(w) recursively.

As B(w) is locally bounded away from 0, the values we find follow an arithmetico-geometric recursion scheme, and their growth rates are bounded by a geometric parameter (λ(w)) n . However, as all terms are continuous of w, λ(w) is a continuous expression of w, thus we may take w ≤ w0 such that • B does not hit 0 over [0, w0] ;

• λ(w) is then bounded by some λ + over [0, w0] [i] all converge to 1, as well as the functions L (a)

[i] . In this special case, we proved that if α(a) = 0 (thus a = 0), then the default probability expressed by the functions L (a)

[i] converges to 1 when C(0) increases ; as it is non-increasing of C(0), this is possible only if default is almost certain as soon as α(0) = 0. Recalling that this happens iff E(C) ≤ 0, we recover a usual property of Lévy processes : a (not globally constant) down-drifted C-process eventually defaults almost surely.

General case

We assume now that α(a) = 0. We shall only look at the aperiodic case, as the regular case is similar when changing

• The integrals over d to sums ;

• The denominator α(a) for e α(a)-1 .

We know that V is continuous at point 0, so for every i ≤ A, V (a) (0) To end the proof of lemma 6.13 and theorem 1, we will prove that K (a)

∞ holds the suitable properties. We shall only consider the case α(a) = 0, as α(a) = 0 is already solved. which yields ∀a ∈ R + , -ΛT 0 (a) = α(a)C(0) + α(a) The additive term α(a) of the affine equation is related with, as said earlier in the study, the "way" C defaults. As it may only default on -1 no matter the starting point, we recover that this term is K(a) = ln e -α(a)(-1) = α(a) which is the "minimum" term required by the lemma 6.13. However, we remark that K∞(a) need not be lower than e -α(a) in general when C is a regular C-process. Taking C whose Laplace matrix function is given by ∀w ∈ R, LC (w) = 0 This is because of the vector w (a) , that makes the "way to default" E w possibly greater than e -α(a) when w (a) 's spread is too high.

Bad approximation

The same idea as for periodic C-processes applies now.

1. Let us take two starting points C1, C2 in the range C0, C0 + 10 -f (k)-1

for some k ∈ N * and C0 that suits the requirement of the lemma 8.4. We know that the Laplace transform of C's default time may be expressed as ∀a ∈ R + , LT 0 (a) = P P(P )e -aT (P )

where the sum runs over all defaulting paths P , P(P ) is the probability of following the defaulting path P (exactly) and T (P ) is P 's length. However, as given, any defaulting path from C1 is a defaulting path from C2, so this formula yields the same values for both starting points, i.e.

L (a) (C1) = L (a) (C2)
This being over the whole range, it follows that L (a) is constant over C0, C0 + 10 -f (k)-1 .

2. Now, the martingale parameter α(a) has a linear (in C0) effect on the main term in the theorem, so the approximation of a constant function over the range C0, C0 + 10 -f (k)-1 by an affine (multiplicative factor α(a)) function of C0 on this interval yields an error of at least y(a, k) = 10 -f (k)-1 α(a)/2

It follows that the error function e at C0 and a cannot decrease permanently below y(a, k) before C0 gets larger than max 0, 10

f (k+1)-f (k) 12 -1 ∩ N + 2 * 10 -f (k)-1 ≥ x(k) = 10 f (k+1)-f (k) 12 - 2
In particular, there is x ≥ x(k) such that e(x, a) ≥ y(a, k).

3. Now, let us take g as in the proposition 2.6, y ∈ R + , and a ∈ R * + . As g converges to 0, then for every k ∈ N * there is z(k) ∈ R + after which g is lower than y(a, k) ; for future purposes, we may assume that the sequence (z(k)) k∈N * goes to infinity. We choose recursively the terms of the function f such that ∀k ∈ N * , x(k) = 10 f (k+1)-f (k) 12 -2 > z(k)

for example f (1) = 1 and then ∀k ∈ N * , f (k + 1) = k + f (k) + ln (12z(k) + 24) ln [START_REF] Klüppelberg | Ruin probabilities and overshoots for general lévy insurance risk processes[END_REF] We verify that L f defined as such is not a rational number, so that the f -Liouville process is aperiodic. If L f is a rational number, its decimal expansion is

• Either finite (but it is not, because there are infinitely many digits 1 given by the successive positions f (k) since f is increasing) ; • Or ultimately periodic, so the successive number of 0 digits between digits 1 cannot go to infinity (but it is the case, because f (k + 1) -f (k) ≥ k)

Hence, the f -Liouville process holds the hypotheses of the theorem 1 :

• It is bounded (by L f ), positive recurrent (because it is a bounded Lévy), not globally increasing by construction ; • We just proved that it is aperiodic.

For every k ∈ N * , there is there is x ≥ x(k) > z(k) such that e(x, a) ≥ y(a, k) > g(x) (since x > z(k)). Hence, as the sequence of successive terms z(k) goes to infinity, there is k ∈ N * such that z(k) > y, and taking this k and an associated x will end the proof.

After an immediate recursion, pn ≥ p0/2 n by choice of each zn. As the series of 2 -n converges, then (zn) n converges to a limit called x ∈ R. We want to prove that x = z holds the desired property, so let u be as described and n = -log 2 (u) . By construction of terms zn and pn, we have P X ∈ zn, zn + 2 -n ≥ pn ≥ p0/2 n However, one has zn ≤ z because the sequence (zn) n is non-decreasing, and zn + 2 2. Let ξ < γ and λ = (γ -ξ)/2 ; we are going to prove that X + Y is locally heavy over (x + y ± ξ).

Let us write that for u ∈ (±ξ) and υ ≤ λ, we have

P (Y ∈ [x + y + u -X ± υ]) ≥ P (Y ∈ [x + y + u -X ± υ] |X ∈ [x ± λ]) P (X ∈ [x ± λ])
As X is a-locally heavy on x, we have

P (X ∈ [x ± λ]) ≥ aλ > 0
As X and Y are independent, we bound this from below by (bυ) aλ = υ(abλ)

P (Y ∈ [x + y + u -X ± υ]) ≥ inf
Finally, setting q = abλ, we proved that for every u ∈ (±ξ) and υ ≤ λ, P (X + Y ∈ [x + y + u ± υ]) ≥ υq which leads, for every υ ≤ 1, to

P (X + Y ∈ [x + y + u ± υ]) ≥ min (υ, λ) q
If λ > 1, this ends the proof ; if not, we take qλ instead of q.

3. Applying the previous result at point x + u (with u ∈ (±η)) for X, we get that for every ξ < , X + Y is c-heavy over (x + y + u ± ξ) with such a c. This being for every u ∈ (±η), we get the result.

These properties will prove useful during the incoming analysis. 131

Construction of heavy payoffs

First, let us prove that we have a descending transition payoff F i n -→j such that • Ai and Aj are in C's descending class A ;

• It is locally heavy over some non-trivial interval I1 ⊂ R (for the descending process C).

By hypothesis, we have a transition payoff Di→j that is locally heavy over some non-trivial interval I1 ⊂ R. When C is positive recurrent and not globally increasing, it is possible to build a cycle of arbitrary low (negative) value, with a length n ∈ N * , starting from Ai and going first through the transition (i → j). Conditionally to M going through this cycle, let us express C(n) -C(0) as the sum of its active increments :

• The first Di→j is locally heavy over I1 = (-y, -x).

• Thanks to the lemma 8.5, the cycle has a finite value thus subsequent transition payoffs must be locally heavy on some points x k ∈ R.

Let us take a state A k ∈ A and a path of finite value from A k to Ai ; in this path, we select its last state in A , called Ay. We look at the path between Ay and Ai, whose length is h and value is called v1 ∈ R.

One must have v1 ≥ 0, else there would be a descending state between Ay and Ai.

• If v1 -y < 0, then we decompose the value v1 over its successive transition payoffs x 1...h . Choosing > 0 small enough and U = (-∞, xt + ) in the lemma 8.5 yields successive points yt ≤ xt + such that the transition payoffs are locally heavy on yt ; thanks to the second part of the lemma 8.5, the concatenation of this path to the transition Di→j yields a value C(h)-C(0) ≤ v2 = v1 -y +h , that may be chosen negative. Hence, this gets a descending payoff between Ay and Aj that is locally heavy over a non-trivial sub-interval of (v2, min(v2 + y -x, 0)) (that run over a duration h + 1).

• If not, we know that from Aj we may build a path of arbitrarily large negative value v < 0. It is decomposed in the same fashion and chosen such that v1 -y + v < 0 (but v1 -y + v ≥ 0 for all partial values of the path), and its concatenation eventually leads to a descending payoff between Ay and its finishing state. Likewise, we control the lengh of the additional path, so we get in both cases a locally heavy descending transition payoff, finally called F t i -→ j.

Now we work on C only. As C admits a cycle going first through the transition (i → j), this allows for the existence of some descending transition payoffs F t k i k -→ i k+1 with k ≤ n. The use of the second part of the lemma 8.5 indicates that C(n) -C(0) (conditionned to the cycle) is locally heavy on some non-trivial interval I2 whose own measure is called m > 0. We have in particular I2 ⊂ R * -by definition of C. We repeat this cycle p = 2AQ/m + 1 times : thanks to the last part of the lemma 8.5, C(np) -C(0) (conditionally to the repeated cycle) is locally heavy on some interval I3 ⊂ R * -whose length is greater than 2AQ.

End of the proof

From this point, we transform the cycle for C into the corresponding cycle for C, so that it has now a total length of t0 = p n k=1 t k There are paths of controlled length Li ≤ A -1 ∈ N going to every state of M , so for every state Ai there is a path Qi of length at most t = t0 + A -1 such that conditionally to Qi, C(t0 + Li) -C(0) is locally heavy on an interval Ji whose length is greater than 2AQ and at most (A -1)Q apart from I3 (because we add up to A -1 times at most Q to I3). It follows that there is an interval J, e.g. [(-A -1)Q, (-A + 1)Q], belonging to the intersection of all intervals Ji,

• Of length at least 2Q (by substraction) ;

• Belonging to R * -. The final part is similar to the proof for aperiodic C-processes : the previous construction yields a "regionhalting time" τ for C itself, defined by stopping at every time period with 1/2 probability (and then systematically after t time periods). Decomposition on the possible paths and stopping times indicates that for every x high enough, K j (C(t))|∀u ≤ t, M (t) = P (t) while this conditioned C(t) -C(0) was proven to be c-locally heavy over J for some c > 0. Therefore, as t is controlled, we have C(t) -C(0) ≥ -Qt ; let us consider the discrepancy δ between K [K -, K -+ δ/2] and (K -+ δ/2, K + ] over I, named m -and m + respectively. As C(t) is c-locally heavy over C(0) + J (hence I), then the masses m -and m + have an effect of at least m ± c/2 on the convex combination evaluating K and as m -+ m + = Q by construction, the new discrepancy is (1 -cQ/4) δ so decreased by an exponential factor. Finally, as one may repeat this operation at an ergodic pace Qt + (A + 1)Q, it follows that the convergence is exponential.

Figure 1 :

 1 Figure 1: Realization for C the graph refer to the active transitions at present time. It comes to no one's surprise that

-

  Occupied state numbers at ≤ A for any t ∈ [|0, T |], with Aa 0 named the starting state and Aa T named the finishing state ; -Payoffs values xt ∈ R for any t ∈ [|1, T |]. These must obey the following constraints : -Possibility of transitions ∀t ∈ [|1, T |] , Pa t-1 →Aa t > 0 Possibility of values ∀t ∈ [|1, T |] , xt ∈ supp D M (t-1)→M (t)

Definition 1 . 6

 16 Cycle support of a C-process Let C be a positive recurrent C-process. For any n ∈ N * , let Vn(C) ⊆ R ∪ {∞} the set of all possible values for all cycles of length n in C. The cycle support of C is the set supp(C) = ∞ n=1 Vn(C)

Figure 2 :

 2 Figure 2: Globally constant C-process

Figure 3 :

 3 Figure 3: Globally increasing C-process

Figure 4 :

 4 Figure 4: Logarithmic eigenvalues for L C (α)

i

  ) are obtained once all values K (a) i (0) are found, so one gets the exact log-Laplace values.

Figure 5 :

 5 Figure 5: Default probability

Definition 5 . 4

 54 Periodically concatenated C-process Let n ∈ N * . The n-periodically concatenated C-process of C is the C-process defined by concatenation of C by the time sequence Tn associated with the binary determination sequence ∀t ∈ N, ρn(t) = 1 t∈nN .

Definition 5 . 5

 55 Concatenation of paths Let us consider two paths of respective lengths Ta, T b ∈ N and values x, y ∈ R ∪ {∞}, defined respectively by • Occupied state numbers at ≤ A for any t ∈ [|0, Ta|], and bt ≤ A for any t ∈ [|0, T b |] ; • Payoffs values xt ∈ R for any t ∈ [|1, T b |], and yt ∈ R for any t ∈ [|1, T b |].

Definition 5 . 6

 56 Rotations of a cycle Let us consider a cycle of C of length T ∈ N * , whose • Occupied state numbers are at ≤ A for t ∈ [|0, T |] ; • Payoffs values are xt ≤ A for t ∈ [|1, T |].

Definition 5 . 7

 57 Universal cyclesLet C be a C-process. • Let us consider a cycle of C of length T ∈ N * , whose occupied state numbers are at ≤ A for t ∈ [|0, T |]. It is said universal iff for every state number i ≤ A, there is t ∈ [|0, T |] such that at = i.

1 .

 1 Let us deem C positive recurrent, and take α ∈ R + and i, j ≤ A. By virtue of the lemma 5.5, there is k ∈ N * allowing for a path of length k of finite value between Ai and Aj. We characterize this path by • Its occupied state numbers Aa u , for u ∈ [|0, k|] and au ≤ A ; • Its payoffs values xu ∈ R, for u ∈ [|u, k|] : they are finite because C is positive recurrent.

2 .

 2 LC is well-defined over R * + , and the dominant eigenvalue d(α) of LC (α) holds ∀α ∈ R * + , d(α) < 1. 1. Let us assume C globally increasing and not globally constant. Thanks to the proposition 1.2, we rewrite it as C = C + + C = with C + non-decreasing and C = globally constant. We know that C + has a cycle of positive value, as else it would be identically 0 so C = C = would be globally constant : this cycle has• A length T ∈ N * ; • A starting state A k with k ≤ A ; • A positive value v ∈ R * +First, considering C = 's increments from Ai to Aj as cj -ci ∈ R thanks to the proposition 1.3, and noting by ∆(α) the diagonal matrix whose entries are ∀α ∈ R, i ≤ A, (∆(α)) i,i = e -αc i the Laplace matrix function of C rewrites as

Definition 6 . 1

 61 Drifted C-process Let C be a C-process whose underlying Markovian process is M . For every d ∈ R, C's drifted process by d is the process C [d] , whose • Underlying Markovian process is the same M ; • Transition payoffs are given through C's ones Di→j by ∀i, j ≤ A, D [d]

Lemma 6 . 4 P k→k 2 .

 642 Distributions of reduced processes Let C be a positive recurrent C-process whose underlying Markovian process is M , A k = M (0) be a state of M , and d ∈ R be a drift. Let M -k and C (-k,d) be the reduced Markovian process and C-process of C with respect to state A k and drift d.1. The transition probabilities of M -k are ∀i, j = k, P -k,i→j = Pi→j + P i→k P k→j 1 -The transition payoffs of C (-k,d) are D (-k,d),i→j such that there are independent random variables • Di→j, D i→k , D

1 .

 1 Let us take u ≤ A and i, j ∈ {kx; x ≤ u}. For α ∈ R * + and d ∈ R, let us note by Pu,i→j and D u,d,i→j the transition probability and payoffs of C (d) u

  5 that L C (d) 1 (α) is given by C's d-restricted Lévy process : noting by τ the random waiting time of return to M (0) and Dτ the increment of C's restricted Lévy process, we have C1(1) -C1(0) = C(τ ) -C(0) = Dτ + τ d so the function f defined by the lemma is actually given by

  αDτ e -τ d τ 2 e -αD τ e -τ d + E D τ 2 e -αD τ e -τ d τ 2 e -αDτ e -τ d -E Dτ τ D τ τ e -αDτ e -τ d e -αD τ e -τ d and this simplifies to 2H(α, d) = E Dτ τ -D τ τ 2 e -αDτ e -τ d e -αD τ e -τ d ≥ 0

•

  The sets Ju(d) are not empty thanks to ∀d ∈ R, ∃α ∈ R * + ; g (α, d) ∈ Su. It follows that for every u and d, there is xu(d) ∈ R * + ∪ {∞} such that Ju(d) = (0, xu(d)) As the sets Su are ordered by inclusion, the sequence (xu(d)) u is non-decreasing of u and J1 = A u=1 Ju By definition of z(d), z(d) = sup (J1) = min u≤A sup (Ju)This minimum is attained for some value of u ≤ A, and we select the largest one among them. This means that for this u,z(d) = sup(Ju) = xu(d) < sup(Ju+1) = xu+1(d) • If z(d) < ∞, this means that z(d) / ∈ Ju but z(d) ∈ Ju+1, so g (z(d), d) / ∈ Su butbelongs to Su+1, which implies that it cannot belong to S u+1 . By definition of S u+1 , we get that hu+1 (g (z(d), d)) ≥ 1 but as z(d) = sup(Ju), we must have ∀x < z(d), hu+1 (g (x, d)) < 1 As hu+1 • g is continuous around (z(d), d), then hu+1 (g (z(d), d)) = 1. Now, like when proving that an entry of L C (d) u diverges, we find i , j ∈ {kx; x ≤ u} such that P u+1,i→k u+1 E e -αD u+1,d/α,i→k u+1 P u+1,k u+1 →j E e -αD u+1,d/α,k u+1 →j is bounded from below when α goes to z(d) (this is true because z(d) < ∞), so multiplication by ∞ n=0 P n u+1,k u+1 →k u+1 E e -αD u+1,d/α,k u+1 →k u+1 n e -nd = 1 1 -hu(g(α, d)) will yield an entry of L C (d/α) u (α) that diverges around z(d). Once again, as z(d) < ∞, further reductions still allow for an entry that diverges here (and L C (d/α) 1 (α) is still well-defined for every α < z(d) because z(d) = sup (J1)). • If z(d) = ∞, we want to prove that lim α→∞ E e -αDτ e -τ d = ∞

L 1 .

 1 r,c = (Li,j) i∈[|1,A|]\{r},j∈[|1,A|]\{c} ∈ Mn-1 (C) We may now rewrite the reduction of C's Laplace matrix function as the Laplace matrix function of C's reduced process. Lemma 6.9 Reduced Laplace matrix function Let C be a positive recurrent C-process, whose Laplace matrix function is LC . Let C (-k,d) be C's reduced process with respect to A k = M (0) a state to eliminate and a drift d ∈ R. Let β ∈ R * + such that (LC (β)) k,k < e βd Thanks to the construction from the lemma 6.8, LC (-k,d) (β) is well-defined. For every (β, d) ∈ S k , LC (-k,d) (β) is the reduced matrix of LC (β) with respect to the dimension k and the parameter βd, i.e. LC (-k,d) (β) = (LC (β)) (-k,βd)

  is a martingale, we compute, for any t ∈ N, E X

C

  (t + 1)|F(t) = E w (a) [M (t+1)] e -α(a)C(t+1) |F(t) e -(t+1)a

C

  (t + 1)|F(t) = e -at e -α(a)C(t) w

  with e.g. λ(a) ≤ λ(b), and c ∈ (a, b) \X, the intermediate value theorem creates a value x ∈ [c, b] such that λ(x) = λ(a) > 0, which contradicts λ's injectivity.

  (a) as in proposition 2.1. There are constants c ∈ R + and n ∈ N such that∀a ∈ R + , δ w (a) ≤ c + naAs a consequence, w (a) 's coordinates are bounded by e c+na . Let us start from the eigenvector equation LC (α(a)) w (a) = e a w (a) Thanks to the proposition 1.5, the n th power of C's Laplace matrix function yields ∀i ≤ A,

µ

  (a) (LC (α(a)) -e a Id) = µ (b) LC (α(b)) -e b Id = 0 by construction. When b converges to a, the terms e b -e a and LC (α(a)) -LC (α(b)) go to zero because α and LC are continuous ; as µ (b) is locally bounded (by 1), we get lim b→a µ (b) -µ (a) (LC (α(a)) -e a Id) = 0 We also know that µ (b) -µ (a) 1 = 0 by the equation of scaling. Hence, let us look at the linear function

  lim b→a Fa µ (b) -µ (a) = 0 so µ (b) must go to µ (a) ; this being for every a ∈ R * + , µ as a function is continuous over R * + . The similar property for w comes from the equality (LC (α(a)) -e a Id) w (b) -w (a) = e b -e a w (b) -(LC (α(a)) -LC (α(b))) w (b) that one may verify, using (LC (α(a)) -e a Id) w (a) = LC (α(b)) -e b Id w (b) = 0 by construction. Likewise, as w (b) is locally bounded (as proved above), we get lim b→a (LC (α(a)) -e a Id) w (b) -w (a)

  Computations lead to the block related with x ∀x, y ∈ R A , ∂fx ∂x (x, y) = (LC (α(a))) * -x 1 e a Id -1 xe a * the block related with y ∀x, y ∈ R A , ∂fy ∂y (x, y) = LC (α(a)) -xye a Id -yxe a but also to ∀x, y ∈ R A , ∂fx ∂y (x, y) = 0 which means that the restriction of f 's Jacobian matrix related to derivatives with respect to (x, y) will be invertible as soon as both matrices ∂fx/∂x and ∂fy/∂y are. Taking them at a point x = µ (a) and y = w (a) leads to ∂fx ∂x µ (a) , w (a) = (LC (α(a))) * -µ (a) 1 e a Id -1 µ (a) e a * and ∂fy ∂y µ (a) , w (a) = LC (α(a)) -µ (a) w (a) e a Id -w (a) µ (a) e a However, the equations of scaling simplify the previous equations to ∂fx ∂x µ (a) , w (a) = LC (α(a)) -e a Id -1 µ (a) e a * ∧ ∂fy ∂y µ (a) , w (a) = LC (α(a)) -e a Id -w (a) µ (a) e a Now, we prove that the matrix LC (α(a)) -e a Id -1 µ (a) e a is injective, thus invertible. If there is a row vector v ∈ R A such that v (LC (α(a)) -e a Id) = e a v 1 µ (a)

  a) da Dτ + τ + 1 e -α(a)Dτ e -a(τ +1) = 0 and from this we get that dα(a) da = E (τ + 1)e -α(a)Dτ e -a(τ +1)E (-Dτ e -α(a)Dτ e -a(τ +1) )

a) da 2 2 e 2 - 2 e

 2222 Dτ e -α(a)Dτ e -a(τ +1) + E dα(a) da Dτ + τ + 1 -α(a)Dτ e -a(τ +1) = 0Thanks to the previous equation, the first term of this sum simplifies, leading to d 2 α(a) da E (τ + 1)e -α(a)Dτ e -a(τ +1) -α(a)Dτ e -a(τ +1)

1 . 1 p

 11 Let us start with C's properties, ensuring the existence of a martingale parameter for C. (a) If C is positive recurrent, then by lemma 5.5, for every states Ai and Aj, there is a path of length T ∈ N and finite value v ∈ R from Ai to Aj, defined by its • Occupied state numbers at ≤ A for t ∈ [|0, T |], with a0 = i and aT = j ; • Payoffs values xt ∈ R for t ∈ [|1, T |]. By construction of C's transition payoffs, the following determinations define a path of finite value for C : • Same occupied state numbers at ≤ A ; • Payoffs values defined by ∀t ∈ [|1, T |] , yt = xt -pa t + pa t-1 p This being for every state numbers i, j ≤ A, C is positive recurrent. (b) We prove that C and C have the same cycle support (up to a scaling by p), so that the lemma 1.7 will lead to the equivalence between global monotonicity of one and the other. Let us start with a cycle of finite value x ∈ R for C, whose successive payoffs values are xt ∈ R. Like previously, we find a cycle for C whose successive payoffs values are ∀t ∈ [|1, T |] , yt = xt -pa t + pa t-Its value y amounts to T t=1

  [s] i breaking points of Λ (a) s . They do not depend on a ∈ R + .

  [s] k (shifted of p) because C's natural offsets are p-periodic in C(0) by construction.

  and a ∈ R + , and look for K(x + p, a) : as the sets I [s] n,i are built p-periodically, then x + p ∈ I [s]

  construction of e that ∀x ∈ R + , ∀a ≤ amax, e(x, a) < Be -xβ

  i→j the convolution transition probabilities of the convolution equation.

  i→j and transition probabilities P (a) i→j of the convolution equation as above. Then the C-process Φ (a) defined by • Its underlying Markovian process M (a) , holding M (a) (0) = M (0) almost surely and whose transition probabilities are, for any t ∈ N, ∀i, j ≤ A, P M (a) (t + 1) = Aj|M (a) (t) = Ai = P (a) i→j This is a correct definition, as the constants P (a) i→j sum to 1 by the previous lemma 7.1 ; • Its transition payoffs between Ai and Aj, being -G (a) i→j ; • Its starting point Φ (a) (0) = C(0).

  β] , the mass rewrites as

  β] ; if they have zero mass, then we set φ (a) i→j,[β] arbitrarily, as they will have no effect on the sequel. This leads to

  β] are probability measures (with support in (0, Q]), they define random variables called G (a) i→j,[β] on the same support. The previous equation may hence be rewritten as

  β] to be of unit masses, but at the expense of parameters m (a) i→j,[β] appearing in the equation. Now, we aim at scaling these parameters to get L (a) i,[β] (x) as a convex combination of expectancies

  α(a)] (x) will be averages of previous values K (a) j,[α(a)] (y) for y < x, which means that the functions K (a) i,[α(a)] solve this lemma 7.1 for P This leads to the desired result by definition of the functions K (a) j,[α(a)] , which ends the proof of the lemma 7.1.

  proof method for the proposition 2.4 and the theorem 1 transforms C into Φ (a)

Lemma 7 . 4

 74 Regularity of Φ (a) C 's transitions Let C be as in lemma 7.1. 1. Viewed as a function of a ∈ R + , the probabilities P (a) i→j are continuous. 2. Viewed as functions of a ∈ R + the random variables G (a)i→j such that Qi→j > 0 are continuous in probabilities, i.e. for every i, j ≤ A with Qi→j > 0, for every measurable set S ⊆ R, i→j , that still solves the convolution property from the lemma 7.2. We prove this lemma 7.4 considering how Φ (a) C 's parameters were built. 1. This comes from their definitions as P

  zero masses. Let S be any measurable subset of R : by definition, n -→j e -an x∈S e xα(a) dν i n -→j m (a) i→j,[α(a)]

•

  Its probability p (b) ( ) of being followed at precision converges to p (a) ( ) when b goes to a.We shall use the lemma 5.2 here. Let P be a path of length T ∈ N, we note• a t∈[|0,T |] ≤ A its successive occupied state numbers ; • v t∈[|0,T |] ∈ Rits successive partial values (sums of the first t payoffs values).

6

 6 Aperiodicity of C's convolution process Let a ∈ R + . 1. If C is aperiodic, then the restriction of Φ(a) C to its descending class A is aperiodic. 2. If C's fundamental period is p ∈ R * + , then the restriction of Φ (a)

C

  's cycle support by definition of its increments -G (a) i→j : for -x ∈ [ -Q, 0), we want to find a sufficient condition that will imply x ∈ supp G (a) i→j . Let i, j ∈ A such that Qi→j > 0 : we recall that the random variables G (a) i→j were defined as the variables G (a) i→j,[α(a)] . By definition, they have the same support as φ (a) i→j,[α(a)] , then as ν (a) i→j,[α(a)] because Qi→j > 0. As a consequence, every x ∈ (0, Q] will belong to G (a) i→j 's support as soon as there is n ∈ N * such that Q i n

i

  (x), for x < C(0). A central idea is to consider the functions K (a) i only over intervals whose forms are [x -Q, x) for x ∈ R + . The reason behind this is that, as the increments G (a) i→j are deemed bounded, any path for Φ (a)C 's successive values must hit some point in this region before going down : as a consequence, the global behaviour of the functions K (a) i may be found looking at one such interval only. For this reason, for every x ∈ R + , we shall name [x -Q, x) the x-interval, and look at the extrema of the functions K

Lemma 7 . 7

 77 and bounded by -Q, this is tantamount to asking if τ (a) < ∞ almost surely. Finiteness of τ(a)

C

  (t) and M (a) (t), P Φ

C

  (t + 1) < Φ

C

  (t) -η = P G

M

  (t)→M (t+1) > η > 1/2This being for every Φ

C

  (t) and M (a) (t), Borel-Cantelli's lemma states that the event G

C

  , so values in this latter set. As a consequence, the useful controls are local extrema given below, where we recall that A is C's descending class. Definition 7.5 Local discrete extrema Let K (a) i

1 .j

 1 They are sorted by ascending order as in ∀x ∈ N, K non-decreasing. 3. All four of them converge to finite limits, respectively called l Sorting these functions in ascending order comes from their definitions and A ⊆ [|1, A|], so we shall focus on the monotonicity property ; since the proofs are almost similar, we shall only present the case of K (a) + . Let x ∈ N : we know that, thanks to the convolution equation from lemma 7.2, every function K (x -d) If i ∈ A , we know that the contributions of values K (a) j (x -d) when j / ∈ A are non-existent because A is closed, thus P (a) i→j = 0. However, the remaining terms K (a) j (x -d) are involved in the local maximum K

i

  's values over whole x-intervals. The local extrema are thus defined as follows. Definition 7.6 Local extrema Let K (a) i be the functions defined by the lemma 7.1. We define • The local maxima of positive recurrent functions K (a) i over the x-interval as a function of x, given by

  They are well-defined and locally bounded, since for every a ∈ R + , i ≤ A and x ∈ R + we haveK (a) i (x) ∈ 0,e α(a)x w

  non-decreasing. 3. All four of them converge to finite limits, respectively called l

+

  (x) + y0 = Y (a) (t) = E Y (a) (t +1) being bounded by hypothesis by P Φ (a) C (t + 1) < x + nη K (a) + (x) + P Φ

C

  (t + 1) ≥ x + nη K (a) + (x) + 2y0 ≤ s(x) + 2p(η)y0 < s(x) + y0 because M (a) (t + 1) ∈ A since A is closed, which would be contradictory (N.B. : for K (a) ++ , one needs no condition on M (a) (t + 1)). Starting over with x1 and i1, one would progressively get sequences indexed by u ∈ N, given by

as given by the lemma 7 . 2 .

 72 Then there are • A function K∞ : R + → R * + ; • Exponential error functionsZ : R + → R + ∧ β : R + → R * + Z and β being continuous over R + , such that ∀i ≤ A, x ∈ N, a ∈ R + , K(a)i (x) ∈ K∞(a) ± Z(a)e -β(a)x When C is aperiodic, we want this one : Lemma 7.11 Weak theorem for aperiodic C-processes Let C be a positive recurrent, bounded and not globally increasing C-process, deemed aperiodic. For every a ∈ R + , we set • C's martingale parameter at point a as α(a) ; • C's dominant eigenvector at point a as w (a) . We define the functions K (a) i as given by the lemma 7.2. Then there are • A function K∞ : R + → R * + ; • A continuous error function e : R + × R + → R + that converges uniformly over any compact set for its first variable, i.e. ∀a ∈ R + , ∀ > 0, ∃x0(a, ); ∀b ∈ [0, a] , ∀x ≥ x0(a, ), e(a, x) ≤

C

  on a specific situationM (a) τ (a) = Ai ∧ Φ (a) C τ (a) = y for i ∈ A and y ∈ N ; 2.Prove that the discrepancy between K

τ

  (a) = y ∧ M (a) τ (a) = Aifor y ∈ [|x -Q, x -1|] and i ∈ A .Lemma 7.12 Uniform references to previous values Let Φ (a)

C

  to be C's convolution process at any point a ∈ R * + , whose• Starting point is C0 ∈ N ; • Starting state is s ≤ A ; • Descending class is A .We shall take an arbitrarily large x0, then 1. Exhibit a path of Φ (a) C leading to any event afterwards noted B(y, i) = Φ (a) C τ (a) = y ∧ M (a) τ (a) = Ai 2. Prove that Φ (a)

C

  (0) and M (a) (0) = Ai. As Y (a) C

1 w

 1 is bounded by the highest value among K (a) i (x) for i ∈ A and x ∈ [| -Q, -1|], amounting to (at most) max i∈A

1 e-

 1 -β (a) 1 x + l-(a)This implies that l+(a) = l-(a), and the convergence speed is driven by the exponential parameters Z (x) being symmetrical).

  ) ≤ l-(a) + Z2(a)e -β (a) 3 nAQ which leads, as K (a) ++ is non-increasing, and for the same reasons as before with Z3(a) = 2Z2(a), to ∀i ≤ A, x ∈ N, K (a) i (x) ≤ l-(a) + Z3(a)e -β (a) 3 x

i 4 x

 4 (x) ≥ l+(a) + Z4(a)e -β (a) However, when x goes to ∞, as we know that l-(a) ≤ l+(a), this is possible only if l-(a) = l+(a) (which is the suitable K∞(a)), so setting ∀a ∈ R + , Z(a) = max (Z3(a), Z4(a)) ∧ β(a) = min β

2 .

 2 Use the density alternative, so that for every x ∈ R + remote enough from C(0) one may ensure thatP M (a) τ (a) ∧ Φ (a) τ (a) ∈ [x ± ]is bounded away from 0 ;3. When expressing the martingale property, write Y (a) (0) as a convex combination involving average values of K (a) i over some sets [x, x + ], with controlled coefficients ; 4. When looking at the values of i ∈ A and y ∈ R where K (a) i (y) is minimal, use the local control to bound the contribution of this set to K (a) i from above ;

i

  (y) is upper bounded by K (a) i (x) + (y -x)Ξ, locally controlling K (a) i around an arbitrary point x. Lemma 7.13 Half-Lipschitz functions Let a ∈ R + , and K (a) i≤A be the functions coming from the convolution equation (lemma 7.1). Let us note by

C

  (τ ) |M (a) (τ ) = Aj By definition of ξi, this comes to zξi(x, x + z) = A j=1 P M (a) (τ ) = Aj E ξj Φ (a) C (τ ), z + Φ (a) C (τ ) |M (a) (τ ) = Aj As Φ (a) C (τ ) < 0, we are driven back to one of the previous cases, however in both of them we know that ∀u < 0, ξj (u, z + u) ≤ zΞ(a)so what finally remains is zξi(x, x + z) ≤ zΞ(a)

Lemma 7 . 14

 714 Uniform references to approximate previous values Let Φ (a)

  (a) As A is the only closed communicating class of Φ (a) C , if s /

C

  has a universal (in A ) cycle of some value in [y ± /2]. In particular, let us take b = (2A + 1)Q Doing this for y = X --k for every k ∈ [|0, b|], we get a universal cycle Q k starting from Ai of some value in X --kη ± η 18 for every k ∈ [|0, b|]. For each k, we note by r k (a, η) > 0 the probability of following such a cycle at precision η/9.

-

  (x0). By definition, setting any > 0 allows us to findx1 ∈ [x0 -Q, x0) and i1 ∈ A such that K (a) i 1 (x1) < y0 + Now, let us take η such that Q/η = m ∈ N * , i2 ∈ A ,and x2 ∈ [x1 + η + e(C, η), x1 + η + e(C, η) + Q) Thanks to the convolution equation, the value K (a) i 2 (x2) may be expressed as a convex combination, involving the intervals from the lemma 7.14. Noting by Φ (a) C the convolution process of C starting from M (a) (0) = Ai 2 and Φ (a) C (0) = x2, and τ (a) its (x2 -e(C, η))-halting time, we have ∀k

τ

  In particular, we consider the term with j0 = i1 andk0 = x2 -e(C, η) -x1 η ∈ [|1, m|]by construction of m. Noting by I the intervalI = [x2 -e(C, η) -k0η, x2 -e(C, η) -(k0 -1)η)the conditional expected value to this B(k0, j0) rewrites asE K (a) ∈ I ∧ M (a) τ (a) = Aj 0However, by definition of k0, I's lower bound is at least x1 and at most x1 + η. Thanks to the half-Lipschitz property from lemma 7.13, the function K (a) j 0 is upper bounded over the involved set, as it is included in [x1, x1 + 2η), by K (a) j 0 (x1) + 2ηΞ(a) < y0 + + 2ηΞ(a) Now, we know by definition of τ (a) that almost surely, Φ (a) C

+

  Hence, let us note by β1(η, a) = -ln 1 -h (a) (C, η) η + e(C, η) + 3Q and as earlier, thanks to h (a) (C, η) ≤ 1/2, Z2( , η, a) = 2Z1( , η, a)e 2Qβ 1 (η,a) so that the inequation implies∀x ≥ 2Q, K (a) + (x) ≤ l (a)-+ + 2ηΞ(a) + Z2( , η, a)e -β 1 (η,a)x Let us take ζ > 0 and a ∈ R + . We set = ζ/3 ; as Ξ is continuous, we may define its maximum Ξ(a) over [0, a], then take η(a, ζ) to be any constant in 0This η allows us to define in turn :• Since β1 is continuous of its second variable, a term β1(ζ, a) = inf b∈[0,a] (β1(η(a, ζ), b)) > 0 • Since Z2 is continuous of its third variable, another term Z2(ζ, a) = sup b∈[0,a] Z2(ζ/3, η(a, ζ), b) < ∞Hence, provided that proved that for every ζ > 0, a ∈ R + , there is a x(ζ, a) such that for every x ≥ x(ζ, a) and every b ∈ [0, a], K To get the symmetrical property, the only changes in this proof are that• To x0 ∈ R + is associated y0 = K(a) + (x0), and x1, i1 such that K (a) i 1 (x1) > y0 -• One takes x2 ∈ [x1 -η + e(C, η), x1 -η + e(C, η) + Q) • The set in which B(k0, j0) falls lies in [x1 -2η, x1), and K (a) j 0 is bounded from below on this set by K (a) j 0 (x1) -2ηΞ thanks to the lemma 7.13 ; • So, we end up with every K (a) i 2 (x2), and thus K (a) -(x1 -η + e(C, η) + Q), being lower bounded by a convex combination consisting of -A term y0 -, of weight at least h (a) (η, C) ; -Remaining terms l (a) + . Solving this ultimately leads, the similar way, to the symmetrical property : for every ζ > 0, a ∈ R + , there is a x(ζ, a) such that for every x ≥ x(ζ, a) and every b ∈ [0, , this is possible only if l (a) -= l (a)

+

  (x) ≤ K∞(b) + /2 Hence, let us consider n ( , a) ∈ N such that ∀b ≤ a, K (b) ++ (0) (1 -g(b)) n( ,a) < /2 This is possible because K (a) ++ and g(a) are continuous expressions of a. When starting Φ (a) C from x + n ( , a) AQ and any Ai, the convolution equation after a waiting time of n ( , a) A time periods yields, for every b ≤ a, K (b) i

C

  (n ( , a) A) ≥ x0, so • If M (b) (n ( , a) A) ∈ A , then the term in the conditional expectancy is upper bounded by K (b) + (x0) as we just proved ; • If not, we use the universal bound K (b) ++ (0) < ∞ and the control of the corresponding probability (choice of n ( , a)). Hence, one eventually gets that K (b) i (x + n ( , a) AQ) ≤ K∞(b) + Doing this for every x ∈ [x0, x0 + Q) and i ≤ A yields K (b) ++ (x0 + Q + n ( , a) AQ) ≤ K∞(b) + So, we proved that for every a ∈ R + , > 0, there is m = x0 + Q + n ( , a) AQ such that for every x ≥ m and b ≤ a, K (b) ++ (x) ≤ K∞(b) + As the symmetrical inequality holds the same way, this ends the proof.

2 .

 2 Get an equation for K (a) i 's Laplace transform around 0 + , so that we shall use the final value theorem to find K∞(a) later : this is possible because the functions K (a) i converge.

Lemma 8 . 1 1 .

 811 Properties of the DLT Let u : Z → R + be a non-negative sequence, and ψ be a probability distribution over N * . Final value theorem : if u(∞) ∈ R + exists, then lim w→0 + (wû(w)) = u(∞) 2. Convolution : for every w ∈ R, we have u * ψ(w) = ∞ x=0 ∞ d=1 u(x -d)ψ(d)e -wx = u(w) ψ(w) +

P

  term ultimately simplifies (when w = α(a)) to A j=1 i→j = d e -wd -e -α(a)d e α(a)-w -1 We introduce the vector function V (a) , defined by its coordinates : i→j = d e -wd -e -α(a)d e α(a)-w -1 i≤A   

  term of a matrix product.Hence, noting K (a) the vector of discrete Laplace transforms, we haveK (a) (w) = L Φ (a) C (-w) K (a) (w) + V (a) (w) which leads to K (a) (w) = Id -L Φ (a)

Lemma 8 . 2 1 .

 821 Properties of the Laplace transform Let u : R → R + be a non-negative measurable function, and ψ be a probability distribution over R * + . Final value theorem : if u(∞) ∈ R + exists, then lim w→0 + (wLu(w)) = u(∞) 2. Convolution : for every w ∈ R, we have L u * ψ (w) = ∞ x=0 ∞ d=0 u(x -d)e -wx dψ(d)dx = Lu(w)L ψ (w) + )e -wx dxdψ(d) (possibly +∞).

e

  so the rightmost term ultimately simplifies (when w = α(a)) to -wd -e -α(a)d α(a) -w dφ

1 V

 1 vector function V , defined by its coordinates : When w = α(a), V is again continuously prolongated byV i→j e -α(a)G (a) i→j • We still recognize ∀w ∈ R, P (a) i→j L G (a) i→j (w) = L Φ (a) C (-w) i,jbeing the general term of a matrix product.Hence, noting L K (a) the vector of Laplace transforms, we haveL K (a) (w) = L Φ (a) C (-w)L K (a) (w) + V (a) (w) which leads to L K (a) (w) = Id -L Φ (a) (a) (w)whenever 1 is not an eigenvalue of L Φ (a)C (-w), which is automatic if w > 0. This is the same equation as for regular C-processes, as only the definition of V has changed.8.2 Final value theoremWe want the value lim w→0 +(wL K (a) (w)) = lim w→0 + w Id -L Φ (a)

B(w)U0(w) = 1 B•

 1 (w)V0(w) = 0 T W1(w) + D(w)U0(w) = 0 T X1(w) + D(w)V0(w) = Id B(w) is locally non-zero, because B(w) = rH(w)c where r and c are positive, and H(w) is non-zero and non-positive. As a consequence, we get U0(w) = 1/B(w), V0(w) = 0, and then W1(w) and X1(w) because T is invertible. At further orders n ∈ N * , we have B(w)Un(w) + C(w)Wn(w) = 0 B(w)Vn(w) + C(w)Xn(w) = 0 T Wn+1(w) + D(w)Un(w) + E(w)Wn(w) = 0 T Xn+1(w) + D(w)Vn(w) + E(w)Xn(w) = 0

Taking

  w1 < min(1/λ + , w0) guarantees absolute and uniform convergence of the sum M (w) over [0, w1]. Now, we found by construction of M (w) that around w = 0, ∀i, j ≤ A, M (w) = B(w)1i=j=1 + O(w) so that P -1 f (w)P = M (w) leads tof (w) = cr 1 B(w) + O(w)However, we also know that -B(0) = rH(0)c, so we finally get the value of f (0) :f (0) = crr dL(w) dw (0)c which ends the proof.

8. 2 . 2

 22 Case α(a) = 0 When α(a) = 0, which may happen only if a = 0, the above formula simplifies. Indeed, we get ∀i, j ≤ A, L Φ (0) i→j E e wG (0) i→j so L Φ (0) C coincides with L C (without a state A∞). We also have 1. In the regular case, ∀i ≤ A, V

1 V

 1 (0) = 0 means w (0) = 1 thanks to the proposition 2.1 ; when w goes to 0, we also have that lim(a) (w) = limw→0 + w (Id -L C (-w)) -1 V(0)[i] (0 + ) thanks to the lemma 8.3, the limit matrix is (since w (0) (0 + ) is precisely dL C (w) dw

1 -

 1 e -α(a)d e α(a) -1 dφ (a) i→j (d) = 1 -E e -α(a)G (a)Noting by v (a) ∈ (R * + ) A the vector whose coordinates are∀i ≤ A, v v (a) -L Φ (a) C (-α(a))v (a)Besides, the lemma 8.3 states that limw→0 + Id -L Φ (a) C (-w) -1 w = c (a) r (a)r (a) dL(w) dw (0)c (a) 124 where r (a) and c (a) are the dominant eigenvectors of L Φ (a) C (0), i.e. of the matrix whose general entry (i, j) is P (a) i→j . However, this means that c (a) = 1 and r (a) is M (a) 's invariant distribution, so by the definition 1.11 of Φ (a) C 's mean expectancy, r (a) dL(w) dw (0)c (a) = -E Φ ) v (a) -L Φ (a) C (-α(a))v (a) which rewrites as K (a) ∞ = r (a) P (a) -L Φ (a) C (-α(a)) v (a)

1 .

 1 It is positive, because by construction K non-decreasing. 2. It is continuous of a over R * + by definition of its expression, as all terms are continuous and -E Φ

∞ 1 -

 1 and α(a) are positive. The only case we should look at is what happens at a = 0 when α(0) = 0. • The vectors c (a) = 1 and r (a) are the dominant eigenvectors of L Φ (a) C (0), this matrix being P (a) i→j , and we write L Φ (a) C (-α(a)) as ∀i, j ≤ A, L Φ (a) i→j E e -α(a)G i→j So when α(a) goes to 0, ∀i, j ≤ A, P (a) -L Φ (a) i→j (α(a)E (Gi→j) + o (α(a))) As v (a) converges to 1 by definition, we recover that r (a) P (a) -L Φ (a) C (-α(a)) v (a) = α(a) -E Φ when a goes to 0. 3. When C is an aperiodic Lévy process, the matrix P (a) is the unit and the vectors v (a) and r (a) are 1 . Noting by G the negated only random increment of Φ E e -α(a)GE(G)α(a)We ensure it is less than 1 using that ∀x ∈ R * , 1 -e -α(a)x < α(a)x 125 so as G > 0 almost surely, one gets K C is a regular Lévy process, the matrix P (a) is again the unit and the vectors v (a) and r (a) are 1 . Noting by G the negated only random increment of Φ (a) C , we have this timeK (a) ∞ = e -α(a) 1 -E e -α(a)G E(G) (1 -e -α(a) )Let us set f (x) = 1 -e -α(a)x for x ∈ R. As α(a) > 0, f is strictly concave ; from f (0) = 0, it comes that ∀x > 1, f (x) < xf (1) Hence, unless G = 1 almost surely (and then we have an exact equality),E 1 -e -α(a)G < E 1 -e -α(a) G Thus, we proved that K (a)∞ ≤ e -α(a) , with equality iff G = 1 almost surely. On a side note, the fact that C is a regular Lévy process with G = 1 almost surely indicates that its default time T0 may be expressed asT0 = C(0)+1 i=1Tiwhere the random variables Ti describe the successive descent times for C (necessarily involving a loss of 1 per descent). These are i.i.d. (whose common distribution is named T 's one), so one has ∀a ∈ R + , E e -aT 0 = E e -aT 1+C(0) However, we know that then-ln E e -aT = α(a)thanks to the martingale property E e -α(a)C(T ) e -aT = e -α(a)C(0)and because C(T ) must be C(0) -1 since G = 1 almost surely. So, ∀a ∈ R + , E e -aT 0 = e -α(a)(1+C(0)) = K (a) ∞ e -α(a)C(0)

M

  (T 0 ) e -α(a)C(0)

  -n ≥ z because by construction z ≤ zn + ∞ k=n+1 2 -k = zn + 2 -n It follows that zn ≥ z -2 -n ≥ z -u and zn + 2 -n ≤ z + u, so zn, zn + 2 -n ⊂ [z ± u] which implies that P (X ∈ [z ± u]) ≥ p0 2 -log 2 (u) ≥p0u2 so η = p0/2 works.

  Y ∈ [y + u -z ± υ])) aλSince z and are bounded by λ and u strictly bounded by ξ, then[y + u -z ± υ] ⊂ (y ± γ)and as Y is b-locally heavy around y for γ, this tells thatP (Y ∈ [x + y + u -X ± υ]) ≥ inf z∈[±λ]

i

  (x) is a convex combination of other values K (a) j (y).The key point is that the chosen paths have a positive contribution to K

  local extrema K -and K + over [C(0) -Qt, C(0)]. Now, let us take an interval I = [x, x + Q] and a C(0) such that I ⊂ C(0) + J, i.e. C(0) ∈ [x + AQ, x + (A + 1)Q] ; let us compute the measures of K

  i (C(0)) ∈ K -+ (δ/2)m + (c/2), K + -(δ/2)m -(c/2)This holds over C(0) ∈ [x + AQ, x + (A + 1)Q] for every i, which means that the discrepancy has becomeK + -(δ/2)m -(c/2) -K -+ (δ/2)m + (c/2) = δ -(δ/2)(c/2) m -+ m +

  When C is p-periodic, let us consider the support modulo p of paths from Ai to Aj for i, j ≤ A (excluding values +∞), herein named Si→j ⊆ R/pZ. Let us take a cycle of finite value for C, whose occupied state numbers are kn for n ∈ [|0, T |]. By definition of a periodic C-process, we must have

	T
	n=1

  Vn(C), so there are both a universal cycle P1 of length m, occupied state numbers a t∈[|0,m|] , and value x, and a cycle P2 of length n, occupied state numbers b t∈[|0,n|] , and value y. As P1 is universal, we rotate it to A b 0 , getting a universal cycle P3 of identical value x. Concatenating P3 and P2 yields a path by lemma 5.5 because P3's finishing state is P2's starting state ; it is a universal cycle because P3 is universal ; its value is x + y by addition. So, x + y must belong to ucs(C), which ends the proof.

x ∈ ucs(C) and y ∈ supp(C). By construction, there are m, n ∈ N * such that x ∈ Um(C) and y ∈

5.2.7 Remote density

We present a result stating that, provided that supp(C) contains two "close" values, then one gets a density-like property for ucs(C), i.e. large negative values of x are approximated by values in ucs(C). It will be used to create paths of C going to (approximations of) low enough arbitrary values.

Lemma 5.8 Remote density

Let C be a positive recurrent C-process, deemed not globally increasing, whose universal cycle support is ucs(C). We assume the existence of two values x1

This paragraph introduces useful lemmata to subsequent paragraphs, indicating properties of paths of a C-process, that will be used in further proofs.

Successive reductions and restricted Lévy process

We proved that the reduction of a positive recurrent C-process over A ≥ 2 states exists and is a positive recurrent C-process over A -1 states. Hence, we may apply the reduction again, until we finally hit a C-process over one single state, i.e. a Lévy process. It turns out that the result of these successive reductions coincides with C's restricted Lévy process, as we only select times t ∈ N holding M (t) = M (0). Lemma 6.5 Reductions up to the restricted Lévy process Let C be a positive recurrent C-process, and (ki) i≤A be an enumeration of the state numbers with A k 1 = M (0). Let us take d ∈ R a drift, and build the sequence of successive reduced C-processes recursively, with CA = C and ∀u ≤ A -1, Cu = (Cu+1) (-k u+1 ,d)

Let C * be C's restricted Lévy process and τ * be its time sequence. Then ∀t ∈ N, C1(t) = C * (t) + (τ * (t) -t) d

We call it C's d-restricted Lévy process.

We are going to iterate reductions, choosing successively state numbers kA to k2 to eliminate (other than M (0)).

• First, these successive reductions are possible, as we proved that the reduction of a positive recurrent, non-degenerated C-process (i.e. A ≥ 2) is still a C-process in lemma 6.4. As reductions each eliminate one state, we correctly do A -1 of them so only one state is left.

• For every u ≤ A, we look at -Mu, being Cu's underlying Markovian process (so MA = M ) ; ρu, being Cu's binary determination sequence, as being a reduction of Cu+1 when u ≤ A -1 ; τu, being ρu's associated time sequence when u ≤ A -1.

We also define chained time sequences υu by ∀s ∈ N, υ1(s) = s and

In particular, we want to prove that υA is τ * , the time sequence of C's restricted Lévy process.

• We know that (for any s ∈ N) ρu(s) = 1 iff Mu(s) = A ku . As C is positive recurrent, return to M (0) is almost certain, so no τu(s) amounts to +∞ almost surely, and we have

so by induction on u, we get ∀s ∈ N, M1(s) = MA (υA(s))

We prove that ∀s ∈ N, Mu (υu(s)) = M (0). This is done by induction on u : when u = 1, M1 has M (0) as only state, and we know that if Mu (υu(s)) = M (0), then

by definition of Mu. Hence, since M = MA, we proved that υA only hits states equal to M (0).

• Finally, we prove that all values of t ∈ N such that M (t) = M (0) are hit by υA, i.e. there is s ∈ N with υA(s) = t. To do this, let us assume that M (t) = M (0) and let us start with tA = t. For u from A -1 down to 1, we do the following loop (A -1 times) :

0) = A k 1 and by hypothesis, and for further steps thanks to the last part of the previous loop. ρu indicates times x ∈ N such that Mu+1(x) = A k u+1 . In particular, this holds for x = tu+1, so there is tu ∈ N such that tu+1 = τu (tu).

-We select such a tu, and we have by construction of Mu

and provided that u = 1, this is not A ku , which enables the verification for the next loop (and when u hits 1, looping stops here).

• If E (|Di→j|) = ∞, then (i, j) / ∈ Γ (else C would not be integrable), so P (X1 = 1) = 0. It follows that the first term of the sum is integrable, be it naturally or because it is irrelevant.

• For the same reason, if either one of the transition payoffs D i→k or D k→j is not integrable, then the corresponding (i, k) or (k, j) would not be in Γ so P (X1 = 0) = 0. • Finally, if D k→k is not integrable, then (k, k) / ∈ Γ, which implies that X2 = 0 almost surely and the effect of D k→k is nullified.

Hence, the only needed verification is that the series converges. We know that

because P k→k < 1 since C is positive recurrent, then all terms are integrable so RC -k (0) is welldefined.

2. Let M -k be the reduced Markov chain ; as it is still positive recurrent, it has a single invariant distribution. We note by P and P -k the transition matrices of M and M -k ; we prove that µ -k as given is M -k 's invariant distribution by testing if

The first constraint comes from the identity i =k

To get the other one, we evaluate for every j = k

Thanks to the lemma 6.4, this rewrites as

Decomposing between "direct" and "indirect" transitions, we have

that simplifies to

So, µ -k is the sought invariant distribution.

We aim at finding

We are going to use the successive reductions from lemma 6.5. According to its notations, we name

• The state to be removed is A ku = M (0) for u ∈ [|2, A|] ; for convenience, we shall note A k 1 = M (0) the last state (the starting state) that will not be removed. • The reduced processes before reduction number u, for u ∈

• The invariant measure of its underlying Markovian process is

Our idea is to prove by induction that for every u ≤ A, andµ sums to 1 by definition.

• Taking Cu into account for u ∈ [|2, A|], we have -For every x ≤ u -1, we know that

By induction hypothesis, this amounts to

that simplifies to

By virtue of the induction hypothesis and the value of µu[ku],

It follows from this that, taking u = 1,

which ends the proof.

This result is not surprising, as the mean expectancy of a C-process is its expected variation divided by the required amount of time to get it : a reduction means an acceleration of time by cancellation of the steps t ∈ N where M (t) = A k , which happens roughly at a fraction µ [k] of time, which leads to an acceleration of E by a factor 1/(1 -µ [k] ).

Conservation of global monotonicity

When discarding drifts (i.e. d = 0), global monotonicity is conserved upon reductions.

Lemma 6.7 Global monotonicity and reductions

Let C be a positive recurrent C-process, and C -k be C's reduced process with respect to a state

As the case C globally decreasing is symmetrical to the other one, we shall only consider the case of globally increasing C-processes. We may note that this symmetry holds because C is positive recurrent, so C -k cannot produce a "by default" (no further realization of M (t) = A k ) increment +∞, because τ is almost surely finite. The main idea is to remark that C's and C -k 's restricted Lévy processes are one and the same : as the binary determination sequence of the restricted Lévy process retains only times t ∈ N such that M (t) = M (0), and none of them are removed by reduction with respect to a state A k = M (0), then their restricted Lévy processes will coincide. The lemma 1.2 ends the proof.

Integrability of a reduced Laplace matrix function

Since we computed the transition probabilities and payoffs of C's reduced process C (-k,d) in lemma 6.4, we may calculate its Laplace matrix function. However, this must be done with care, since C (-k,d) may not be exponentially integrable on the same domain as C. We provide a lemma indicating the domain of integrability of C's d-restricted Lévy process, as defined by lemma 6.5.

Lemma 6.8 Domain of integrability

Let C be a positive recurrent, sEI C-process, deemed not globally increasing. Let d ∈ R be a drift and C (d) be C's d-restricted Lévy process. We define the function

where λ(M ) is M 's dominant eigenvalue if M is well-defined, and +∞ otherwise. It holds these properties :

1. The subset S of (R * + × R) defined by S = f -1 (R) is opened and non-empty. 2. Let (α, d) ∈ S. For every α ∈ (0, α] and d ∈ [d, ∞), we also have (α , d ) ∈ S.

3. f is C ∞ and convex over S.

For every

To prove this lemma, we shall use the successive reductions leading to C (d) : reusing the notations from lemma 6.5, we shall name them

for u from A down to 1. Defining g, the function given by

we are going to prove some translation of the properties for

1 Descending process Let C be a C-process. We define its sequence τ of all-time lows recursively as τ (0) = 0 and ∀u ∈ N, τ (u + 1) = min ({t > τ (u); C(t) < C(τ (u))})

C's descending process is the concatenated process of C associated with this time sequence τ .

• It is noted C ;

• Its underlying Markovian process is noted M .

To verify that C is a C-process, one uses the proposition 5.1 only requiring τ to be canonical. It is, because the time increment τ (u + 1) -τ (u) only relies on the moves of M and C between time τ (u) and time τ (u + 1), and thanks to the Markovian property. As a consequence, we may define its transition payoffs and transition probabilities. However, we are also interested in the time increment n = τ (u + 1) -τ (u) taken by every transition : for this reason, we shall decompose C with respect to n.

Definition 7.2 Distribution of C's increments

Let C be a C-process, C be its descending process, i, j ≤ A and n ∈ N * . For any t ∈ N,

• The transition probability of M going from Ai to Aj while M goes through exactly n steps is noted

• The distribution of C(t + 1) -C(t) conditionally to the former transition is called

Its support lies in [ -Q, 0) by construction.

• Its mirror distribution is called

We shall name F i n -→j a random variable whose distribution is ν i n -→j . In particular, F i n -→j > 0 almost surely.

When considering the transitions going to the "new" state A∞ (when the binary determination sequence of the concatenation gets stuck on 0), we have

• When i = ∞, then for every n ∈ N, we have Q i n -→∞ = 0 and we set for any t ∈ N

is then +∞ almost surely by definition.

• When i = ∞, we need not define Q ∞ n -→∞ , as we may assume C(t + 1) -C(t) = +∞ almost surely. We note that when the conditions are empty because P i n -→j = 0, then these definitions may be taken arbitrarily, as they have no effect on C.

Solved convex combination

We still aim at controlling the function LT 0 : this is tantamount to controlling the value of one among the Laplace transforms L Let C be a C-process, C be its descending process, a ∈ R * + , and i, j ≤ A. Thanks to the lemma 7.2, we define

-→j the mirror conditional transition payoffs of C.

Recapitulation

We review the statements we proved during this study to end with the proposition 2.4 and the theorem 1. Starting with C a positive recurrent, bounded, not globally increasing C-process, we did the following :

1. We defined its martingale parameter α(a) and dominant eigenvectors µ (a) and w (a) at any point a ∈ R + (first a ∈ R * + , then we extended to a = 0). 2. For C periodic only :

(a) We changed C into a regular process C, with the idea of getting integer increments. As shown, α(a) is unchanged, and µ (a) and w (a) are scaled by C's offsets (the correction to apply so that it becomes regular). (b) We ensured that solving the case of regular processes allows to get the theorem 2.4 : this used the notions of breaking points, transforming the error parameters given by the lemma 6.13 into suitable terms to the theorem 2.4.

Periodic processes are solved using regular processes, so we may deem that C is either aperiodic or regular.

3. We transformed C into a decreasing process satisfying the martingale property : this is the purpose of the descending process Φ

C , whose main property is driving the convolution equation for the twisted Laplace transforms K (a) i . 4. In both cases, we proved that they converge to a common positive limit K (a) ∞ , that is continuous of a ∈ R + . We also ensured that the convergence is of the desired form :

• Uniformly exponential for regular C-processes ;

• Uniform for aperiodic C-processes.

Computing -ln K (a) i (x) leads to the final results (respectively lemma 6.13 and theorem 1) ; since K (a) ∞ is bounded away from 0 on every compact set, this preserves the convergence properties, so this ends the proofs.

Slow convergence for Liouville processes

To prove the proposition 2.6, we will use the same idea as with periodic process : we exhibit intervals over which L (a) i (herein renamed L (a) , as since C is a Lévy process, there is only one i) is constant. As a constant function is badly approximated by an affine non-constant function, this will lead us to the proposition 2.6.

Defaulting paths

To get the value of L (a) (C0), we analyze all paths C may follow on its way to default.

Definition 8.2 Defaulting paths of a Liouville process

Let C be a Liouville process.

• Its paths are determined by a length T ∈ N and a sequence P : [|1, T |] → {1, L} indicating their negated successive payoffs values.

• Such a path P is said to be a defaulting path for the starting point x ∈ R + iff following it from C(0) = x leads to T0 = T , i.e. both 1. C(T ) < 0, i.e.

127

In the sequel, we will assume that L / ∈ Q. In particular, we have L = 1, so the probability of following a path of length T exactly (at precision 0) is

Moreover, every non-trivial path may be described as follows :

• It consists in n1 ∈ N transition payoffs of -1 and n2 ∈ N transition payoffs of -L ;

-. We shall use these notations in the next paragraph.

Preservation of defaults

We want to prove the following lemma.

Lemma 8.4 Preservation of defaults

Let C be any f -Liouville process. Let k ∈ N * , and C0 ∈ R + such that

Let P be any path for C : if it is a defaulting path for the starting point C0, then it is also a defaulting path for all starting points in C0, C0 + 10 -f (k)-1

First, as C0 is the lower starting point, one is not concerned about defaults before time T , so it suffices to prove that C(T ) = C0 + v < 10 -f (k)-1 when starting from C0. Noting L = L f the f -Liouville number, the possible negated values for defaulting paths write as -v = n1 + n2L for n1, n2 ∈ N. In particular, as -v ≤ C0 + L by definition of the default time thanks to L > 1, and n1, n2 ≥ 0, then

Let us look at these -v modulo 10 -f (k) : as n1 ∈ N and

then thanks to the bound for C0, we get that

In particular, no defaulting path has a negated value in 10 -f (k)-1 , 10 -f (k) modulo 10 -f (k) . Now, as P is a defaulting path for C0, we have -v > C0 ; however, as we just proved that

and we recall that

, which ends the proof.

4. Finally, if α(0) > 0 (which is automatic if E(C) > 0), we recall that α is increasing so one may set

and then work with y(k) instead of y(a, k).

This ends the proof.

Quick convergence

We aim at proving the proposition 2.7 for a positive recurrent C-process C. Hence, let T , i, j ≤ A, u ∈ R * + and η > 0 be as given.

Local weight

During this proof, we shall use the following definitions to state the property given by hypothesis.

Definition 8.3 Local weight

Let X be a real (or ∞) random variable, η > 0 and x ∈ R.

• X is said to be η-locally heavy on x iff for every u ∈ (0, 1],

• X is said to be η-locally heavy around x iff there is γ > 0 such that X is η-locally heavy on every y ∈ (x ± γ). We also say that it is η-heavy over the interval (x ± γ).

In both cases, it is said locally heavy iff η-locally heavy for some η > 0.

We want to prove the following properties about local weight.

Lemma 8.5 Operations on local weight Let X and Y be independent random variables over R ∪ {∞}.

1. Let U ⊆ R be a compact interval such that P(X ∈ U ) > 0. There are x ∈ U and η > 0 such that X is η-locally heavy on x.

2. Let us deem that X is a-locally heavy on x and Y is b-locally heavy around y over (y ± γ). For every ξ < γ, X + Y is c-locally heavy around x + y over (x + y ± xi) with c = ab min λ, λ 2

where λ = (γ -ξ)/2.

3. If X is a-heavy over the interval (x ± η) and Y is b-heavy over the interval (y ± ), then for every ξ < η + , X + Y is heavy over the interval (x + y ± ξ).

In particular, X + Y is not necessarily heavy over the whole interval (x + y ± (η + )), take e.g. X and Y uniform distributions over [0, 1].

1. Let us set p0 = P(X ∈ U ) > 0. First, U cannot be empty, and if it a singleton then X is p0-locally heavy on its only point. Without loss of generality, we may now deem that U = [0, 1] after an affine transformation (that only modifies η by a non-zero multiplicative factor). Let us set z0 = 0 ; we define recusively the sequences (zn) n and (pn) n by