Valerie Girardin
email: valerie.girardin@unicaen.fr

Victor Konev

Serguei Pergamenchtchikov

Valérie Girardin

V Girardin

Lmno

Kullback-Leibler Approach to CUSUM Quickest Detection Rule for Markovian Time Series

Keywords: Change-point detection, CUSUM detection procedure, Ergodic Theorem of information theory, Kullback-Leibler divergence, Relative entropy, Sequential detection, Sparre Andersen models Classifications: 62L10, 62L15, 60G40, 60J10, 62B10, 94A17

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INTRODUCTION

The problem of quick detection of an abrupt change in time series arises in many different areas, related to automatic control, segmentation of signals, biomedical signal processing, quality control engineering, finance, link failure detection in communication networks, intrusion detection in computer systems, and target detection in surveillance systems; see [START_REF] Page | Continuous Inspection Schemes[END_REF], [START_REF] Basseville | Detection of Abrupt Changes: Theory and Applications[END_REF], [START_REF] Kent | On the Trial of Intrusions into Information Systems[END_REF], [START_REF] Tartakovsky | Sequential Analysis: Hypothesis Testing and Changepoint Detection[END_REF] for details and further references. A challenging application area is intrusion detection in distributed computer networks; see [START_REF] Kent | On the Trial of Intrusions into Information Systems[END_REF] and [START_REF] Tartakovsky | Change-Point Detection in Multichannel and Distributed Systems with Applications, Applications of Sequential Methodologies[END_REF]. Large scale Andersen model -widely used in insurance. In Section 5, we study the problem of the quickest detection in autoregressive models with Gaussian noise. Note that for sake of clarity and ease of reading, long proofs are postponed to the appendix.

OPTIMALITY PROPERTIES IN DETECTION PROCEDURES

In this section, we present the quickest change-point detection problems in the pointwise and minimax settings. The presentation follows the lines of the modified CUSUM procedures developed for general stochastic models in [START_REF] Lai | Informations Bounds and Quick Detection of Parameters Changes in Stochastic Systems[END_REF].

Let the time series (X k ) k≥0 be specified by the conditional density function of X k given X k-1 0 = (X 0 , . . . , X k-1 ) for k ≥ 1. The model with a disruption at time ν is given by the parametric family of conditional densities

f (ν) (x k |x k-1 0 ) = f 0 (x k |x k-1 0 )1 {k≤ν} + f 1 (x k |x k-1 0 )1 {k>ν} , (2.1)
where x k-1 0 = (x 0 , . . . , x k-1 ), together with the distribution of X 0 . The disruption time ν will be assumed to be a nonrandom unknown integer. We will denote by P (ν) the distribution of (X k ) k≥0 defined by the two families of conditional densities f 0 and f 1 and the disruption time ν, and E (ν) will denote mean under P (ν) .

The issue of sequential detection is to detect the change-point as soon as possible after it occurs. The considered procedures involve the system of CLLRS

Z j = log f 1 (X j |X j-1 0 ) f 0 (X j |X j-1 0 )
, j ≥ 1.

(2.2)

In [START_REF] Lai | Informations Bounds and Quick Detection of Parameters Changes in Stochastic Systems[END_REF], the classical CUSUM procedure is changed into the so-called window-limited CUSUM rule. For each 0 < α < 1, let M α,m be the class of admissible stopping times τ such that sup k≥1 P (0) (k ≤ τ < k + m) ≤ α.

Note that P (0) stands for the distribution with density f 0 , that α is a preassigned upper bound for false alarm probability, and m a suitably chosen window size function of α such that The problem of optimal change-point detection thus becomes the determination of a procedure which minimizes in the class M α,m the positive part detection delay risk R ν (τ ) = E (ν) (τ -ν) + , that is the mean time between the detection time τ and the true disruption time ν; note that (x) + = max(0, x).

The window-limited CUSUM procedure is defined by the stopping time

T α = inf    n ≥ 1 : max (n-mα) + ≤k≤n n j=k Z j ≥ c α    , where c α = log 2m α α . (2.3)
The optimization problem is studied under the following conditions on the CLLRS defined in (2.2).

Condition 1. Some I ∈ R * + exists such that, for any ν ≥ 0, the following almost sure (a.s.) convergence holds:

lim n→∞ 1 n ν+n j=ν+1 Z j = I, P (ν) -a.s. (2.4)
As established in Pergamenchtchikov and Tartakovsky (2016, Theorem 5), Condition 1 implies for the delay risk the sharp lower bound

lim inf α→0 1 | ln α| inf τ ∈Mα R ν (τ ) ≥ 1 I . (2.5)
In order to derive the upper bound for the detection procedure (2.3), a stronger condition is required.

Condition 2. Some I ∈ R * + exists such that, for any δ > 0,

lim n→∞ sup k≥ν essup P (ν)   k+n j=k+1 Z j < (I -δ)n F k   = 0. (2.6)
where

F k = σ{X 1 , . . . X k }.
As proven in Lai (1998, Theorem 4 (ii)), Condition 2 provides the following sharp upper bound for the average time between the true disruption time ν and the detection time T α ,

lim sup α→0 1 | ln α| sup ν≥0 R ν (T α ) ≤ 1 I .
(2.7)

KLD APPROACH TO THE LIMIT OF THE NORMALIZED CLRRS

Here naturally arise questions about the assumptions to be fulfilled by a time series to ensure the validity of Conditions 1 and 2, and in which way the limit I can be evaluated from observations. This section aims at providing conditions of existence and closed-form formulae for the case of Markov chains, and more particularly for autoregressive models. We will mainly use the geometric ergodicity properties of Markov chains together with concepts of entropy and information theory; see [START_REF] Gray | Entropy and Information Theory[END_REF] and [START_REF] Girardin | On the Different Extensions of the Ergodic Theorem of Information Theory, Recent Advances in Applied Probability[END_REF]. Even though autoregressive models constitute a subclass of Markov chains, specific methods and tools have to be developed in this case; see [START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF] and [START_REF] Galtchouk | Geometric Ergodicity for Classes of Homogeneous Markov Chains[END_REF]. Section 3.1 will develop an approach based on information theory for investigating the asymptotic convergence of the normalized CLLRS. Condition 1 will be shown to amount to a particular case of the classical AEP -also called Shannon-McMillan-Breiman theorem, or ergodic theorem of information theory; see [START_REF] Cover | Elements of Information Theory[END_REF]. Thus, for Markov chains of any fixed order, the limit I will appear as the KLD rate between two random sequences with the conditional distributions before and after the disruption. In Section 3.2, the limit will be shown to take the form of an explicit function of the model parameters. In Section 3.3, for autoregressive models, assumptions on the existence of the limit I will be specifically checked, yielding a closed-form formula for I too. The change-point detection procedure will then be studied in full details, for Sparre Andersen models in Section 4 and autoregressive models in Section 5.

The Ergodic Theorem of Information Theory

Let us first recall some basics concerning entropy and divergence for random sequences. Entropy measures randomness or uncertainty in a random phenomenon. Its introduction into probability theory in [START_REF] Shannon | A Mathematical Theory of Communication[END_REF] gave rise to information theory, by measuring the information content of sources. The Shannon (or differential) entropy of a random vector X n-1

0 is S(X n-1 0 ) = S(f ) = - R n f (x) log f (x)dx
, if its distribution is absolutely continuous with respect to the Lebesgue measure Λ, with density f . An extension of the definition leads to consider the relative entropy

S g (f ) = - R n log f (x) g(x) f (x)dx,
if the support of f is included in the support of g, otherwise set as +∞. This is the entropy of a measure with density f relative to a measure with density g -both absolutely continuous with respect to Λ. When both f and g are densities of probability measures on R n , the Kullback-Leibler divergence (KLD) of f with respect to g, also called neg-entropy and introduced in [START_REF] Kullback | On Information and Sufficiency[END_REF], is the opposite number of the entropy of f relative to g, precisely

K(f |g) = R n log f (x) g(x) f (x)dx = -S g (f ).
Thus, the entropy of a random vector X n-1 0 with density f relative to a measure with density g with respect to Λ on R n is defined as

S g (X n-1 0 ) = S g (f ).
Similarly, the KLD of a random vector X n-1 0 with density f with respect to a random vector Y n-1 0 with density g on R n is defined as K(X n-1

0 |Y n-1 0 ) = K(f |g).
The KLD measures dissimilarity between two random phenomena. Although it is not a true distance because of lack of symmetry and triangular inequality -see [START_REF] Cover | Elements of Information Theory[END_REF], it constitutes a good tool for discriminating two distributions because

K(f |g) ≥ 0, with K(f |g) = 0 if and only if f = g. (3.1)
Let f (x, y) and g(x, y) denote densities (with respect to Λ) on R m+n , with conditional densities denoted by f (y|x) and g(y|x). The conditional KLD of f with respect to g is usually defined as the real number

K(f (y|x)|g(y|x)) = R m+n f (x, y) log f (y|x) g(y|x) dxdy (3.2) = R m f (x) R n f (y|x) log f (y|x) g(y|x) dydx = E f [K(f, g; x)],
where the function of x

K(f, g; x) = R n f (y|x) log f (y|x) g(y|x) dy (3.3) is called local divergence (in x).
The marginal Shannon entropy of order n of a sequence X = (X n ) n≥0 taking values in (R, B(R)) is defined as the entropy of its n-dimensional marginal distribution, namely

S(X n-1 0 ) = -E f [log f (X n-1 0 )] = - R n f (x n-1 0 ) log f (x n-1 0 )dµ(x n-1 0 ),
where f is the density of the random vector X n-1 0 with respect to the n-dimensional marginal of a reference measure µ on the infinite product space (R N , B(R N )). Further, the Shannon entropy rate of a process X is defined by H(X) = lim n 1 n S(X n-1 0 ) when the limit exists; see Cover and Thomas (1991, Chapter 9) and Gray (2011, Chapter 8). The convergence in probability of -1 n log f (X n-1 0

) to H(X) constitutes the weak AEP while the almost sure convergence is called the strong AEP -or ergodic theorem of information theory, or Shannon-MacMillan-Breiman theorem from the authors who originally proved it.

For any ergodic random sequence X whose distribution P X is absolutely continuous with respect to Λ on R N , with density f , the extension of AEP to relative entropy says that

-1 n log[ f g (X n-1 0 )] converges to H g (X)
, where convergence holds in probability, in mean, or a.s. with respect to P X , for any reference measure absolutely continuous with respect to Λ on R N , with density g; see [START_REF] Barron | The strong Ergodic Theorem for Densities: Generalized Shannon-McMillan-Breiman Theorem[END_REF] and [START_REF] Girardin | On the Different Extensions of the Ergodic Theorem of Information Theory, Recent Advances in Applied Probability[END_REF] and the references therein for details. Note that, for sake of simplification, f or g here denote as well the density of measures on R N as the density of all their marginals on R n for n ∈ N * .

Similarly, let X = (X n ) n≥0 and Y = (Y n ) n≥0 be two ergodic random sequences whose distributions are absolutely continuous with respect to Λ on R N , with respective densities f and g. Averaging per time unit leads to the definition of the KLD rate

D(X | Y) = lim n→∞ 1 n K(X n-1 0 |Y n-1 0 ) = -H g (X),
when the limit exists. Extended forms of AEP have been proven to hold for large classes of ergodic processes; see [START_REF] Girardin | On the Different Extensions of the Ergodic Theorem of Information Theory, Recent Advances in Applied Probability[END_REF] for a detailed review. In particular, the strong AEP was extended to Borel state spaces independently by [START_REF] Barron | The strong Ergodic Theorem for Densities: Generalized Shannon-McMillan-Breiman Theorem[END_REF] and [START_REF] Orey | On the Shannon-Perez-Moy Theorem[END_REF]. For full details on AEP and KLD rates in the special case of of Markovian sequences, see Algoet and Cover (1988, Theorem 4) and Gray (2011, Corollary 8.4.1; in short, if both X and Y are ergodic Markov chains of finite order, with respective transition densities f and g, then their KLD rate is well-defined and the following a.s. convergence holds with respect to the distribution of X:

1 n log f g (X n-1 0 ) -→ D(X | Y).
(3.4)

Application to the CLLRS for Markov Chains

In Section 2, the quantity I of interest in both Conditions 1 and 2 is the limit of the normalized CLLRS 1 n ν+n j=ν+1 Z j , with Z j defined in (2.2). As remarked in [START_REF] Lai | Informations Bounds and Quick Detection of Parameters Changes in Stochastic Systems[END_REF], for the trivial case of i.i.d. observations, I is the KLD K(f 1 |f 0 ) between the post-change and pre-change distributions f 1 (x) and f 0 (x) of X n . For Markovian dependent observations, the next result states that I is the KLD rate between two well-identified sequences.

Lemma 3.1. Let the time series X = (X n ) n≥0 be a model with a disruption at time ν, given by the parametric family of conditional densities (2.1). Let X be a time series with distribution given by the conditional densities f 1 (x k |x k-1 0 ). Let X be another time series, with conditional densities f 0 (x k |x k-1 0 ). Suppose that both X and X are ergodic homogeneous Markov chains of order d.

Then Condition 1 is fulfilled, with I = D( X| X).

Proof. We can write

ν+n j=ν+1 Z j = log   ν+n j=ν+1 f 1 (X j |X j-1 0 ) f 0 (X j |X j-1 0 )   = log n j=1 f 1 (X ν+j |X ν+j-1 0 ) n j=1 f 0 (X ν+j |X ν+j-1 0 ) .
For any density f defining a Markov chain of order d, we compute

n j=1 f (x ν+j |x ν+j-1 0 ) = n j=1 f (x ν+j |x ν+j-1 ν+j-d ) = n j=v+1 f (x ν+j |x ν+j-1 ν+1 ) d j=1 f (x ν+j |x ν+j-1 ν+j-d ).
For any density f and time ν, we can also write

f (x ν+n ν ) = f (x ν+n |x ν+n-1 ν )f (x ν+n-1 ν ) = f (x ν+n |x ν+n-1 ν )f (x ν+n-1 |x ν+n-2 ν )f (x ν+n-2 ν ) = • • • = n j=1 f (x ν+j |x ν+j-1 ν ) • f (x ν ).
Therefore, for a Markov chain of order d, we get n j=d+1 f (

x ν+j |x ν+j-1 ν+1 ) = f (x ν+n ν+1 )/f (x ν+d ν+1 ). Specifi- cally ν+n j=ν+1 Z j = log f 1 (X ν+n ν+1 ) f 0 (X ν+n ν+1 ) + h(ν, d),
where h(ν, d) is a quantity depending on ν and d but not on n, and hence h(ν, d)/n converges to 0. Therefore, 1 n ν+n j=ν+1 Z j converges to the same limit as 1 n log f 1 (X ν+n ν+1 )/f 0 (X ν+n ν+1 ) . By the strong AEP -see Gray (2011, Corollary 8.4.1) and (3.4), the latter converges to I = D(X| X), which proves the result.

2 The first explicit formula for entropy rates was already derived in [START_REF] Shannon | A Mathematical Theory of Communication[END_REF] for ergodic homogeneous Markov chains of order 1 with finite state spaces. Further, let X be a Markov chain taking values in R, with transition density f (x|y) and stationary distribution with density π(x). Then -1 n log f (X n-1 0 ) converges almost surely to the entropy rate H(X) = -R π(y) R f (x|y) log f (x|y)dx dy; see for instance [START_REF] Girardin | Entropy for Semi-Markov Processes with Borel State Spaces: Asymptotic Equipartition Properties and Invariance Principles[END_REF], where the extension to KLD is also proven to hold for semi-Markov processes. Considering straightforward extension of the above formula for Markov chains of order d together with Lemma 3.1 yields the following result for a Markov chain with a disruption.

Theorem 3.1. Let X be an ergodic real Markov chain of order d with disruption time ν, with transition density

f (ν) (x n |x n-1 0 ) = f 0 (x n |x n-1 n-d )1 {k≤ν} + f 1 (x n |x n-1 n-d )1 {k>ν} , n ≥ d.
Let π 1 (x) denote the density of the stationary distribution for the Markov chain X of order d, with transition density f 1 (x|y d-1 0 ). Let similarly X be an ergodic homogeneous Markov chain of order d, with transition density f 0 (x|y d-1 0 ). Then Condition 1 holds true, with

I = D( X | X) = R d π 1 (y d-1 0 ) R f 1 (x|y d-1 0 ) log f 1 (x|y d-1 0 ) f 0 (x|y d-1 0 ) dx dy 0 . . . dy d-1 . (3.5)
Note that in [START_REF] Fuh | SPRT and CUSUM in hidden Markov models[END_REF] a KLD number between Markov transition kernels is defined for proving optimality properties of CUSUM procedures in hidden Markov models of order 1. In particular, a formula similar to (3.5) is given for a Markov chain of order 1, but no link to the AEP and KLD rates is highlighted.

Autoregressive Models

For a scalar autoregressive sequence of order p with a disruption, we will provide an explicit expression for I in terms of the Fisher matrix of the model. In this aim, we will consider the associated vector valued autoregressive sequence of dimension p, that is a Markov chain of order 1.

Precisely, let X = (X k ) k≥1 be a random sequence governed by the autoregressive recursive equations of order p

X k = α 1,k X k-1 + . . . + α p,k X k-p + ε k k ≥ 1, (3.6)
where α i,k = a i before the change-point time ν (for k ≤ ν), and α i,k = θ i after ν, that is to say for k > ν. Here (ε k ) k≥1 is Gaussian white noise -an i.i.d. standard Gaussian sequence, and the initial vector

(X 1-p , . . . , X 0 ) is independent of (ε k ) k≥1 . Setting Φ k = (X k , . . . , X k-p+1
) , a = (a 1 , . . . , a p ) and θ = (θ 1 , . . . , θ p ) , the model (3.6) becomes

X k = Φ k-1 a 1 {k≤ν} + θ 1 {k>ν} + ε k , k ≥ 1. (3.7)
In this case the conditional densities (2.1) take the form

f 0 (x k |x k-1 1 ) = 1 √ 2π e - (x k -Φ k-1 a) 2 2 and f 1 (x k |x k-1 1 ) = 1 √ 2π e - (x k -Φ k-1 θ) 2 2 .
Since the CLLRS (2.2) can be written

Z k = X k Φ k-1 (θ -a) - 1 2 (Φ k-1 θ) 2 -(Φ k-1 a) 2 , (3.8)
the ergodic properties of Φ = (Φ k ) k>ν will induce a closed-form formula for I, involving the Fisher matrix of the model. Indeed, the vector valued process Φ is a Markov chain of order 1, and is governed after the change-point by the multivariate autoregressive equation of order 1,

Φ k = AΦ k-1 + ξ k k ≥ ν + 1 , (3.9)
where ξ k = (ε k , 0, . . . , 0) ∈ R p and

A = A(θ) =        θ 1 . . . . θ p 1 0 . . . . 0 . . . . . . 0 . . . 0 1 0       
.

Applying (3.9) repeatedly yields

Φ k = A k-ν Φ ν + k j=ν+1 A k-j ξ j for k ≥ ν + 1, and hence, for any v ∈ R p , E Φ k Φ k |Φ ν = v = A k-ν v v (A ) k-ν + k-ν-1 l=0 A l B (A ) l , (3.10)
where the prime denotes transpose, and B = (b ij ) 1≤i,j≤p is the matrix with b ij = 1 if (i, j) = (1, 1) and 0 otherwise.

In the sequel we will denote by Θ stb the set of all θ ∈ R p for which all eigenvalues of A are less than one in modulus, and say accordingly that Φ -or X -is a stable sequence. If θ ∈ Θ stb , then Φ is ergodic, the Fisher matrix F (θ) = F of the model (3.9) is well-defined, and, for any v ∈ R p ,

lim k→∞ E Φ k Φ k | Φ ν = v = E Φ ∞ Φ ∞ = l≥0 A l B (A ) l = F, (3.11)
where Φ ∞ = l≥1 A l-1 ξ l ∼ N p (0, F (θ)) is the ergodic distribution of Φ k ; see for example Basseville and Nikiforovbook (1993). Moreover the following strong law of large numbers, proven in the appendix, holds true. Φ j Φ j = F, P νa.s..

Condition 1 follows, with explicit limit.

Theorem 3.3. Let X be governed by the model (3.6). Then Condition 1 holds true for all θ ∈ Θ stb and any ν ≥ 0. Moreover, the limit in (2.4) is

I = K(f θ (x|v)|f a (x|v)) = 1 2 (θ -a) F (θ) (θ -a).
(3.12)

with the KLD defined in (3.2) for the conditional densities defined in (3.8).

Proof. Equation (3.9) implies that

ν+n j=ν+1 Z j (θ) = (θ -α) M n + 1 2 (θ -a) ν+n-1 j=ν Φ j Φ j (θ -a) , (3.13)
where M n = ν+n j=ν+1 Φ j-1 ε j . The Chebyshev inequality easily induces that for any δ > 0

n≥1 P (|M n | > δn) ≤ δ -4 sup k≥1 E M 4 k k 2 n≥1 1 n 2 .
Since the process (M n ) n≥1 is a martingale, using successively the Burkholder-Gundy inequality and the Cauchy-Schwarz inequality proves that some constant C * > 0 exists such that

E M 4 n ≤ C * E   ν+n j=ν+1 |Φ j-1 | 2 ε 2 j   2 ≤ C * n ν+n j=ν+1 E |Φ j-1 | 4 ε 4 j = C * E   ε 4 1 n ν+n j=ν+1 E |Φ j-1 | 4   .
From (3.11), it is easy to deduce that sup k≥ν+1 E |Φ k | 4 is finite, where |x| denotes the Euclidean norm of the vector x ∈ R p . Therefore,

lim n→∞ |Φ n | 2 n = 0 and lim n→∞ 1 n M n = 0, P ν -a.s., (3.14)
where the second limit follows from sup n≥1 (E M 4 n )/n 2 < ∞, thanks to the Borel-Cantelli lemma. Equation (2.4) follows directly from Theorem 3.2, and the convergence is proven. Finally, (3.12) is given in Example 6.5 of [START_REF] Pergamenchtchikov | Asymptotically Optimal Pointwise and Minimax quickest Change-Point Detection for Dependent Data, Statistical inference for stochastic processes[END_REF], with the link to KLD detailed for an autoregressive model of order 1 in Example 6.2. 2

SPARRE ANDERSEN MODELS

In this section we study the quickest detection problem for a specific case of the discrete time Sparre Anderson models that is widely used in insurance; see for example Asmussen and Albrecher (2010, p. 427) and [START_REF] Back | Stochastic Methods in Finance[END_REF]. We will show that, after the disruption, this observation process is a geometric ergodic homogeneous Markov chain under some technical assumptions. We will explicitly compute its stationary distribution, and finally obtain a closed form expression for the limit of the normalized CLLRS through specific means. Let (J k ) k≥0 be an autoregressive process of order 1 with disruption time ν, governed by the recursive equation

J k = (a 0 1 {k≤ν} + a 1 1 {k>ν} )J k-1 + ε k , k ≥ 1, (4.1)
where the real parameters a 0 = a 1 are known and the initial variable J 0 is independent of the Gaussian white noise (ε k ) k≥1 . Let N k = m≥1 1 (Sm≤k) , where N 0 = 0 and the S m are sums of m i.i.d. exponential random variables with parameter λ > 0. The renewal process (N k ) k≥0 and the sequence (J k ) k≥0 are independent. The process is assumed to be observed only through

J k = J N k for k ≥ 0.
The problem is to construct a detection rule for ν on the basis of the observations

X k = J σ k , k ≥ 0, (4.2)
where σ 0 = 0 and

σ k = inf{l ≥ σ k-1 + 1 : N l -N σ k-1 > 0} for k ≥ 1. Note that (σ k ) k≥0
is an increasing sequence of stopping times with respect to the filtration (σ(N k , k ≤ n)) n≥0 . The shift from the sequence ( J k ) to the subsequence ( J σ k ) is necessary because, in the case of equal values of N k , the corresponding values of J k coincide, and hence carry no information on the disruption; therefore such J k must be excluded from the detection rule.

For this construction, we need to study the observation process (4.2) after ν, that is (X k ) k>ν . Applying (4.1) repeatedly for k > ν, we get J m+k = a k 1 J m + k j=1 a k-j 1 ε j+m , and hence (X k ) k>ν is governed by the recursive equation

X k = θ k X k-1 + η k , k ≥ ν + 1, (4.3) where η k = s k j=1 a s k -j 1 ε j+σ k-1 , s k = σ k -σ k-1 and θ k = a s k 1 . The associated conditional densities are f (x, y; a i ) = f i (y|x) = k≥1 γ k φ i,k (y, x) , i = 0, 1 , (4.4)
where γ k = (1 -e -λ )e -λ(k-1) and

φ i,k (y, x) = 1 ς i,k √ 2π e - (y-a k i x) 2 2ς 2 i,k with ς 2 i,k = 1 -a 2k i 1 -a 2 i . (4.5)
Now let us compute the transition probability function for the random sequence governed by (4.3).

Theorem 4.1. The sequence (X k ) k>ν is a homogeneous Markov chain of order 1, with transition function given for any Borel set B ⊂ R by

P (x, B) = B f 1 (y|x)dy, (4.6)
where f 1 is given by (4.4) for i = 1.

Proof. First, let us check that (s k ) is an i.i.d. sequence. We compute

P(s 1 = m 1 , . . . , s j = m j ) = E P(s 1 = m 1 , . . . , s j = m j |F σ j-1 ) = E [1 (s 1 =m 1 ) . . . 1 (s j-1 =m j-1 ) ] P(s j = m j |F σ j-1 ),
where F 0 = {∅, Ω}, and

F k = σ{X 1 , . . . , X k } for k ≥ 1. It follows from the definition of σ j that P(s j = m j |F σ j-1 ) = P(N σ j-1 = • • • = N σ j-1 +m j -1 , N σ j-1 +m j > N σ j-1 |F σ j-1 ) = (1 -e -λ )e -λ(m j -1) .
Thus, X is a homogeneous Markov chain with transition probability function given by

P(x, B) = P(a s 1 1 x + t 1 j=1 a s 1 -j 1 ε j+1 ) = k≥1 P(a k 1 x + k j=1 a k-j 1 ε j+1 ∈ B) P(s 1 = k) = k≥1 γ k B φ 1,k (y, x)dy ,
and the desired result follows. 2

Lemma 4.1. The Markov chain X is geometrically ergodic if and only if |a 1 | < 1.

Proof. Let us check the assumptions of Theorem A.1 from the appendix. Let V be defined by

V (x) = c * (1 + x 2 ) , (4.7)
for some parameter c * ≥ 1.

Then E x [V (X 1 )] = c * 1 + E x X 2 1 = c * 1 + E x (θ 1 x + η 1 ) 2 = c * 1 + x 2 E (θ 2 1 ) + E (η 2 1 ) .
It is easy to check that E (η 2 1 ) = (1 -a 2 e -λ ) -1 and

E (θ 2 1 ) = m≥1 a 2m 1 P(s 1 = m) = (1 -e -λ ) a 2 1 1 -a 2 1 e -λ := a 1 < 1 . 10 Therefore, for any a 1 < ρ < 1, E x [V (X 1 )] = c * a 2 1 e λ -a 2 1 + a 1 x 2 ≤ c * + a 1 V (x) ≤ ρV (x), x ∈ C,
where c * = c * (e λ -a 2 1 ) -1 a 2 1 and

C = x ∈ R : |x| > c * c * (ρ -a 1 ) -1 .
Consequently, both Conditions D 1 and D 2 of Theorem A.1 hold for the set C. Applying this theorem yields that X is geometrically ergodic.

2 Lemma 4.2. If |a 1 | < 1, then the stationary distribution of the Markov chain (X k ) k>ν is centered Gaussian with variance (1 -a 2 1 ) -1 .
Proof. Let S = σ{N t , t ≥ 0} be the σ-algebra generated by the renewal counting process (N t ) t≥0 . We get by induction from (4.3) that

X n =   n j=ν+1 θ j   X ν + n k=ν+1 η k n j=k+1 θ j , n > ν, (4.8)
where an empty product is set to 1 by convention, and η k is S-conditionally centered Gaussian distributed with variance

∆ k = s k j=1 a 2(s k -j) 1 = 1 -a 2s k 1 1 -a 2 1 .
Since (η n ) n>ν is a sequence of S-conditionally independent random variables, the distribution of X n for n > ν is S-conditionally Gaussian distributed, with mean

m n = E (X n |S) = n j=ν+1 θ j E (X ν |S) = a 2(σn-σν ) 1 E (X ν |S),
and variance

d n = E X n -E (X n |S) 2 S = n k=ν+1 ∆ k n j=k+1 θ j = 1 -a 2σn-σν 1 1 -a 2 1 . (4.9)
Moreover m n converges a.s. to 0 and d n to (1 -a 2 1 ) -1 when n tends to infinity. Thus, both the conditional distribution of X n and its ordinary one are asymptotically centred Gaussian distributed with variance (1a 2 1 ) -1 . Therefore the candidate for the stationary distribution of the process is the probability measure defined by

π(B) = 1 -a 2 1 2π A e -(1-a 2 1 )u 2 /2 du, B ∈ B(R). (4.10)
Finally, it is straightforward to show that π(B) = R P (x, B)π(dx) for any Borel set B ∈ B(R), where P (x, B) is given in (4.6).

2 Thanks to (4.6), the conditional densities in (2.1) take the form (4.4), or f i (x j |x j-1 0 ) = f (x j-1 , x j ; a i ), for i = 0, 1, and the CLLRS in (2.2) takes the form Z j = g(X j-1 , X j ), where g(x, y) = log f (x, y; a 1 ) f (x, y; a 0 ) . (4.11)

The following properties of the conditional KLD are necessary for obtaining a closed-form formula for I.

The proof is postponed to the appendix.

Lemma 4.3. Let f 1 (y|x) and f 0 (y|x) be given by (4.4).

1. If a 1 = a 0 , then the local divergence K(f 1 , f 0 ; x) defined by (3.3) is positive for all x ∈ R.

2. Let V be defined by (4.7). If

c * ≥ 6(1-a 2 1 ) -1 +2 log √ 2π-2 log(1-e -λ ), then K(f 1 , f 0 ; x) ≤ V (x) for all x ∈ R.
It follows from Lemma 4.3 that the local divergence satisfies

K(f 1 , f 0 ; x) ≤ 6(1 -a 2 1 ) -1 + 2 log √ 2π -2 log(1 -e -λ ) .
Condition 1 can then be deduced from Theorem 3.1 and a closed-form expression for the limit derives from (3.5). Still, the following given proof is specific to the case under study.

Theorem 4.2. Let X be a Sparre-Anderson model governed by (4.3). Then for any ν ≥ 0, Condition 1 is satisfied, with

I = R π(x)K(f 1 (y|x) | f 0 (y|x))dx = 1 -a 2 1 2π R K(f 1 , f 0 ; x)e -(1-a 2 1 )
x 2 /2 dx, P (ν)a.s..

where the conditional densities are given in (4.4).

Proof. We can write

ν+n j=ν+1 Z j = ν+n j=ν+1 U j + ν+n j=ν+1 g(X j-1 ) = A n + B n ,
where U j = g(X j-1 , X j ) -g(X j-1 ) and g(x) = E x [g(x, X 1 )] = R g(x, y)f 1 (y|x)dy. Let us show that, P (ν) -a.s., B n /n converges to K(f 1 |f 0 ) and A n /n converges to 0 as n tends to infinity.

First, applying the strong law of large numbers -see Meyn and Tweedie (1993, Theorem 17.1.7) -to the series B n = ν+n j=ν+1 g(X j-1 ), for n ≥ 1, yields

lim n→∞ 1 n B n = R g(y)dπ(y) = K(f 1 |f 0 ) P (ν) -a.s..
Second, (U j ) j>ν is a martingale-difference with respect to the natural filtration F j = σ(X j 0 ) for j > ν. Therefore A n /n will converge to 0 if j>ν 1 j 2 E (U 2 j |F j-1 ) < ∞ P (ν)a.s.. (4.12)

Thanks to the bound (A.5), we get

E (U 2 j |F j-1 ) ≤ E [g 2 (X j-1 , X j )|F j-1 ] ≤ E [(u * + X 2 j-1 + X 2 j ) 2 |F j-1 ] ≤ 3 (u * ) 2 + X 4 j-1 + E X 4 j |F j-1 .
Thanks to the latter inequality, it is sufficient for proving (4.12) to show that

sup n>ν E (X 4 n |F ν ) < ∞ . ( 4
E (Y 4 n |F ν ) < ∞ .
For n > ν, applying the Hölder inequality yields

Y 4 n ≤   n k=ν+1   |η k | n j=k+1 |θ j | 1/2     n j=k+1 |θ j | 1/2     4 ≤ c n n k=ν+1 η 4 k n j=k+1 θ 2 j ,
where c n = n k=ν+1 n j=k+1 |θ j | 2/3 3 . Therefore, thanks to (4.9),

E Y 4 n |S ≤ 3c n n k=ν+1 ∆ 2 k Π n j=k+1 θ 2 j ≤ 3c n 1 -a 2 1 n k=1 ∆ k Π n j=k+1 θ 2 j ≤ 3c n (1 -a 2 1 ) 2 • Since σ j -σ j-1 ≥ 1 in (4.2), we get c n ≤ ( k≥1 |a 1 | 2k/3
) 3 < ∞, and (4.13) is proven. Finally, the series (4.12) is convergent, and hence A n /n converges P ν -a.s. to 0. 2

OPTIMALITY PROPERTIES FOR AUTOREGRESSIVE MODELS

We will here prove the optimality of the window-limited CUSUM procedure (2.3) for the autoregressive model (3.6) for which both pre-change and post-change parameters are known. We will apply the detecting procedures proposed in Lai (1998) and shortly described in Section 2. In this aim, we will develop analytical tools leading to the conditions that provide the lower and upper bounds for the detection delay. First the detection delay has an asymptotic upper bound, because Condition 2 is satisfied, as stated in the following proposition, whose proof is given in the appendix.

Theorem 5.1. If the autoregressive model (3.6) is stable -that is for θ ∈ Θ stb , then Condition 2 is satisfied.

In Lai (1998, Theorem 4 (ii)), the upper bound (2.7) is proven to hold true provided that the system of statistics (3.8) satisfies (2.6). Therefore, the lower bound (2.5) together with the upper bound (2.7) imply the minimax property of the CUSUM procedure, as stated below.

Theorem 5.2. If the autoregressive model (3.6) is stable, then the detection delay risk of the windowlimited procedure (2.3) based on the statistics (3.8) is upper bounded as in (2.7), with I given by (3.12), and the procedure is asymptotically minimax with

lim α→0 inf τ ∈Mα sup ν≥0 R ν (τ ) sup ν≥0 R ν (T α ) = 1 .
Moreover, this procedure is pointwise optimal. In mathematical words, for any ν ≥ 0,

lim α→0 inf τ ∈Mα R ν (τ ) R ν (T α ) = 1.
we obtain for the matrix Ψ n the linear equation

Ψ n -AΨ n A = D n . (A.1)
Thanks to both (3.14) and the strong law of large numbers, we get that 1 n D n converges P ν -a.s. to the matrix B in (3.10).

A vector operation vec(•) is defined by setting vec(V ) = (v 1,1 , . . . , v p,1 , . . . , v 1,p , . . . , v p,p ) ∈ R p 2 , for any p × p matrix V . Thus we can write (A.1) as

(I -A ⊗ A)vec(Ψ n ) = vec(D n ), (A.2)
where U ⊗ V = (u i,j v k,l ) 1≤i,jk,l≤p is the p 2 × p 2 matrix called the Kronecker product of the two p × p matrices U = (u i,j ) 1≤i,j≤p and V = (v i,j ) 1≤i,j≤p ; see, for example, Feigin and Tweedie (1985, Section 3.3) One can check directly that the eigenvalues of the matrix A ⊗ A are (λ i λ j ) 1≤i,j≤p , where (λ i ) 1≤i≤p are the eigenvalues of A. If |λ i | < 1 for all i, then |λ i λ j | < 1 for all (i, j) too. Therefore, (A.2) has a unique solution which can be written 1. Thanks to (3.1), it is sufficient to show that, for all x ∈ R, some set Γ(x) with positive Lebesgue measure exists such that f 1 (y|x) = f 0 (y|x) in (4.4) for all y ∈ Γ(x). Let us suppose, without loss of generality, that a 1 > a 0 and show that φ 1,k (y, x) > φ 0,k (y, x), k ≥ 1.

(A.4)

According to (4.5), the above inequality amounts to y 2 A 1,k + yA 2,k + A 3,k > 0, where

A 1,k = 1 2 1 ς 2 0,k - 1 ς 2 1,k , A 2,k = x a k 0 ς 2 0,k - a k 1 σ 2 1,k
, and

A 3,k = x 2 2 a k 0 ς 2 0,k - a k 1 ς 2 1,k -log ς 2 1,k ς 2 0,k .
Since A 1,k > 0 for all k, (A.4) will follow for example if y > y k (x), where

y k (x) = |A 2 2,k -4A 1,k A 3,k | -A 2,k 2A 1,k .
Note that, if the discriminant A 2 2,k -4A 1,k A 3,k is negative, then the inequality (A.4) holds for any y ∈ R. Taking into account that lim k→+∞ A 1,k = (a 2 1 -a 2 0 )/2, lim k→∞ A 2,k = 0 and lim k→∞ A 3,k = -log(1a 2 0 )/(1 -a 2 1 ), we get lim k→∞ y k (x) = 2 2(a 2 1 -a 2 0 ) log(1 -a 2 0 )/(1 -a 2 1 ) a 2 1 -a 2 0 .

Therefore, for any x ∈ R, z * (x) = sup k≥1 y k (x) < ∞, and the interval Γ(x) = (z * (x), +∞) is a suitable choice. On this set, (A.4) holds, and hence, f 1 (y|x) > f 0 (y|x), as required. Point 1. follows.

It is easy to check that g(•) is lower bounded, that is g(t) ≥ - Further, since θ ∈ Θ stb , the limit of V n exists when n tends to infinity. Precisely, V n converges to l≥0 (A ) l θ θ A l , and, for each 1 ≤ j ≤ p, the sequence (v j,n ) n≥1 is increasing with a finite limit.

Finally,

E sup n≥1 |U n | ≤ |θ| k≥1 E |u k | |(A ) k-1 | ≤ |θ| sup k≥1 E (|u k |) k≥1 |(A ) k-1 | < ∞ .
Together with (A.11), this yields the desired result with

ζ * = -inf n≥1 p j=1 β 2 j,n
v j,n 1 {v j,n >0} .

2

Theorem 3. 2 .

 2 If θ ∈ Θ stb for the model (3

  vec 1 n Ψ n = (I -A ⊗ A) -1 vec 1 n D n . (A.3)Obviously, the Fisher matrix F defined in (3.11) satisfies the linear equation F -AF A = B, that can be rewritten as vec(F ) = (I -A ⊗ A) -1 vec(B) . Letting n tend to infinity in (A.3) yields the desired convergence. 2A.3. Proof of Lemma 4.3

ACKNOWLEDGEMENTS

This work is partly supported by the French Federation Normandie Mathématiques, by the RSF grant 14-49-00079 (National Russian Research University "MPEI" 14 Krasnokazarmennaya, 111250 Moscow, Russia), and by the RFBR Grant 16-01-00121 A.

APPENDIX

A.1. Geometric Ergodicity for Homogeneous Markov Processes

We will follow the so-called Meyn-Tweedie approach; some definitions from [START_REF] Meyn | Markov Chains and Stochastic Stability[END_REF] and [START_REF] Galtchouk | Geometric Ergodicity for Classes of Homogeneous Markov Chains[END_REF] are necessary.

For a homogeneous Markov chain (X n ) n≥0 with measurable state space (X , B(X )), the transition probability of the chain is defined by P (x, A) = P(X 1 ∈ A|X 0 = x) for all x ∈ X and A ∈ B(X ). The n-step transition probability is P n (x, A) = P(X n ∈ A|X 0 = x). A measure π on B(X )) is said to be invariant -or stationary, or ergodic -for the chain if π(A) = X P (x, A)π(dx) for all A ∈ B(X ). If an invariant positive measure π with µ(X ) = 1 exists, then the chain is said to be positive.

The following conditions are minorization and drift conditions: D 1 . There exist δ > 0, a set C ∈ B(X ), and a probability measure

The following theorem is used in the proof of Lemma 4.1 in Section 4; see [START_REF] Feigin | Random Coefficient Autoregressive Processes: a Markov chain Analysis of Stationarity and Finiteness of Moments[END_REF], and also [START_REF] Galtchouk | Geometric Ergodicity for Classes of Homogeneous Markov Chains[END_REF].

Theorem A.1. Let (X n ) n≥0 be a homogeneous Markov chain satisfying conditions D 1 and D 2 with the same set C ∈ B(X ). Then (X n ) n≥0 is a positive geometric ergodic process, i.e.,

Note that the positive constants κ * and R * are determined in [START_REF] Galtchouk | Geometric Ergodicity for Classes of Homogeneous Markov Chains[END_REF].

A.2. Proof of Theorem 3.2

Setting Ψ n = ν+n j=ν+1 Φ j Φ j , we obtain from (3.9) that

where M n = ν+n j=ν+1 Φ j-1 ξ j is the p × p matrix whose first column is the vector M n defined in (3.13) and all other columns are null. So, if we set

2. Since ς i,1 = 1 and ς 2 i,k ≥ 1 for all k ≥ 2, we get for all x and y,

where u * = 2 log √ 2π -2 log(1 -e -λ ) and g is defined in (4.11). Further,

The above equality jointly with (A.6) yield the desired result, precisely

First note that, thanks to the time homogeneity of the process (X j ) j>ν , for any k ≥ ν,

where P (0) denotes the probability measure when the disruption occurs at time 0. In order to simplify the analysis of the right hand side conditional probability in (A.7), let us introduce, for any initial value v ∈ R p , the autoregressive process

Note that ( Φ k ) k≥1 satisfies (3.9) with initial value zero. Substituting X k from (3.7) into (3.8), we get

where θ = θ -a. For the process ( Z k ) k≥1 with initial value zero associated to (Z k ) k≥1 , we have

Moreover, the function g is bounded as stated in Lemma A.1 below.

Using Theorem 3.3 and this bound will allow us to prove that for any δ > 0 lim n→∞ sup

Indeed, thanks to Lemma A.1 below,

Thus

Letting n tend to infinity and applying Theorem 3.3 yield (2.6). Thus Theorem 5.1 is proven. 2

Lemma A.1. The function g in (A.9) is lower bounded by some integrable random variable ζ * , that is

Proof. Since the matrix V n is symmetric and nonnegative definite, an orthogonal matrix Q n exists such that

A non-degenerate transformation is defined onto R p by setting t = (t 1 , . . . , t p ) = Q n v . Therefore, the function g(•) can be viewed as a function of t, precisely

where < x > j denotes the j-th component of the vector x. For any j, we can write v j,n = e j V n e j with e j = Q n e j , where e j = (0, . . . , 1, . . . , 0) ∈ R p . Note that β j,n = U n e j .

If v V n v = n k=1 (θ A k-1 v) 2 = 0, (where θ = θ -a as in (A.8)), then θ A k-1 v = 0 for all 1 ≤ k ≤ n, and hence U n v = n k=1 u k θ A k-1 v = 0, and finally g(v) is null. Therefore, if v j,n = 0 for some 1 ≤ j ≤ p, then β j,n = 0 .

Thus the function g(•) in (A.10) can be rewritten as

v j,n t 2 j 2 + β j,n t j 1 {v j,n >0} .


Ornstein-Uhlenbeck Process

Yuri Kabanov
email: youri.kabanov@univ-fcomte.fr

Serguei Pergamenshchikov
email: serge.pergamenchtchikov@univ-rouen.fr

Ruin probabilities for a Lévy-driven generalized

Keywords: Ruin probabilities, Dual models, Price process, Renewal theory, Distributional equation, Autoregression with random coefficients, Lévy process Mathematics Subject Classification (2000) 60G44 JEL Classification G22, G23

We study the asymptotic of the ruin probability for a process which is the solution of linear SDE defined by a pair of independent Lévy processes. Our main interest is the model describing the evolution of the capital reserve of an insurance company selling annuities and investing in a risky asset. Let β > 0 be the root of the cumulant-generating function H of the increment of the log price process V 1 . We show that the ruin probability admits the exact asymptotic Cu -β as the initial capital u → ∞ assuming only that the law of V T is non-arithmetic without any further assumptions on the price process.

Introduction

The general ruin problem can be formulated as follows. We are given a family of scalar processes X u with the initial values u > 0. The object of interest is the exit probability of X u from the positive half-line as a function of u. More formally, let τ u := inf{t : X u t ≤ 0}.

The question is to determine the function Ψ (u, T ) := P(τ u ≤ T ) (the ruin probability on a finite interval [0, T ]) or Ψ (u) := P(τ u < ∞) (the ruin probability on [0, ∞[). In the particular case where X u = u + X 0 the function Ψ (u) is the tails of distribution function of random variable sup t≥0 (-X 0 ), respectively.

The exact solution of the problem is available only in a few rare cases. For instance, for X u = u + W where W is the Wiener process Ψ (u, T ) = P(sup t≤T W t ≥ u) and it remains to recall that the explicit formula for the distribution of the supremum of the Wiener process was obtained already in the Louis Bachelier thesis of 1900 which is, probably, the first ever mathematical study on continuous stochastic processes. Another example is the well-known explicit formula for Ψ (u) in the Lundberg model of the ruin of insurance company with exponential claims, i.e. when X u = u + P and P is a compound Poisson process with drift and exponential jumps. Of course, for more complicated cases the explicit formulae are not available and only asymptotic results or bounds can be obtained as it is done, e.g., in the Lundberg-Cramér theory. In particular, if E P 1 > 0 and the size of jumps are random variables satisfying the Cramér condition (i.e. with finite exponential moments), then Ψ (u) is exponentially decreasing as u → ∞.

In this paper we consider the ruin problem for a rather general model, suggested by Paulsen in [START_REF] Paulsen | Risk theory in stochastic economic environment[END_REF], in which X u (sometimes called the generalized Ornstein-Uhlenbeck process) is given as the solution of linear stochastic equation

X u t = u + P t + ]0,t] X u s-dRs, (1.1)
where R and P are independent Lévy processes with their Lévy triplets (a, σ 2 , Π) and

(a P , σ 2 P , Π P ), respectively. There is a growing interest in models of this type because they describe the evolution of reserves of insurance companies investing in a risky asset with the price process S. In the financial-actuarial context R is interpreted as the relative price process with dR t = dS t /S t-, i.e. the price process S is the stochastic (Doléans) exponential E(R). The equation (1.1) means that the (infinitesimal) increment of the capital reserve dX u t is the sum of the increment dP t due to the insurance business activity and the increment due to the risky placement which is the product of number of owned shares X u t-/S t-and the price increment of a share dS t , that is X u t-dR t . In this model the log price process V = ln E(R) is also a Lévy process with the triplet (a V , σ 2 , Π V ). Recall that the behavior of the ruin probability in such models is radically different from that in the classical actuarial models. For instance, if the price of the risky asset follows a geometric Brownian motion, that is, R t = at + σW t , and the risk process P is as in the Lundberg model, then Ψ (u) = O(u 1-2a/σ 2 ), u → ∞, if 2a/σ 2 > 1, and Ψ (u) ≡ 1 otherwise, [START_REF] Frolova | In the insurance business risky investments are dangerous[END_REF], [START_REF] Kabanov | In the insurance business risky investments are dangerous: the case of negative risk sums[END_REF], [START_REF] Pergamenshchikov | Ruin probability in the presence of risky investments[END_REF].

We exclude degenerate cases by assuming that Π(]-∞, -1]) = 0 (otherwise Ψ (u) = 1 for all u > 0, see discussion in Section 2) and P is not a subordinator (otherwise Ψ (u) = 0 for all u > 0 because X u > 0, see (3.2), (3.1)). Also we exclude the case R = 0 well studied in the literature, see [START_REF] Kl Üppelberg C | Ruin probabilities and overshoots for general Lévy insurance risk processes[END_REF].

We are especially interested in the case where the process P describing the "business part" of the model has only upward jumps (in other words, P is spectrally positive). In the classical actuarial literature such models are referred to as the annuity insurance models (or models with negative risk sums), [START_REF] Grandell | Aspects of Risk theory[END_REF], [START_REF] Én | On the probability of ruin in the collective risk theory for insurance enterprises with only negative risk sums[END_REF], while in modern sources they serve also to describe the capital reserve of a venture company investing in development of new technologies and selling innovations; sometimes they are referred to as the dual models, [START_REF] Albrecher | On the dual risk model with taxation[END_REF] - [START_REF] Avanzi | Optimal dividends in the dual model[END_REF], [START_REF] Bayraktar | Optimizing venture capital investments in a jump diffusion model[END_REF], etc.

The mentioned specificity of models with negative risk sums leads to a continuous downcrossing of the zero level by the capital reserve process. This allows us to obtain the exact (up to a multiplicative constant) asymptotic of the ruin probability under weak assumptions on the price dynamics.

Let H : q → ln E e -qV1 be the cumulant-generating function of the increment of log price process V on the interval [0, 1]. The function H is convex and its effective domain dom H is a convex subset of R containing zero.

If the distribution of jumps of the business process has not too heavy tails, the asymptotic of the ruin probability Ψ (u) as u → ∞ is determined by the strictly positive root β of H, assumed existing and laying in the interior of dom H. Unfortunately, the existing results are overloaded by numerous integrability assumptions on processes R and P while the law L(V T ) of the random variable V T is required to contain an absolute continuous component where T is independent random variable uniformly distributed on [0, 1], see, e.g., Th. 3.2 in [START_REF] Paulsen | On Cramér-like asymptotics for risk processes with stochastic return on investments[END_REF] whose part (b) provides an information how heavier tails may change the asymptotic.

The aim of our study is to obtain the exact asymptotic of the exit probability in this now classical framework under the weakest conditions. Our main result has the following easy to memorize formulation.

Theorem 1.1 Suppose that H has a root β > 0 not laying on the boundary of dom H and

|x| β I {|x|>1} Π P (dx) < ∞. Then 0 < lim inf u→∞ u β Ψ (u) ≤ lim sup u→∞ u β Ψ (u) < ∞.
(1.2)

If, moreover, P jumps only upward and the distribution

L(V 1 ) is non-arithmetic 1 , then Ψ (u) ∼ C∞u -β where C∞ > 0.
In our argument we are based, as many other authors, on the theory of distributional equations as presented in the paper by Goldie, [START_REF]Implicit renewal theory and tails of solutions of random equations[END_REF]. Unfortunately, Goldie's theory does not give a clear answer when the constant defining the asymptotic of the tail of the solution of an affine distributional equation is strictly positive. The striking simplicity of our formulation is due to a recent progress in this theory, namely, due to the criterion by Guivarc'h and Le Page, [START_REF] Guivarc'h Y | On the homogeneity at infinity of the stationary probability for affine random walk[END_REF], which simple proof can be found in the paper [START_REF] Buraczewski | A simple proof of heavy tail estimates for affine type Lipschitz recursions[END_REF] by Buraczewski and Damek. This criterion gives a necessary and sufficient condition for the strict positivity of the constant in the Kesten-Goldie theorem determining the rate of decay of the tail of solution at infinity. Its obvious corollary allows us to simplify radically the proofs and get rid of additional assumptions presented in the earlier papers, see [START_REF] Kalashnikov | Power tailed ruin probabilities in the presence of risky investments[END_REF], [START_REF] Bankovsky | On the ruin probability of the generalised Ornstein-Uhlenbeck process in the Cramér case[END_REF], [START_REF] Nyrhinen | On the ruin probabilities in a general economic environment[END_REF] - [START_REF] Paulsen | Ruin theory with stochastic return on investments[END_REF] and references therein. Our technique involves only affine distributional equations and avoids more demanding Letac-type equations.

The question whether the concluding statement of the theorem holds when P has downward jumps remains open.

The structure of the paper is the following. In Section 2 we formulate the model and provide some prerequisites from Lévy processes. Section 3 contains a well-know reduction of the ruin problem to the study of asymptotic behavior of a stochastic integral (called in the actuarial literature continuous perpetuity, see [START_REF] Dufresne | The distribution of a perpetuity, with applications to risk theory and pension funding[END_REF]). In Section 4 we prove moment inequalities for maximal functions of stochastic integrals needed to analyze the limiting behavior of an exponential functional in Section 5. The latter section is concluded by the proof of the main result and some comments on its formulation. In Section 6 we establish Theorem 6.4 on the ruin with probability one using the technique suggested in [START_REF] Pergamenshchikov | Ruin probability in the presence of risky investments[END_REF]. This theorem implies, in particular, that in the classical model with negative risk sums and investments in the risky asset with price following a geometric Brownian motion the ruin is imminent if a ≤ σ 2 /2, [START_REF] Kabanov | In the insurance business risky investments are dangerous: the case of negative risk sums[END_REF]. In Section 7 we discuss examples.

Our presentation is oriented towards the reader with preferences in the Lévy processes rather than in the theory of distributional equations (called also implicit renewal theory). That is why in Section 8 (Appendix) we provide a rather detailed information on the latter covering the arithmetic case. In particular, we give a proof of a version of the Grincevicius theorem under slightly weaker conditions as in the original paper.

We express our gratitude to E. Damek, D. Buraczewski, and Z. Palmowski for fruitful discussions and a number of useful references on distributional equations.

2 Preliminaries from the theory of Lévy processes Let (a, σ2 , Π) and (a P , σ 2 P , Π P ) be the Lévy triplets of the processes R and P corresponding to the standard 2 truncation function h(x) := xI {|x|≤1} . Putting h(x) := xI {|x|>1} we can write the canonical decomposition of R in the form

R t = at + σW t + h * (µ -ν) t + h * µ t (2.1)
where W is a standard Wiener process, the Poisson random measure µ(dt, dx) is the jump measure of R having the deterministic compensator of the form ν(dt, dx) = dtΠ(dx). For notions and results see [START_REF] Jacod | Limit theorems for stochastic processes[END_REF], Ch. 2, [START_REF] Bertoin | Lévy Processes[END_REF], and also [START_REF] Cont R | Financial Modelling with Jump Processes[END_REF], Chs. 2 and 3.

As in [START_REF] Jacod | Limit theorems for stochastic processes[END_REF], we use * for the standard notation of stochastic calculus for integrals with respect to random measures. For instance,

h * (µ -ν) t = t 0 h(x)(µ -ν)(ds, dx).
We hope that the reader will be not confused that f (x) may denote the whole function f or its value at x; the typical example is ln(1+x) explaining why such a flexibility is convenient. The symbols Π(f ) or Π(f (x)) stands for the integral of f with respect to the measure Π.

Recall that

Π(x 2 ∧ 1) := (x 2 ∧ 1)Π(dx) < ∞
and the condition σ = 0 and Π(|h|) < ∞ is necessary and sufficient for R to have trajectories of (locally) finite variation, see Prop. 3.9 in [START_REF] Cont R | Financial Modelling with Jump Processes[END_REF]. The process P describing the actuarial ("business") part of the model admits a similar representation:

P t = a P t + σ P W P t + h * (µ P -ν P ) t + h * µ P t .
(2.

2)

The Lévy processes R and P generate the filtration F R,P = (F R,P t ) t≥0 .

Standing assumption S.0 The Lévy measure Π is concentrated on the interval ] -1, ∞[; σ 2 and Π do not vanish simultaneously; the process P is not a subordinator.

Recall that if Π charges ] -1, ∞[, then the ruin happens at the instant τ of the first jump of the Poisson process I {x≤-1} * µ of strictly positive intensity. Indeed, the independence of processes P and R implies that their trajectories have no common instants of jumps (except a null set). Note that τ = inf{t ≥ 0 :

xI {x≤-1} * µ t ≤ -1} < ∞, when Π(] -∞, -1]) > 0, and ∆Rτ ≤ -1 . According to (1.1) ∆Xτ = X τ -∆Rτ , that is Xτ = X τ -(∆Rτ + 1). It follows that τ u ≤ τ < ∞.
If Π does not charge ]-1, ∞[, but P is a subordinator, that is an increasing Lévy process, the ruin never happens. According to [START_REF] Cont R | Financial Modelling with Jump Processes[END_REF], Prop. 3.10, the process P is not a subordinator if and only if σ 2 P > 0, or one of the following three conditions hold: 1)

Π P (] -∞, 0[) > 0, 2) Π P (] -∞, 0[) = 0, Π P (xI {x>0} ) = ∞, 3) Π P (] -∞, 0[) = 0, Π P (xI {x>0} ) < ∞, a P -Π P (xI {0<x≤1} ) < 0.
Under the first assumption of S.0 ∆R > -1 and the stochastic exponential, solution of the linear equation dZ = Z -dR, Z 0 = 1, has the form

E t (R) = e Rt-1
2 σ 2 t+ s≤t (ln(1+∆Rs)-∆Rs) .

In the context of financial models it stands for the price of a risky asset (e.g., stock). The log price V := ln E(R) is a Lévy process and can be written in the form

V t = at - 1 2 σ 2 t + σW t + h * (µ -ν) t + (ln(1 + x) -h) * µ t .
(2.3)

Its Lévy triplet is (a V , σ 2 , Π V )
where

a V = a - σ 2 2 + Π(h(ln(1 + x)) -h)
and

Π V = Πϕ -1 , ϕ : x → ln(1 + x).
The cumulant-generating function H : q → ln E e -qV1 of the random variable V 1 admits an explicit expression. Namely,

H(q) := -a V q + σ 2 2 q 2 + Π e -q ln(1+x) -1 + qh(ln(1 + x)) .
(

Its effective domain dom H = {q : H(q) < ∞} is the set {J(q) < ∞} where J(q) := Π I {| ln(1+x)|>1} e -q ln(1+x) = Π I {| ln(1+x)|>1} (1 + x) -q .

(2.5)

Its interior is the open interval ]q, q[ with q := inf{q ≤ 0 : J(q) < ∞}, q := sup{q ≥ 0 : J(q) < ∞}.

Being a convex function, H is continuous and admits finite right and left derivatives on ]q, q[. If q > 0, then the right derivative

D + H(0) = -a V -Π( h(ln(1 + x))) < ∞,
though it may be equal to -∞.

In formulations of our asymptotic results we shall always assume that q > 0 and the equation H(q) = 0 has a root β ∈]0, q[. Since H is not constant, such a root is unique. Clearly, it exists if and only if D + H(0) < 0 and lim sup q↑q H(q)/q > 0. In the case where q < 0 the condition D -H(0) > 0 is necessary to ensure that H(q) < 0 for sufficiently small in absolute value q < 0.

If J(q) < ∞, then the process m = (m t (q)) t≤1 with m t (q) := e -qVt-tH(q) (2.6) is a martingale and E e -qVt = e tH(q) , t ∈ [0, 1].
(2.7)

In particular, we have that H(q) = ln E e -qV1 = ln EM q where M := e -V1 . For the above properties see, e.g., Th. 25.17 in [START_REF] Sato | Lévy processes and Infinitely Divisible Distributions[END_REF].

Note that

E sup t≤1 e -qVt < ∞ ∀ q ∈]q, q[. (2.8)
Indeed, let q ∈]0, q[. Take r ∈]1, q/q[. Then E m r 1 (q) = e H(qr)-rH(q) < ∞. By virtue of the Doob inequality the maximal function m * 1 (q) := sup t≤1 m t (q) belongs to L r and it remains to observe that e -qVt ≤ Cqm t (q) where the constant Cq = sup t≤1 e tH(q) . Similar arguments work for q ∈]q, 0[.

Ruin problem: a reduction

Let us introduce the process

Y t := - ]0,t] E -1 s-(R)dPs = - ]0,t]
e -Vs-dPs.

(3.1)

Due to independence of P and R the joint quadratic characteristic [P, R] is zero, and the straightforward application of the product formula for semimaringales shows that the process

X u t := E t (R)(u -Y t ) (3.2)
solves the non-homogeneous linear equation (1.1), i.e. the solution of the latter is given by this stochastic version of the Cauchy formula. The strict positivity of

E(R) = e V implies that τ u = inf{t ≥ 0 : Y t ≥ u}.
The following lemma is due to Paulsen, see [START_REF] Paulsen | Risk theory in stochastic economic environment[END_REF].

Lemma 3.1 If Y t → Y∞ almost surely as t → ∞
where Y∞ is a finite random variable unbounded from above, then for all u > 0

Ḡ(u) ≤ Ψ (u) = Ḡ(u) E Ḡ(X τ u ) | τ u < ∞ ≤ Ḡ(u) Ḡ(0) , (3.3)
where Ḡ(u) := P(Y∞ > u).

In particular, if

Π P (] -∞, 0]) = 0, then Ψ (u) = Ḡ(u)/ Ḡ(0).
Proof. Let τ be an arbitrary stopping time with respect to the filtration (F P,R t

). As we assume that the finite limit Y∞ exists, the random variable

Yτ,∞ := -lim N →∞ ]τ,τ +N ] e -(Vt--Vτ ) dP t , τ < ∞, 0, τ = ∞, is well defined. On the set {τ < ∞} Yτ,∞ = e Vτ (Y∞ -Yτ ) = X u τ + e Vτ (Y∞ -u).
(3.4)

Let ξ be a F P,R τ -measurable random variable. Since the Lévy process Y starts afresh at τ , the conditional distribution of Yτ,∞ given (τ, ξ) = (t, x) ∈ R + × R is the same as the distribution of Y∞. It follows that

P (Yτ,∞ > ξ, τ < ∞) = E Ḡ(ξ) 1 {τ <∞} . Thus, if P(τ < ∞) > 0, then P (Yτ,∞ > ξ, τ < ∞) = E Ḡ(ξ) | τ < ∞ P(τ < ∞) .
Noting that Ψ (u) := P(τ u < ∞) ≥ P(Y∞ > u) > 0, we deduce from here using (3.4) that

Ḡ(u) = P Y∞ > u, τ u < ∞ = P Y τ u ,∞ > X u τ u , τ u < ∞ = E Ḡ(X u τ u ) | τ u < ∞ P(τ u < ∞)
implying the equality in (3.3). The result follows since X u τ u ≤ 0 on the set {τ u < ∞} and, in the case where Π P (] -∞, 0]) = 0, the process X u crosses zero in a continuous way, i.e.

X u τ u = 0 on this set. 2
In view of the above lemma the proof of Theorem 1.1 is reduced to establishing of the existence of finite limit Y∞ and finding the asymptotic of the tail of its distribution.

Moments of the maximal function

In this section we prove a simple but important result implying the existence of moments of the random variable Y * 1 . Here and in the sequel we use the standard notation of stochastic calculus for the maximal function of a process:

Y * t := sup s≤t |Ys|.
Before the formulation we recall the Novikov inequalities, [START_REF] Novikov | On discontinuous martingales[END_REF], also referred to as the Bichteler-Jacod inequalities, see [START_REF] Bichteler | Calcul de Malliavin pour les diffusions avec sauts: existence d'une densité dans le cas unidimensionnel[END_REF], [START_REF] Marinelli | On maximal inequalities for purely discontinuous martingales in infinite dimensions[END_REF], providing bounds for moments of the maximal function I * 1 of stochastic integral I = g * (µ P -ν P ) where g 2 * ν P 1 < ∞. In dependence of the parameter α ∈ [START_REF] Albrecher | On the dual risk model with taxation[END_REF][START_REF] Asmussen | Ruin Probabilities[END_REF] they have the following form:

EI * p 1 ≤ Cp,α E |g| α * ν P 1 p/α , ∀ p ∈]0, α], E |g| α * ν P 1 p/α + E |g| p * ν P 1 , ∀ p ∈ [α, ∞[. (4.1)
Let U be a càdlàg process adapted with respect to a filtration under which the semimartingale P has deterministic triplet (a P , σ 2 P , Π P ) and let

Υ t := [0,t] U s-dPs. Lemma 4.1 If p > 0 is such that Π P (| h| p ) < ∞, Kp := EU * p 1 < ∞, then E Υ * p 1 < ∞.
Proof. The elementary inequalities |x + y| p ≤ |x| p + |y| p for p ∈]0, 1] and

|x + y| p ≤ 2 p-1 (|x| p + |y| p ) for p > 1
allows us to treat separately the integrals corresponding to each term in the representation

P t = a P t + σ P W P t + h * (µ P -ν P ) t + h * µ P t ,
that is, assuming that other terms are zero.

The case of the integral with respect to dt is obvious (we dominate U by U * . The estimation for integral with respect to dW P is reduced, by applying the Burkholder-Davis-Gundy inequality reduces to the estimation of the integral with respect to dt.

Let p < 1. In the more detailed notations f * µ P 1 = {s≤1: ∆Ps>0} f (s, ∆Ps) and U -= (U t-). Therefore,

E (|U -|| h| * µ P 1 ) p ≤ E |U -| p | h| p * µ P 1 = E |U -| p | h| p * ν P 1 ≤ Π P (| h| p )Kp.
Using the Novikov inequality (with α = 2) we have:

E U -h * (µ P -ν P ) * p 1 ≤ C p,2 (Π P (h 2 )) p/2 E 1 0 U 2 t dt p/2 ≤ C p,2 (Π P (h 2 )) p/2 Kp. Let p ∈]1, 2[
. By the Novikov inequality with α = 1 and we have:

E U -h * (µ P -ν P ) * p 1 ≤ C p,1 E(|U -|| h| * ν P 1 p + E|U -| p | h| p * ν P 1 ≤ Cp,1 Kp,
where Cp,1 :

= C p,1 Π P (| h|) p + Π P (| h| p ) .
Using again the Novikov inequality but with α = 2 we obtain that

E U -h * (µ P -ν P ) * p 1 ≤ C p,2 E (U 2 -h 2 * ν P 1 ) p/2 ≤ C p,2 (Π P (h 2 )) p Kp.
Finally, let p ≥ 2. Using the Novikov inequality with α = 2, we have:

E U -x * (µ P -ν P ) * p 1 ≤ C p,2 Π P (|x| 2 ) p/2 E 1 0 U 2 dt p/2 +C p,2 Π P (|x| p ) E 1 0 |U | p dt ≤ C p,2 Π P (|x| 2 ) p/2 + Π P (|x| p ) Kp.
Combining the above estimate we conclude that

Y∞ d = Y 1 + M 1 Y∞, Y∞ independent of (M 1 , Y 1 ), (5.1)
where

M 1 := e -V1 .
Proof. If the hypotheses hold for some p, they hold also for smaller values. We assume without loss of generality that p < 1 and H(p+) < ∞. For any integer j ≥ 1 we have the identity

Y j -Y j-1 = M 1 . . . M j-1 Q j , .
where (M j , Q j ) are independent random vectors with components

M j := e -(Vj -Vj-1) , Q j := - ]j-1,j]
e -(Vv--Vj-1) dPv

(5.2)

having distributions L(M j ) = L(M 1 ) and L(Q j ) = L(Y 1 ).
By assumption, the value

ρ := EM p 1 = e H(p) < 1 and E|Y 1 | p < ∞ in virtue of (2.8) and Lemma 4.1. Since EM 1 ...M j-1 |Q j | = ρ j E|Y 1 | p we have that E j≥1 |Y j -Y j-1 | p < ∞ and, therefore, j≥1 |Y j -Y j-1 | p < ∞ a.s. But then also j≥1 |Y j -Y j-1 | < ∞
a.s. and, therefore, the sequence Yn converges almost surely to the random variable Y∞ := j≥1 (Y j -Y j-1 ).

Put

∆n := sup n-1≤v≤n ]n-1,v]
e -Vs-dPs , n ≥ 1.

Note that

E ∆ p n = E n-1 j=1 M p j sup n-1≤v≤n ]n-1,v] e -(Vs--Vn-1) dPs p = ρ n-1 E Y * p 1 < ∞.
For any ε > 0 we get using the Chebyshev inequality that n≥1

P(∆n > ε) ≤ ε -p E Y * p 1 n≥1 ρ n-1 < ∞.
By the Borel-Cantelli lemma ∆n(ω) ≤ ε for all n ≥ n 0 (ω) for each ω ∈ Ω except a null-set. This implies the convergence Y t → Y∞ a.s., t → ∞.

Let us consider the sequence

Y 1,n := Q 2 + M 2 Q 3 + • • • + M 2 . . . MnQ n+1
converging almost surely to a random variable Y 1,∞ distributed as Y∞. Passing to the limit in the obvious identity

Yn = Q 1 + M 1 Y 1,n-1 we obtain that Y∞ = Q 1 + M 1 Y 1,∞ . For finite n the random variables Y 1,n and (M 1 , Q 1 ) are independent, L(Y 1,n ) = L(Yn). Hence, Y 1,∞ and (M 1 , Q 1 ) are independent, L(Y 1,∞ ) = L(Y∞) and L(Y∞) = L(Q 1 + M 1 Y 1,∞ ).
This is exactly the properties abbreviated by (5.1).

Note that our hypothesis insures the uniqueness of the affine distributional equation (5.1). Indeed, any its (finite) solution Ỹ∞ can be realized on the same probability space as Y∞ as a random variable independent on the sequence (M j , Q j ). Then

L( Ỹ∞) = L(Q 1 + M 1 Ỹ∞) = L(Q 1 + M 1 Q 2 + ... + M 1 . . . M n-1 Qn + M 1 ...Mn Ỹ∞).
Since the product M 1 ...Mn → 0 in L p as n → ∞, hence, in probability, the residual term also tends to zero in probability. Thus, L( Ỹ∞) = L(Y∞).

It remains to check that Y∞ is unbounded from above. For this it is useful the following simple observation. Lemma 5.2 If the random variables Q 1 and Q 1 /M 1 are unbounded from above, then Y∞ is also unbounded from above.

Proof. Since Q 1 /M 1 is unbounded from above and independent on Y 1,∞ , we have that

P(Y 1,∞ > 0) = P(Y∞ > 0) = P(Q 1 /M 1 + Y 1,∞ > 0) > 0. Take arbitrary u > 0. Then P(Y∞ > u) ≥ P(Q 1 + M 1 Y 1,∞ > u, Y 1,∞ > 0) ≥ P(Q 1 > u, Y 1,∞ > 0) = P(Q 1 > u)P(Y 1,∞ > 0) > 0
and the lemma is proven. 2 Notation: J θ := [0,1] e -θVv dv, Q θ := -[0,1] e -θVv-dPv where θ = ±1.

Lemma 5.3 L(Q -1 ) = L(Q 1 /M 1 ).
Proof. We have:

]0,1] n k=1 e V k/n-I ](k-1)/n,k/n] (v)dPv = n k=1 e V k/n (P k/n -P (k-1)/n ), e V1 ]0,1] n k=1 e -V k/n-I ](k-1)/n,k/n] (v)dPv = n k=1 e V1-V k/n (P k/n -P (k-1)/n ).
Note that V and P are independent, the increments P k/n -P (k-1)/n are independent and identically distributed, and

L(V 1 -V k/n ) = L(V (n-k)/n ).
Thus, the right-hand sides of the above identities have the same distribution. The result follows because the left-hand sides tend in probability, respectively, to -Q -1 and -Q 1 /M 1 . 2 Thus, Y∞ is unbounded from above if so are the stochastic integrals Q θ . Lemma 5.4 below shows that Q θ are unbounded from above if the ordinary integrals J θ are unbounded from above. For the latter property we prove necessary and sufficient conditions in terms of defining characteristics (Lemma 5.7). The case where these conditions are not fulfilled we treat separately (Lemma 5.8).

Lemma 5.4 If J θ is unbounded from above, so is Q θ .

Proof. We argue using the following observation: if f (x, y) is a measurable function and ξ, η are independent random variables with distributions P ξ and Pη, then the distribution of f (ξ, η) is unbounded from below if the distribution of f (ξ, y) is unbounded from below on a set of y of positive measure Pη.

In the case σ2 P > 0, we use the representation

Q θ = -σ P ]0,1] e -θVv-dW P v + ]0,1]
e -θVv-d(σ P W P v -Pv).

Applying the above observation (with ξ = W P and η = (R, P -σ P W P )) and taking into account that the Wiener integral of a strictly positive deterministic function is a nonzero Gaussian random variable, we get that Q θ is unbounded. Consider the case where σ 2

P = 0.
For ε > 0 we denote by ζ ε the locally square integrable martingale with

ζ ε t := e -θV-I {|x|≤ε} x * (µ P -ν P ) t .
(5.3)

Since ζ ε

Note that

Q θ = (Π P (xI {ε≤|x|≤1} ) -a P )J θ -ζ ε 1 -e -θV-I {|x|>ε} x * µ P 1 .
Take N > 1. Since J θ is unbounded from above, there is

N 1 > N + 1 such that the set {N ≤ J θ ≤ N 1 , inf t≤1 e -Vt ≥ 1/N 1 } is non-null. Then Γ ε := N ≤ J θ ≤ N 1 , inf t≤1 e -Vt ≥ 1/N 1 , |ζ ε 1 | ≤ 1
is also a non-null set for all sufficiently small ε > 0.

The process P is not a subordinator and, therefore, we have only three possible cases. 1)

Π P (] -∞, 0[) > 0. Then Π P (] -∞, -ε 0 [) > 0 for some ε 0 > 0.
Due to independence, the intersection of Γ ε with the set

{|I {x<-ε} x * µ P 1 | ≥ N 1 (a + P N 1 + N ), I {x>ε} * µ P 1 = 0}
is non-null when ε ∈]0, ε 0 [. On this intersection we have that

Q θ ≥ -a P J θ -ζ ε 1 -e -θV-I {x<-ε} x * µ P 1 ≥ -a + P N 1 -1 + a + P N 1 + N ≥ N -1.
2)

Π P (] -∞, 0[) = 0, Π P (h) = ∞.
Diminishing in the need ε to ensure the inequality

Π P (xI {x>ε} ) ≥ N 1 (a + P N 1 + N ), we have on the non-null set Γ ε ∩ {I {x>ε} * µ P 1 = 0} that Q θ = -a P J θ -ζ ε 1 + e -θV-I {x>ε} * ν P 1 ≥ -a + P N 1 -1 + a + P N 1 + N ≥ N -1.
3) Π P (] -∞, 0[) = 0, Π P (h) < ∞, and Π P (h) -a P > 0. Then on the non-null set

{J θ ≥ N } ∩ {I {x>0} * µ P 1 = 0} we have that Q θ = (Π P (h) -a P )J θ ≥ (Π P (h) -a P )N.
Since N is arbitrary, in all three cases Q θ is unbounded from above. 2 Remark 5.5 If J 1 I {V1<0} is unbounded from above, so is Q 1 I {V1<0} .

Remark 5.6 The proof above shows that in the case where σ P = 0 there is a constant κ > 0 such that if the set {J θ > N } is non-null, then Q θ > κN on its F R,P

1

-measurable non-null subset. The statement remains valid with obvious changes if the integration over the interval [0, 1] is replaced by the integral over arbitrary finite interval [0, T ]. Lemma 5.7 (i) The random variable J 1 is unbounded from above iff

σ 2 + Π(] -1, 0[) > 0 or Π(xI {0<x≤1} ) = ∞.
(ii) The random variable J -1 is unbounded from above iff

σ 2 + Π(]0, ∞[) > 0 or Π(xI {x<0} ) = -∞.
Proof. In the case where σ 2 > 0 the "if" parts of the statements are obvious: W is independent of the jump part of V and the distribution of the random variable

1 0 e -σθWv g(v)dv,
where g > 0 is a deterministic function, has a support unbounded from above. So, suppose that σ 2 = 0 and consider the "if" parts separately.

(i) Let Π(] -1, 0[) > 0, i.e. Π(] -1, -ε[) > 0 for some ε ∈]0, 1[.
Then the process V given by (2.3) admits the decomposition

V t = at + h * (µ -ν) t + (ln(1 + x) -h) * µ t = (a -Π(xI {-1<x≤-ε} ))t + V t + V t ,
where

V t := I {-ε<x≤1} x * (µ -ν) t + (ln(1 + x) -x)I {-ε<x≤1} * µ t + ln(1 + x)I {x>1} * µ t , V t := ln(1 + x)I {-1<x≤-ε} * µ t .
The processes V and V are independent. The decreasing process V has jumps of the size not less than | ln(1 -ε)| and the number of jumps on the interval [0, t] is a Poisson random variable with parameter tΠ(] -1, -ε[) > 0. Hence, V t is unbounded from below for any t ∈]0, 1[. In particular, for any N > 0, the set where e -V ≥ N on the interval [1/2, 1] is non-null. The required property follows from these considerations.

Let Π(h(x)I {x>0} ) = ∞. We assume without loss of generality that Π(] -1, 0[) = 0. In this case, the process V has only positive jumps. Take arbitrary N > 1 and choose ε > 0 such that Π(xI {ε<x≤1} ) > 2N and Π(I {0<x≤ε} ln 2 (1 + x)) ≤ 1/(32N 2 ). We have the decomposition

V t = ct + V (1) t + V (2) t + V (3) t ,
where the processes

V (1) := I {0<x≤ε} ln(1 + x) * (µ -ν), V (2) := I {ε<x≤1} ln(1 + x) * (µ -ν), V (3)
:= I {x>1} ln(1 + x) * µ are independent and the constant

c := a + Π((ln(1 + x) -x)I {0<x≤1} ) < ∞.
By the Doob inequality P (sup t≤1 V

(1) t < N/2) > 1/2. The processes V (2) and V (3) have no jumps on [0, 1] on a non-null set. In the absence of jumps the trajectory of V (2) is the linear function

y t = -Π(ln(1 + x)I {ε<x≤1} )t ≤ -2N t.
It follows that sup 1/2≤t≤1 V t ≤ c -N/2 on the set of positive probability. This implies that J 1 is unbounded from above.

(ii) Let Π(]0, ∞[) > 0, i.e. Π(]ε, ∞[) > 0 for some ε > 0. Then V t = at + h * (µ -ν) t + (ln(1 + x) -h) * µ t = (a -Π(hI {x>ε} ))t + Ṽ t + Ṽ t ,
where

Ṽ t := I {x≤ε} h * (µ -ν) t + (ln(1 + x) -h)I {x≤ε} * µ t , Ṽ t := ln(1 + x)I {x>ε} * µ t .
The processes Ṽ and Ṽ are independent. The increasing process Ṽ has jumps of the size not less than ln(1 + ε) and the number of jumps on the interval [0, t] is a Poisson random variable with parameter tΠ(]ε, ∞[) > 0. Hence, V t is unbounded from above for any t ∈]0, 1[. In particular, for any N > 0, the set where e V ≥ N on the interval [1/2, 1] is non-null. These facts imply the required property.

It remains to consider the case Π(xI {x<0} ) = -∞ and Π(]0, ∞[) = 0. The process V has only negative jumps. Take arbitrary N > 1 and choose

ε ∈]0, 1/2[ such that -Π(ln(1 + x)I {-1/2<x≤-ε} ) > 2N, Π(I {-ε<x<0} ln 2 (1 + x)) ≤ 1/(32N 2 ).
This time we use the representation

V t = ct + Ṽ (1) t + Ṽ (2) t + Ṽ (3) t ,
where the processes

Ṽ (1) := I {-ε<x<0} ln(1 + x) * (µ -ν), Ṽ (2) := I {-1/2<x≤-ε} ln(1 + x) * (µ -ν), Ṽ (3) := I {-1<x≤-1/2} ln(1 + x) * µ
are independent and the constant

c := a + Π(ln(1 + x) I {-1/2<x<0} -h).
By the Doob inequality P(sup t≤1

Ṽ (1) t < N/2) > 1/2.
The processes Ṽ (2) and Ṽ (3) have no jumps on [0, 1] with strictly positive probability. In the absence of jumps the trajectory of Ṽ (2) is the linear function

y = -Π(ln(1 + x)I {-1/2<x≤-ε} )t ≥ 2N t.
It follows that sup 1/2≤t≤1 V t ≤ c + N/2 on a non-null set. This implies that J -1 is unbounded from above.

The "only if" parts of the lemma are obvious. 2 Summarizing, we conclude that Q 1 and Q -1 (hence, Y∞) are unbounded from above if

σ 2 > 0, or σ 2 P > 0, or Π(|h|) = ∞, or Π(] -1, 0[) > 0 and Π(]0, ∞[) > 0.
The remaining cases are treated in the following:

Lemma 5.8 Let σ = 0, Π(|h|) < ∞, σ P = 0. If Π(] -1, 0[) = 0 or Π(]0, ∞[) = 0,
then the random variable Y∞ is unbounded from above.

Proof. By our assumptions V t = ct + L t where the constant c := a -Π(h), Π = 0, and L t := ln(1 + x) * µ t . The assumption β > 0 implies that V 1 < 0 with strictly positive probability and V cannot be increasing or decreasing process. So, there are two cases which we consider separately.

(i) c < 0 and Π(]0, ∞[) > 0. Take any T > 1. Then [0,T ] e -Vt-dt ≥ T /e on the non-null set {L T ≤ 1}. By virtue of Remark 5.6 on a non-null

F R,P T -measurable subset Γ T ⊆ {L T ≤ 1} we have that -[0,T ] e -Vt-dP t ≥ K T where K T → ∞ as T → ∞. For every T > 1 P(Γ T ∩ {L T +1 -L T ≥ |c|(T + 1)}) = P(Γ T )P(L T +1 -L T ≥ |c|(T + 1)) > 0.
Let ζ ε be the square integrable martingale given by (5.3) with θ = 1. Take N > 1 sufficiently large and ε > 0 sufficiently small to ensure that the set Γ ε,N T defined as the intersection

Γ T ∩ {L T +1 -L T ≥ |c|(T + 1)}, sup s∈[T,T +1] e -Vs ≤ N, inf s∈[T,T +1] e -Vs ≥ 1/N , and {|ζ ε T +1 -ζ ε T | ≤ 1} is non-null. Let us consider the representation Y∞ = - [0,T ] e -Vt-dP t + a ε P ]T,T +1] e -Vt-dt -ζ ε T +1 + ζ ε T -I ]T,∞[ e -V-xI {|x|>ε} * µ P T +1 + e -V T +1 Y T +1,∞ .
Take arbitrary y < 0 such that the set {Y T +1,∞ > y} is non-null.

Since the process P is not a subordinator with σ P = 0, it must satisfy one of the characterizing conditions 1), 2), 3) of Section 2. Let us consider them consecutively.

Suppose that

Π P (] -∞, 0[) > 0. Then there is ε 0 > 0 such that Π P (] -∞, -ε 0 [) > 0. Due to the independence, the intersection of Γ ε,N T with the set Γ ε,N T := {I [T,∞[ I {x<-ε} * µ P T +1 ≥ -(1/ε)N 2 a ε P , I [T,∞[ I {x>ε} * µ P T +1 = 0} is non-null when ε ∈]0, ε 0 [. Due to independence, the intersection of Γ ε,N T ∩ Γ ε,N
T and {Y T +1,∞ > y} also is a non-null set. But on this intersection we have inequality Y∞ ≥ K T -1 + y implying that Y∞ is unbounded from above.

Suppose that Π P (] -∞, 0[) = 0, Π P (h) = ∞. Thus, for sufficiently small ε > 0 we have a ε P > 0. On the non-null set

Γ ε,N T ∩ {I [T,∞[ I {x>ε} * µ P T +1 = 0} ∩ {Y T +1,∞ > y}
the inequality Y∞ ≥ K T -1 + y holds and we conclude as above. Finally, suppose that Π P (] -∞, 0[) = 0, Π P (h) < ∞, and Π P (h) -a P > 0. In this case we can use the representation

Y∞ = - [0,T ] e -Vt-dP t + (Π P (h) -a P ) ]T,T +1] e -Vt-dt -I ]T,∞[ e -V-xI {x>0} * µ P T +1 + e -V T +1 Y T +1,∞ .
On the non-null set Γ ε,N T ∩ {I ]T,∞[ I {x>0} * µ P T +1 = 0} ∩ {Y T +1,∞ > y} we have that Y∞ ≥ K T + y implying that Y∞ is unbounded from above.

(ii) c > 0 and Π(] -1, 0[) > 0. In this case there are γ, γ

1 ∈]0, 1[, γ < γ 1 , such that the theee sets {I ]-1,-γ[ * µ 1 = 0}, {I [-γ,-γ1[ * µ 1/2 = I ]-γ,-γ1[ * µ 1 = N }, and {ln(1 + x)I ]-γ1,0[ * µ 1 ≥ -1} are non-null. Due to independence, their intersection A N is also non-null.
On A N we have the bounds

c + N ln(1 -γ) -1 ≤ V 1 ≤ c + N ln(1 -γ 1 )
and

J 1 := [0,1] e -Vt-dt t ≥ e -c [0,1/2] e -ln(1+x) * µt dt ≥ 1 2 e -c (1 -γ 1 ) -N .
In virtue of Remark 5.6 there is a constant

κ N an F R,P 1 -measurable non-null subset B N of A N such that Q 1 ≥ κ N on B N and κ N → ∞ as N → ∞. Take T = T N > 0 such that cT + N ln(1 -γ) -2 ≥ 0. Then the set {I ]1,1+T [ ln(1 + x) * µ 1+T ≥ -1}
is non-null and its intersection with B N is also non-null. On this intersection e -V 1+T ≤ 1 and

c 1 (N ) ≤ V t-≤ c 2 (N ) where c 1 (N ) := c + N ln(1 -γ) -2, c 2 (N ) := c(T + 1) + N ln(1 -γ 1 ).
With this we accomplish the arguments by considering the cases corresponding to the properties 1), 2), and 3) with obvious modifications. 2

With the above lemma the proof of Proposition 5.1 is complete. 2 Proof of the main theorem. First, we relate the notations and hypotheses of Theorem 1.1 with those used in results from the implicit renewal theory summarized in Theorem 8.6 of Appendix. The hypothesis that H(β) = 0 means that EM β = 1 with M = M 1 = e -V1 . Also, EM β+ε < ∞ for some ε > 0, since β does not belong to the boundary of the effective domain of the function H. In view of (2.8) and Lemma 4.1 we have that E |Q| β < ∞ where Q = Q 1 = ]0,1] e -Vv-dv. Proposition 5.1 provides us the information that the almost sure limit Y∞ of the process Y given by (3.1) does exist, it is finite, unbounded from above, and has the law which solving the distributional equation L(Y∞) = L(Q + M Y∞) which can be written in the form (8.1). Thus, all the condition of Theorem 8.6 are fulfilled. The latter gives the statements on the asymptotic behavior of the tail function Ḡ(u) = P (Y∞ > u) as u → ∞. The reference to Lemma 3.1 allows us to transform them into statements on the asymptotic behavior of the ruin probability Ψ (u) and complete the proof. 2 Remark 5.9 The constant C∞ in Theorem 1.1 is of the formC∞ = C + / Ḡ(0) where C + is given in (8.3).

Remark 5.10 Note that the hypothesis β ∈ int dom H can be replaced by the slightly weaker assumption Ee -βV1 V - 1 < ∞. Remark 5.11 The hypothesis L(V 1 ) is non-arithmetic also can be replaced by a weaker one: one can assume that L(V T ) is non-arithmetic for some T > 0. Indeed, due to the identity ln Ee -βV T = T H(β) the root β does not depend on the choice of the time unit.

The following lemma shows that the condition on L(V 1 ) can be formulated in terms of the Lévy triplets. Lemma 5.12 The (non-degenerate) distribution of V 1 is arithmetic if and only if σ = 0, Π(R) < ∞, and there is d > 0 such that Π V is concentrated on the lattice Π(h) -a + Zd.

Proof. Recall that σ V = σ and Π V = Πϕ -1 where ϕ : x → ln(1 + x). So, we have [START_REF] Dufresne | The distribution of a perpetuity, with applications to risk theory and pension funding[END_REF] The property that Y∞ is unbounded from above can be deduced from much more general Th.1 on support of the exponential functionals from the paper [START_REF] Behme | On the range of exponential functionals of Lévy processes[END_REF]. However, the results for the supports of J θ and Q θ and arguments presented here have its own interest and can be used also without assuming, as in [START_REF] Behme | On the range of exponential functionals of Lévy processes[END_REF], that the limit Y∞ exists.

Π V (R) = Π(R). If σ V > 0 or Π V (R) = ∞, the distribution of V 1 has a density, see Prop. 3.12 in [12]. If σ = 0 and 0 < Π V (R) < ∞, then V is a compound Poisson process with drift c = a -Π(h) and distribution of jumps F V := Π V /Π V (R). In such a case L(V 1 ) is concentrated on the lattice Zd if and only if Π V is concentrated on the lattice -c + Zd. 2 Remark 5.

By construction, Ãδ

1 < 1 for any δ > 0. Using the definition of Q j given by (5.2) we have that

| B1 | ≤ T1 j=1 e V T 1 -Vj-1 |Q j | ≤ T1 j=1 |Q j |.
According to Lemma 4.1 E|Q 1 | p < ∞ for some p ∈]0, 1[. Taking r ∈]0, p/5[ and defining the sequence ln := [n 4r ], we have, using the Chebyshev inequality and (6.3), that

E | B1 | r ≤ 1 + r n≥1 n r-1 P T1 j=1 |Q j | > n ≤ 1 + r n≥1 n r-1 P ln j=1 |Q j | > n + r n≥1 n r-1 P T 1 > ln ≤ 1 + rE|Q 1 | p n≥1 lnn r-1-p + rc n≥1 n r-1 l -1/2 n < ∞.
To apply Corollary 6.2(ii) it remains to check that Y T1 is unbounded from above. Since {Q 1 > N , V 1 < 0} ⊆ {Y T1 > N }, it is sufficient to check that the probability of the set in the left-hand side is strictly positive for all N > 0, or, by virtue of Remark 5.5, that

P(J 1 > N, V 1 < 0) > 0 ∀ N > 0.
(6.4)

Let σ 2 > 0.
The conditional distribution of the process (Ws) s≤1 given W 1 = x coincides with the (unconditional) distribution of the Brownian bridge B x = (B x s ) s≤1 with

B x s = Ws + s(x -W 1 )
. Using this we easily get that for any bounded positive function g and any y, M ∈ R the probability

P 1 0 e -σWv g(v)dv > y , W 1 < M > 0,
cf. with Lemma 4.2 in [START_REF] Kabanov | In the insurance business risky investments are dangerous: the case of negative risk sums[END_REF]. This implies (6.4).

Suppose that σ 2 = 0, but Π(] -1, 0[) > 0, i.e. Π(] -1, -ε[) > 0 for some ε ∈]0, 1[. In the decomposition V = V (1) + V (2) , where

V (1) t = I {-1<x≤-ε} ln(1 + x) * µ t , V (2) t = (a -Π(hI {-1<x≤-ε} ))t + I {x>-ε} h * (µ -ν) t +I {x>-ε} (ln(1 + x) -h) * µ t ,
the processes V (1) and V (2) are independent. The process V (1) is decreasing by negative jumps whose absolute value are larger or equal than | ln(1 -ε)| and the number of jumps on the interval [0, 1/2] has the Poisson distribution with parameter (1/2)Π(] -1, -ε[) > 0. Thus, P(V (1) 1/2 < -n) > 0 for any real n. It follows that

P(J 1 > N, V 1 < 0) ≥ P 1 0 e -Vt dt > N, V 1 < 0, V (1) 1/2 < -n ≥ P e n 1 1/2 e -V (2) t dt > N, V (2) 1 < n, V (1) 1/2 < -n = P 1 1/2 e -V (2) t dt > N e -n , V (2) 1 < n P(V (1) 1/2 < -n).
in fact, a two-dimensional distribution L on ]0, ∞[×R not concentrated on {1} × R and the problem is to find a probability space with random variables Y∞ and (M, Q) on it such that Y∞ and (M, Q) are independent, L(M, Q) = L, and L(Y∞) = L(Q + M Y∞). The uniqueness in this problem means the uniqueness of the distribution of Y∞.

In the sequel (M j , Q j ) will be an i.i.d. sequence whose generic term (M, Q) has the distribution L and Z j := M 1 . . . M j , Z * n := sup j≤n Z j . If there is p > 0 such that EM p < 1 and E|Q| p < ∞, then the solution Y∞ of (8.1) can be easily realized on the probability space (Ω, F, P) where the sequence (M j , Q j ) is defined -just as the limit in L p of the series j≥0 Z j-1 Q j , see the beginning of the proof of Proposition 5.1.

The following classical result of the renewal theory is the Kesten-Goldie theorem, see Th. 4.1 in [START_REF]Implicit renewal theory and tails of solutions of random equations[END_REF]: Theorem 8.1 Suppose that (M, Q) is such that the distribution of ln M is non-arithmetic and, for some β > 0,

E M β = 1, E M β (ln M ) + < ∞, E |Q| β < ∞. (8.2) Then lim u→∞ u β P(Y∞ > u) = C + < ∞, lim u→∞ u β P(Y∞ < -u) = C -< ∞,
where C + + C -> 0.

Theorem 8.1 leaves open the question when the constant C + is strictly positive. The expression

C + = E ((Q + M Y∞) + ) β -((M Y∞) + ) β β EM β ln M (8.3)
given in [START_REF]Implicit renewal theory and tails of solutions of random equations[END_REF] and involving the unknown distribution Y∞ is not helpful. How to check that the right-hand side of this formula is strictly positive? Recently, Guivarc'h and Le Page showed for the above case where the distribution of ln M is non-arithmetic that C + > 0 if and only if Y∞ is unbounded from above, see [START_REF] Guivarc'h Y | On the homogeneity at infinity of the stationary probability for affine random walk[END_REF] and also the paper [START_REF] Buraczewski | A simple proof of heavy tail estimates for affine type Lipschitz recursions[END_REF] for simpler arguments. Of course, this criterion is not a result formulated in terms of the give data: it involves a property of the unknown distribution of Y∞, namely, that the support is unbounded. But this property can be checked in the considered model.

The remaining part of the appendix deals is a compendium of facts needed to cover also the arithmetic case.

Grincevicius theorem

The theorem below is a simplified version of Th.2(b), [START_REF] Grincevicius | One limit theorem for a random walk on the line[END_REF], but with a slightly weaker assumption on Q, namely, E|Q| β < ∞, used in our study. For the reader convenience we give its complete proof after recalling some concepts and facts from the renewal theory. We consider the convolution-type linear operator which is well-defined for all positive as well as for (the Lebesgue) integrable functions by the formula

ψ(x) = x -∞
e -(x-y) ψ(y)dy. (8.5) Clearly, the functions ψ and ψ are integrable or not simultaneously and

R ψ(x)dx = R ψ(x)dx.
Suppose that ψ ≥ 0 is integrable. Then ψ(x + δ) ≥ e -δ ψ(x) for any δ > 0 and

δ inf x∈[jδ,(j+1)δ] ψ(x) ≥ δe -δ ψ(jδ) ≥ e -2δ jδ (j-1)δ ψ(x)dx implying that U ( ψ, δ) := δ j∈Z inf x∈[jδ,(j+1)δ] ψ(x) ≥ e -2δ R ψ(x)dx.
Similarly,

Ū ( ψ, δ) := δ j∈Z sup x∈[jδ,(j+1)δ] ψ(x) ≤ e 2δ R ψ(x)dx.
Thus, Ū ( ψ, δ) < ∞ and Ū ( ψ, δ) -U ( ψ, δ) → 0 as δ → ∞. These two properties mean, by definition, that the function ψ is directly Riemann integrable. Arguing with the positive and negative parts, we obtain that if ψ is integrable, then ψ is directly Riemann integrable. We shall use in the sequel the following renewal theorem for the random walk Sn := n i=1 ξ i on a lattice, see Prop. Proof of Theorem 8.2. Let the solution of (8.1) be realized on some probability space (Ω, F, P). We shall use the notation (M, Q) instead of (M 1 , Q 1 ). As usual, the tail function Ḡ(u) := P(Y∞ > u). Define the function g(x) := e βx Ḡ(e x ). Since Y∞ and M are independent, P(M Y∞ > e x ) = E Ḡ(e x-ln M ). Introducing the new probability measure P := M β P and noting that

e βx P(M Y∞ > e x ) = EM β e β(x-ln M ) Ḡ(e x-ln M ) = Ẽg(x -ln M ),
we obtain the following identity (called renewal equation):

g(x) = D(x) + Ẽg(x -ln M ), (8.6)
where D(x) := e βx (P(Y∞ > e x ) -P(M Y∞ > e x )). The Jensen inequality for the convex function x → x ln x implies that Ẽ ln M = EM β ln M > 0 and, hence, Ẽ| ln M | < ∞.

Let us check that the function x → D(x) is integrable. To this aim, we note that for any random variables ξ, η |P(ξ > s) -P(η > s)| ≤ P(η + ≤ s < ξ + ) + P(ξ + ≤ s < η + ).

Using the Fubini theorem we obtain that

∞ 0 P(η + ≤ s < ξ + )s β-1 ds = EI {η+<ξ+} ξ + η + s β-1 ds = 1 β E (ξ + ) β -(η + ) β ) + .
Applying this bound with ξ :

= Q + M Y∞ d = Y∞ and η := M Y∞ we get that R |D(x)|dx = ∞ 0 |P(ξ > s) -P(η > s)|s β-1 ds ≤ 1 β E (ξ + ) β -(η + ) β )
and it remains to verify that

E|((Q + η) + ) β -(η + ) β | < ∞ (8.7) when E|Q| β < ∞. But |((Q + η) + ) β -(η ) β | = ζ 1 + ζ 2 with positive summands ζ 1 := I {-Q<η≤0} (Q + η) β + I {0<η≤-Q} η β ≤ |Q| β , ζ 2 := I {Q+η>0, η>0} |(Q + η) β -η β |. If β ≤ 1, then the random variable ζ 2 is also dominated by the random variable |Q| β . If β > 1, then the inequality |x β -y β | ≤ β|x -y|(x ∨ y) β-1 for x, y ≥ 0 combined with the inequality (|a| + |b|) β-1 ≤ 2 (β-2) + (|a| β-1 + |b| β-1 ) leads to the estimate ζ 2 ≤ 2 (β-2) + β|Q|(|η| β-1 + |Q| β-1 ).
Using the independence of (M, Q) and Y∞, the Hölder inequality, and taking into account that EM β = 1 and E|Y∞| p < ∞ for p ∈ [0, β[ we get that

E|Q||η| β-1 = E|Q|M β-1 E|Y∞| β-1 ≤ (E|Q| β ) 1/β E|Y∞| β-1 < ∞.
Thus, (8.7) holds. The integrability of D allows us to transform (8.6) into the equality ǧ(x) = Ď(x) + Ẽǧ(x -ln M ).

Iterating it, we obtain that

ǧ(x) = N -1 n=0 Ẽ Ď(x -Sn) + Ẽǧ(x -S N ), (8.8)
where Sn := n i=1 ξ i for n ≥ 1, (ξ i ) is a sequence of independent random variables on (Ω, F, P) independent on Y∞ such that the distribution L(ξ i , P) = L(ln M, P). In particular, Ẽe -βξi = 1.

By the strong law of large numbers S N /N → Ẽ ln M > 0 P-a.s., N → ∞, and, therefore, y -ln S N → -∞ P-a.s. for every y. Since Ẽe -βS N = 1, we have by dominated convergence that Ẽg(y -S N ) = Ẽe β(y-S N ) Ḡ(e y-S N ) → 0.

It follows that the remainder term Eǧ(x -S N ) in (8.8) tends to zero, thus, and, therefore, Theorem 8.2 is proven. 2

Buraczewski-Damek approach

The following result, usually formulated in terms of the supremum of the random walk Sn := n i=1 ln M i , is well-known (see, e.g., Th. A in [START_REF] Kesten | Random difference equations and renewal theory for products of random matrices[END_REF] for much more general setting). Putting Z(x) := e βx H(x), z(x) := e βx F (x), and P := e βξ1 P, we obtain from here that Z(x) = z(x) + Ẽ Z(x -ξ 1 )I {ξ1≤x} . (8.12) The same arguments as were used in deriving (8.8) The function ẑ(x) := z(x)I {x≥0} is directly Riemann integrable. Indeed, for j ≥ 0 we have that We get from here that Ū (ẑ, δ) < ∞ and Ū (ẑ, δ) -U (ẑ, δ) → 0 as δ → 0.

Using the renewal theory, we obtain, if the law of ξ is non-arithmetic, that (8.15)

The equalities (8.14) and (8.15) implies the statement. 2

The proof of the result below, formulated in a form to cover our needs, follows the same line as in Lemma 2.6 of the Buraczewski-Damek paper [START_REF] Buraczewski | A simple proof of heavy tail estimates for affine type Lipschitz recursions[END_REF] with minor changes to include also the arithmetic case.

Theorem 8 . 2

 82 Suppose that (8.2) holds and the distribution of ln M is concentrated on the lattice Zd = {0, ±d, ±2d, . . . } where d > 0. Then lim sup u→∞ u β P(Y∞ > u) < ∞.

2 . 1 ,F

 21 [START_REF] Iksanov | Tail behavior of supreme of perturbed random walks[END_REF]. Proposition 8.3 Let ξ i be i.i.d. random variables taking values in the lattice Zd, d > 0, and having finite expectation m := Eξ i > 0. Let F : R → R be a measurable function. If x ∈ R is such that j∈Z |F (x + jd)| < ∞, then lim n→∞ E k≥0 (x + nd -S k ) = d m j∈Z F (x + jd).

  x -S k ).

( 8 . 9 )

 89 Using Proposition 8.3 (with F = Ď) we obtain that for any x > 0lim n→∞ ǧ(x + dn) = d Ẽ ln M j∈Z Ď(x + jd) ≤ Ū ( Ď, d) < ∞.

(8. 10 )

 10 Replacing in the integrant the function Ḡ(e y ) by its smallest value G(e x ) we obtain thatǧ(x) := x -∞e -(x-y) e βy Ḡ(e y )dy ≥ 1 β + 1 g(x)

  lim sup u→∞ u β P(Y∞ > u) = lim sup x→∞ g(x) ≤ (β + 1) lim sup x→∞ ǧ(x) < ∞.

Proposition 8 . 4

 84 If M satisfies (8.2), then lim inf u→∞ u β P(Z * ∞ > u) > 0.

(8. 11 )

 11 Proof. Let F (x) := P(ln M ≤ x), F (x) := 1 -F (x), Sn := n i=1 ξ i where ξ i := ln M i . The function H(x) := P(sup n Sn > x) admits the representationH(x) = P(ξ 1 > x) + E I {ξ1≤x} H(x -ξ 1 ) = F (x) + x -∞H(x -t)dF (t).

  lead to the representationZ(x) = Ẽ k≥0 z(x -S k )I {S k ≤x} . (8.13)

  sup x∈[jδ,(j+1)δ] z(x) ≤ e β(j+1)δ F (jδ) ≤ e 2βδ jδ (j-1)δ e βv F (v)dv and, therefore,Ū (ẑ, δ) = δz(0) + δ j≥0 sup x∈[jδ,(j+1)δ] z(x) ≤ δz(0) + e 2βδ ∞ -δ e βv F (v)dv.In the same spirit inf x∈[jδ,(j+1)δ] z(x) ≥ e βjδ F ((j + 1)δ) ≥ e -2βδ (j+2)δ (j+1)δ e βv F (v)dv and U (ẑ, δ) = δ j≥0 sup x∈[jδ,(j+1)δ] z(x) ≥ e -2βδ ∞ δ e βv F (v)dv.

  .g., Ch. XI, 9,[START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF]. If the law of ξ is arithmetic with the step d > 0, then, according to Proposition 8.3 for any x > 0 lim n→∞ e β(x+nd) H(x + nd) = d Ẽξ j∈Z z(x + jd) I {x+jd≥0} .

Theorem 8 . 5

 85 Suppose that (8.2) hold. If the support of distribution of Y∞ is unbounded from above thenlim inf u→∞ u β P(Y∞ > u) > 0 . let Z * n := sup j≤n Z j . Theorems 8.1, 8.2 imply that P( Ȳ∞ < -u) ≤ C 1 u -β with C 1 > 0.On the other hand, by Proposition 8.4 we have thatP(Z * ∞ > u) ≥ C 2 u -β with C 2 > 0.Of course, in both cases the inequalities hold when u is sufficiently large.Put Un := {Zn > u, Ȳn > -Cu} where C β := 4C 1 /C 2 . The process Ȳ decreases. Therefore, we have the inclusion {Zn > u} ⊆ { Ȳ∞ ≤ -Cu} ∪ Un. It follows that for sufficiently large u > 0 (3/4)C 2 u -β ≤ P(Z * ∞ > u) = P(∪n{Zn > u}) ≤ P( Ȳ∞ ≤ -Cu) + P(∪nUn) ≤ 2C 1 C -β u -β + P(∪nUn)

That is, the distribution is not concentrated on a set Zd = {0, ±d, ±2d, . . . }.

Other truncation functions are also used in the literature, see, e.g.,[START_REF] Paulsen | On Cramér-like asymptotics for risk processes with stochastic return on investments[END_REF].

= e -2θV-I {|x|≤ε} x

* ν P 1 → 0 as ε → 0, we have that sup t≤1 |ζ ε t | → 0 in probability.

Acknowledgements

The research is funded by the grant n • 14.A12.31.0007 of the Government of Russian Federation. The second author is partially supported by the Russian Federal Professor program (project n • 1.472.2016/1.4) and by the research project n • 2.3208.2017/4.6 of the Ministry of Education and Science of the Russian Federation.

Ruin with probability one

In this section we give conditions under which the ruin is imminent whatever is the initial reserve.

Recall the following ergodic property of the autoregressive process (X u n ) n≥1 with random coefficients which is defined recursively by the relations

where (An, Bn) n≥1 is a sequence of i.i.d. random variables in R 2 (see [START_REF] Pergamenshchikov | Ruin probability in the presence of risky investments[END_REF], Prop. 7.1 and, for a deeper result, [START_REF]A multiplicative ergodic theorem for Lipschitz maps[END_REF]).

Lemma 6.1 Suppose that E|An| δ < 1 and E|Bn| δ < ∞ for some δ ∈]0, 1[. Then for any u ∈ R the sequence X u n converges in L δ (hence, in probability) to the random variable

and for any bounded uniformly continuous function f

Proof. We get (i) just by the straightforward application of (6.2) to the function

The statement (ii) follows from (i). Indeed, put X 0,1

Since B 1 /A 1 and X 0,1 ∞ are independent and the random variable

Let M j and Q j be the same as in (5.2).

Proposition 6.3 Suppose that EM

Proof. The process X u solving the equation (1.1) and restricted to the integer values of the time scale admits the representation

That is, X u n is given by (6.1) with An = M -1 n and Bn = -M -1 n Qn. The result follows from the statement (ii) of Corollary 6.2. 2

Now we give more specific conditions of the ruin with probability one in terms of the triplets. Theorem 6.4 Suppose that 0 ∈ int dom H and Π

Proof. Note that D -H(0) = -a V -Π( h(ln(1 + x))). If D -H(0) > 0, then for all q < 0 sufficiently close to zero H(q) < 0, that is EM q 1 < 1. By virtue of Lemma 5.3 the law

). If Π P (| h| ε ) < ∞ for some ε > 0, then Lemma 4.1 implies that E|Q -1 | q < ∞ for sufficiently small q > 0. To get the result we can use Proposition 6.3. Indeed, by virtue of Lemmata 5.4 and 5.7(i) the random variable Q 1 is unbounded from above except, eventually, the case where σ 2 = 0, σ 2 P = 0, Π(|h|) < ∞, and

Recall that in this special case V t = ct + L t where c := a -Π(h) and L t := ln(1 + x) * µ t . Note that

where the quality in law holds in virtue of Lemma 5.3 (the latter is formulated for the interval [0, 1] but its extension to arbitrary one is obvious). The random variable Yn is defined by the same formula as Yn with V replaced by -V . As in Proposition 5.3 we show that converges to a finite value Y∞ in probability. It follows that L(X 0 n ) = L(-Yn). As in Lemma 5.8(i) we can show that Yn is unbounded from above.

In the case where D -H(0) = 0 we consider, following [START_REF] Pergamenshchikov | Ruin probability in the presence of risky investments[END_REF], the discrete-time process ( Xu n ) n∈N where Xu n = X Tn and the descending ladder times Tn of the random walk (Vn) n∈N which are defined as follows: T 0 := 0,

The formula (2.3) can be written as

According to Theorem 1a in Ch. XII.7 of Feller's book [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF] and the remark preceding the citing theorem, the above properties imply that there is a finite constant c such that

It follows, in particular, that the differences Tn -T n-1 are well-defined and form a sequence of finite independent random variables distributed as T 1 . The discrete-time process Xu n = X u Tn has the representation

and solves the linear equation

where

and B1 / Ã1 = -Y Tn where Y is given by (3.1).

The right-hand side is strictly positive for sufficiently large n and (6.4) holds.

The case where Π(xI {0<x≤1} ) = ∞ is treated similarly as in the last part of the proof of Lemma 5.7(i).

The exceptional case is treated by a reduction to Corollary 6.2(i). 2

The above theorem implies that in the classical model with negative risk sums (where σ P = 0, the jumps of P are positive and form a compound Poisson process, Π P (|x|) < ∞, trend is negative, i.e. a P -Π P (x) < 0) and investments into a risky asset with the price following a geometric Brownian motion (that is, Π = 0 and σ = 0), the ruin is imminent if

Let us consider the model with negative risk sums in which the Lévy measure Π P (dx) = λF P (dx) where the constant λ > 0 and the probability distribution F P (dx) is concentrated on ]0, ∞[, and

The process P admits the representation as sum of an independent Wiener process with drift and a compound Poisson process:

where the Poisson process N P with intensity λ P is independent of the sequence (ξ j ) j≥1 of positive i.i.d. random variables with common distribution F P . Suppose that the price process is a geometric Brownian motion

that is, σ = 0, Π = 0.

For this model q = -∞, q = ∞. The condition D + H(0) < 0 is reduced to the inequality σ 2 /2 < a and the function H(q) = (σ 2 /2 -a + qσ 2 /2)q has the root β = 2a/σ 2 -1 > 0. Suppose that σ 2 P + (a 0 P ) + > 0. By Theorem 1.1 the exact asymptotic

Since the exponential distribution has the above property, we recover, as a very particular case the asymptotic result of [START_REF] Kabanov | In the insurance business risky investments are dangerous: the case of negative risk sums[END_REF] where it was assumed that σ 2 P = 0 and a 0 P > 0.

If σ 2 P + (a 0 P ) + > 0, σ 2 /2 ≥ a, and Eξ 1 < ∞ for some > 0, then Theorem 6.4 implies that Ψ (u) ≡ 1.

The models with the price process given by a geometric Brownian motion were intensively studied using the representation of Ψ as a solution of integro-differential equations. To the reader interested not only in asymptotical results but also in a behavior of the ruin probabilities for finite values of the initial capital we recommend a very detailed study [START_REF] Belkina | Dynamical insurance models with investment: Constrained singular problems for integrodifferential equations[END_REF] with a number of simulation results.

Example 2. Let the process P be again given by (7.1) and suppose that the price process has a jump component, namely,

where the Poisson process N with intensity λ > 0 is independent on the sequence (η j ) j≥1 of i.i.d. random variables with common distribution F not concentrated at zero and such that F (] -∞, -1]) = 0, see [START_REF] Lamberton | Introduction to Stochastic Calculus Applied to Finance[END_REF], Ch. 7. That is, the log price process is represented as

where Π(dx) = λF (dx). The function H is given by the formula

Suppose that E (1 + η 1 ) -q < ∞ for all q > 0. Then q = ∞.

Let σ = 0. Then lim sup q→∞ H(q)/q = ∞. If

then the root β > 0 of the equation H(q) = 0 does exist. Thus, if Eξ β 1 < ∞, then Theorem 1.1 can be applied to get that Ψ (u) ∼ C∞u -β where C∞ > 0.

If

, the root β is larger (resp., smaller) than 2a/σ 2 -1, the value of the root of H in the model of Example 1 where the price process is continuous.

and lim sup q→∞ q -1 E (1 + η 1 ) -q -1 > a/λ, then the root β > 0 also exists. Theorem 1.1 can be applied when P(η 1 > 0) ∈]0, 1[ and the we have the exact asymptotic if the distribution of ln(1 + η 1 ) is non-arithmetic.

Suppose that E (1 + η 1 ) -q < ∞ for all q ∈ R. Then q = -∞ and q = ∞. If the conditions σ 2 /2 -a -λE ln(1 + η 1 ) ≥ 0, σ 2 + P(η 1 < 0) > 0, and E|ξ 1 | ε < ∞ for some ε > 0 hold, then Ψ (u) ≡ 1 in virtue of Theorems 6.4.

8 Appendix: Tails of solutions of distributional equations

Kesten-Goldie theorem

Here we present a short account of needed results on distributional equations (random equations in the terminology of [START_REF]Implicit renewal theory and tails of solutions of random equations[END_REF])

where (M, Q) is a two-dimensional random variable such that M > 0 and P(M = 1) > 0 Thus, P(Y∞ > u) ≥ (1/4)bC 2 u -β where b := P(Y∞ > C + 1) > 0 by the assumption that the support of L(Y∞) is unbounded from above. The obtained asymptotic bound implies that C + > 0. 2

Summarizing the above results we get for function Ḡ(u) = P(Y∞ > u) the following asymptotic properties when u → ∞: Theorem 8.6 Suppose that (8.2) holds. Then lim sup u β Ḡ(u) < ∞. If Y∞ is unbounded from above, then lim inf u β Ḡ(u) > 0 and in the case where L(ln M ) is non-arithmetic Ḡ(u) ∼ C + u -β where C + > 0.