Pavel V Gapeev
email: p.v.gapeev@lse.ac.uk

Monique Jeanblanc
email: monique.jeanblanc@univ-evry.fr

Dongli Wu
email: wudongli.sh@ccbft.com

Projections in enlargements of ltrations under Jacod's absolute continuity hypothesis for marked point processes *

Keywords: . Secondary 60G35, 60H10, 91G40 Marked point process, compensator, conditional probability density, Jacod's absolute continuity hypothesis, initial and progressive enlargements of ltrations, predictable (martingale) representation property, changes of probability measures

 in the Brownian motion setting to the case of absolute continuity hypothesis.

Introduction

In this paper, we consider the initial (resp. progressive) enlargement of a ltration F (called hereafter the reference ltration) with a strictly positive random variable τ (called hereafter the random time), denoted by F (τ ) (resp. G). We study the case in which F is generated by a marked point process (MPP for short. The reason why we are working with such processes is that a marked point process in F remains a marked point process. in any larger ltration, in particular is a semi-martingale in any enlargement of F with possibly a dierent compensator and under some conditions admits the weak predictable representation property (WPRP for short).

We assume that the law of τ has no atoms and that Jacod's absolute continuity hypothesis introduced in [START_REF] Amendinger | Martingale representation theorems for initially enlarged ltrations[END_REF] and [START_REF] Grorud | Insider trading in a continuous time market model[END_REF] holds (see Section 3 below for details). We recall that, under some hypotheses on τ (see Hypothesis 3.4 below), the weak predictable representation property holds in the ltration F (τ ) and (adding a pure jump martingale) in the ltration G. We study the relationship between the representation of martingales in the initially (resp. progressively) enlarged ltration and the various optional projections. The paper is an extension of our previous paper [START_REF] Gapeev | Projections of martingales in enlargements of Brownian ltrations under Jacod's equivalence hypothesis[END_REF] to the case of models driven by marked point processes. We refer the reader to the monograph [START_REF] Aksamit | Enlargement of ltration with nance in view[END_REF] for results on enlargements of ltrations. Our results can be useful to compare the optimal strategies of investors having dierent information ows (see, e.g. [START_REF] Amendinger | Martingale representation theorems for initially enlarged ltrations[END_REF] and [START_REF] Grorud | Insider trading in a continuous time market model[END_REF]). Note that, without any diculties, one can study models driven by independent Brownian motions and marked point processes, but this would simply lead to longer formulae.

Detailed studies of the weak predictable representation property when the process also has a continuous martingale part were provided in [START_REF] Di Tella | On the weak representation property in progressively enlarged ltrations with an application in exponential utility maximization[END_REF] (see also [START_REF] Di Tella | On the propagation of the weak representation property in independently enlarged ltrations: The general case[END_REF]- [START_REF] Di Tella | Martingale representation in the enlargement of the ltration generated by a point process[END_REF]).

The paper is organised as follows. In Section 2, we recall standard results of stochastic analysis that we use in the paper. In Section 3, we give some basic denitions and results related to the initial and progressive enlargements of a ltration F generated by a marked point process with a random time τ , denoted by F (τ ) and G, respectively, under Jacod's hypothesis.

In Section 4, we recall that the weak predictable representation property holds in the reference ltration with respect to the compensated random measure and prove that the weak predictable representation property holds with respect to an explicit martingale and a compensated random measure in the enlargements of ltration involved. In Section 5, we consider the optional projections of an F (τ ) -martingale on the ltrations G and F. We derive explicit expressions for the coecients in the integral representations of these optional projections in terms of the original F (τ ) -martingale and the components in its representation as a stochastic integral and give analogous results in the case of F-optional projections of a G-martingale. In Section 6, we consider the optional projections of a strictly positive F (τ ) -martingale on G and F and the F-optional projection of a strictly positive G-martingale. We describe the set of equivalent martingale measures in the associated extension of the exponential model driven by a marked point process and enhanced with the random time τ . In particular, we show that the set of equivalent martingale measures in the model with the progressively enlarged ltration G is essentially larger than the one obtained by means of the optional projections on G of the Radon-Nikodym densities in the model with the initially enlarged ltration F (τ ) . Some technical proofs are presented in Appendix.

Preliminary denitions and results

We work on a standard complete probability space (Ω, G, P), on which there exists a sequence (T n , Z n ) n≥1 , where (T n ) n≥1 is a strictly increasing sequence of nite strictly positive random variables with no accumulation points, and (Z n ) n≥1 a sequence of real-valued random variables.

We shall say that the sequence (T n , Z n ) n≥1 is a marked point process (MPP) on R with no accumulation points (see Def. 1.1.6 in [25], Section 1.2, pages 3-4 in [START_REF] Last | Marked Point Processes on the Real Line. The Dynamic Approach[END_REF], Chapter 4 in [START_REF] Björk | Point Processes and Jump Diusions: An Introduction with Finance Applications[END_REF], and Chapter VIII in [START_REF] Brémaud | Point Processes and Queues: Martingale Dynamics[END_REF]).

We denote by B(R + ) the σ-algebra of the Borel sets on R + ≡ [0, ∞) and introduce the associated random measure µ on G × B(R + ) × B(R) which is dened, for any set A ∈ B(R) and any t ≥ 0, by µ(ω; (0, t], A) = n≥1 1 1 {Tn(ω)≤t} 1 1 {Zn(ω)∈A} , which is called the jump measure of the marked point process. We denote by F = (F t ) t≥0 the natural ltration of the MPP given by

F t = σ µ((a, b], A), 0 ≤ a < b ≤ t, A ∈ B(R) , ∀t ≥ 0 ,
which is a right-continuous ltration (see Proposition 3.39 in [START_REF] Jacod | Calcul Stochastique et Problèmes de Martingales[END_REF]). We call F hereafter the reference ltration, and note that all T n , for n ≥ 1, are F-stopping times. We dene the compensator ν of the jump measure µ with respect to F as the unique random measure

ν(ω; (0, t], A) = t 0 A ν(ω; ds, dz), ∀t ≥ 0 , is F-predictable and we introduce µ((0, t], A) := µ((0, t], A) -ν((0, t], A), ∀t ≥ 0 ,
which is an F-martingale. We shall say that µ is the F-compensated martingale of the marked point process, and, by abuse of language, that ν is the compensator of µ. More generally, if K is a ltration larger that F, we say that ν K is the K-compensator of µ if, for any A ∈ B(R), the process µ K ((0, t], A) := µ((0, t], A) -ν K ((0, t], A), ∀t ≥ 0 , is a K-martingale, and the process ν K ((0, •], A) is K-predictable. With an abuse of language, we shall write that µ K is a K-martingale.

Hypothesis 2.1 We assume, as in Chapter VIII, Denition D5, page 236 of [START_REF] Brémaud | Point Processes and Queues: Martingale Dynamics[END_REF] and [START_REF] Runggaldier | Jump-diusion models[END_REF], that the F-compensator ν admits the representation

ν(ω; dt, dz) = η t (ω; dz) dt, ∀t ≥ 0 , (1)
where η(dz) is a transition kernel.

As usual, P(F) (resp. O(F)) is the predictable (resp. optional) σ-algebra on F. For a family of processes ξ(z) = (ξ t (z)) t≥0 parameterized by z ∈ R, we shall say that ξ is P(F) ⊗ B(R)measurable, if the map (t, ω, z) → ξ t (ω; z) is P(F) ⊗ B(R)-measurable, and we dene O(F) ⊗ B(R)-measurable processes in a similar way.

Recall that, under Hypothesis 2.1, if ξ is a P(F) ⊗ B(R)-measurable process such that

t 0 R |ξ s (z)| η s (dz) ds < ∞, ∀t ≥ 0 , (2)
the process Y = (Y t ) t≥0 dened as

Y t = Y 0 + t 0 R ξ s (z) µ(ds, dz), ∀t ≥ 0 , (3)
is an F-martingale (see Chapter VIII, Corollary C4, page 235 in [START_REF] Brémaud | Point Processes and Queues: Martingale Dynamics[END_REF]). Furthermore, any F-martingale Y admits a representation as in (3) with ξ satisfying (2) (see Chapter VIII, Theorem T8, page 239 in [START_REF] Brémaud | Point Processes and Queues: Martingale Dynamics[END_REF] and Theorem 2.2 in [START_REF] Runggaldier | Jump-diusion models[END_REF]). This property is referred to as the weak predictable representation property (WPRP) of the marked point process µ in the ltration F with respect to the compensated jump measure µ -ν (see also Theorem 13.19 in [START_REF] He | Semimartingale Theory and Stochastic Calculus[END_REF], or Th. 1.13.2 in [START_REF] Last | Marked Point Processes on the Real Line. The Dynamic Approach[END_REF], or Theorem 1.1.21 in [START_REF] Prigent | Weak Convergence of Financial Markets[END_REF]). Such a representation is essentially unique (P × η t (dz) × dt-a.s.).

Let X = (X t ) t≥0 be a measurable process and H be a ltration satisfying the usual hypotheses of completeness and right continuity. We denote by p,F X = ( p,F X t ) t≥0 (resp. o,F X = ( o,F X t ) t≥0 ) its F-predictable (resp. optional) projection when they exist (see Chapter V, Th. 5.1 (resp. 5.2) in [START_REF] He | Semimartingale Theory and Stochastic Calculus[END_REF] or Section 1.3.1, page 15 in [START_REF] Aksamit | Enlargement of ltration with nance in view[END_REF]).

Jacod's absolute continuity hypothesis

In the whole paper, we work on a complete probability space (Ω, G, P) which supports a marked point process with a right-continuous and completed natural ltration F = (F t ) t≥0 and a strictly positive random variable τ . Note that the inclusion F ∞ ⊂ G holds and, in general, this inclusion is strict. We recall that any F-martingale admits a càdlàg modication, see Corollary 2.48 in [START_REF] He | Semimartingale Theory and Stochastic Calculus[END_REF]. In what follows, τ is a given random time dened on (Ω, G), i.e. a strictly positive random variable. Hypothesis 3.1 We assume in the whole paper that Jacod's absolute continuity hypothesis (in short Jacod's hypothesis) holds, that is, the regular conditional distributions of τ given F t are absolutely continuous with respect to ρ, the unconditional law of the random variable τ is of the form

P(τ ∈ • | F t ) ≪ P(τ ∈ •), ∀t ≥ 0 (P-a.s.) .
Note that Hypothesis 3.1 was not assumed in [START_REF] Di Tella | On the weak representation property in progressively enlarged ltrations with an application in exponential utility maximization[END_REF]. We assume that Hypothesis 3.1 holds in order to be able to obtain explicit expressions for the coecients of the process in the weak predictable representations given considered initially and progressively enlarged ltrations F (τ ) and G. This assumption implies (see Lemma 2.3 in [START_REF] Fontana | The strong predictable representation property in initially enlarged ltrations under the density hypothesis[END_REF]) that there exists a family of nonnegative processes p(u) = (p t (u)) t≥0 such that the function (ω, t, u) → p t (u; ω) is O(F) ⊗ B(R + )-measurable, and, for each u ≥ 0, the process p(u) is a càdlàg F-martingale. Moreover, for any Borel bounded function f , the following equality holds

E f (τ ) F t = ∞ 0 f (u) p t (u) ρ(du), ∀t ≥ 0 (P-a.s.) . (4)
The expression in (4) implies that

P(τ > s | F t ) = ∞ s p t (u) ρ(du), ∀t, s ≥ 0 (P-a.s.) , so that the equality ∞ 0 p t (u) ρ(du) = 1, (P-a.s.),
is satised, and p 0 (u) = 1, for each u ≥ 0.

We shall call the family of F-optional processes p(u), for each u ≥ 0, the F-conditional density family with respect to ρ(du). Note that, even if p is not strictly positive, p(τ ) is strictly positive (see formula (4.10) in [START_REF] Aksamit | Enlargement of ltration with nance in view[END_REF]).

□

The following proposition is proved as a consequence of the weak predictable representation property in [24, Pro. 2.1] (see also Chapter II, Denition 1.27 in [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF] and Chapter III, Theorem 4.24 in [START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF]). Proposition 3.2 For each u ≥ 0, the F-martingale p(u) admits the representation

dp t (u) = R f t (u, z) µ(dt, dz), ∀t ≥ 0, p 0 (u) = 1 , (5)
for a P(F) ⊗ B(R + ) ⊗ B(R)-measurable process f , and

t 0 R |f s (u, z)| η s (dz) ds < ∞, ∀t ≥ 0 .
Proof: The existence of f is due to the weak predictable representation property in the ltration F. Note that at the jump times of the process we have ∆p t (u) = f t (u, z), and hence, using the non-negativity of p, we see that f t (u, z) ≥ -p t-(u), ∀t ≥ 0, ∀z ∈ R. We shall prove latter on that this inequality holds everywhere.

□

Let us denote by H = (H t ) t≥0 with H t = 1 1 {τ ≤t} , for all t ≥ 0, the indicator default process, where τ denotes the time at which a default occurs. Moreover, since H is a G-adapted càdlàg process, we can introduce the F-supermartingale G = (G t ) t≥0 dened by G = o,F (1 -H), that is, the F-optional projection of 1 -H satisfying the property G t = P(τ > t | F t ), ∀t ≥ 0 (P-a.s.) , [START_REF] Björk | Point Processes and Jump Diusions: An Introduction with Finance Applications[END_REF] which, according to the equality (4), can be represented in the form

G t = ∞ t p t (u) ρ(du), ∀t ≥ 0 (P-a.s.) .
Note that G 0 = 1. Hypothesis 3. [START_REF] Amendinger | Martingale representation theorems for initially enlarged ltrations[END_REF] We assume that G, G -are strictly positive.

The F-supermartingale G is called the Azéma supermartingale of the random time τ . Hypothesis 3.4 We assume that the distribution law ρ of the strictly positive random variable τ avoids F-stopping times (in particular, ρ is non-atomic).

Enlargements of ltrations and martingales

We will consider two enlarged ltrations: the initial enlargement of F obtained by adding the σ-eld σ(τ ) at time 0 and denoted F (τ ) , and the progressive enlargement of F obtained by progressively adding information of σ(τ ∧ t) at time t ≥ 0, or, more precisely, the smallest right-continuous ltration G containing F and turning out τ into a stopping time.

The aim of the paper is to explicitly compute the components in the integral representations of the optional projections of the F (τ ) -martingales and of the G-martingales. In this section, we recall some well known results. We give the form of the F (τ ) -semimartingale decomposition and G-semimartingale decomposition of µ dened in [START_REF] Aksamit | Integral representations of martingales for progressive enlargements of ltrations[END_REF] as well as the G-semimartingale decomposition of H. We underline that the martingale part µ (τ ) of the F (τ ) -semimartingale decomposition of µ enjoys the F (τ ) -predictable representation property, while the pair ( µ G , M G ) of the martingale parts of the G-semimartingale decompositions of µ and H enjoys the Gpredictable representation property, where the integral with respect to the pair is understood componentwise.

The initially enlarged ltration

As in the introduction, let us denote by F (τ ) = (F (τ ) t ) t≥0 = (F t ∨σ(τ )) t≥0 the initial enlargement of the ltration F with the random time τ . Note that F (τ ) 0 = σ(τ ). We recall that, under Jacod's hypothesis, any F-local martingale is an F (τ ) -special semimartingale (see, e.g., Theorem 2.1 in [START_REF] Jacod | Grossissement initial, hypothèse (H ′ ) et théorème de Girsanov[END_REF] or Proposition 5.30, page 116 in [START_REF] Aksamit | Enlargement of ltration with nance in view[END_REF]). Note that, according to Proposition 4.20 in [START_REF] Aksamit | Enlargement of ltration with nance in view[END_REF], the ltration F (τ ) is right-continuous.

We further denote F (τ ) -optional processes with the superscript (τ ) as in Y (τ ) . We denote F-adapted processes by capital letters as X, or lower case x, or φ, or even x 0 .

We also recall that, for any t ≥ 0 xed, any F (τ )

t -measurable random variable Y (τ ) t is of the form Y t (ω, τ (ω)), for some F t ⊗ B(R + )-measurable function (ω, u) → Y t (ω, u) (see, e.
g., Proposition 2.7, part (i) in [START_REF] Callegaro | Carthagian enlargement of ltrations[END_REF]). In particular, any F (τ ) 0 -measurable random variable is a Borel function of τ . Recall that any F (τ ) -predictable process can be represented in the form Y t (ω, τ (ω)), for all t ≥ 0, where the mapping (ω, t, u)

→ Y t (ω, u) dened on Ω × R + × R + and valued in R is P(F) ⊗ B(R + )-measurable. Moreover, under Jacod's hypothesis, any F (τ ) - optional process Y (τ ) = (Y (τ ) t ) t≥0 can be written as Y (τ ) t = Y t (τ ), for all t ≥ 0, where the process Y = (Y t (u), t ≥ 0, u ∈ R + ) is O(F) ⊗ B(R + )-measurable (see Theorem 6.9 in [28]).
As an immediate consequence of Jacod's hypothesis, we observe that, for each t ≥ 0, if the

F (τ ) t -measurable random variable Y t (τ ) is integrable, then the following representation holds E Y t (τ ) F t = ∞ 0 Y t (u) p t (u) ρ(du), ∀t ≥ 0
(see, e.g., Proposition 4.18 (b), page 85 in [START_REF] Aksamit | Enlargement of ltration with nance in view[END_REF]).

In the following proposition, we give the F (τ ) -semimartingale decomposition of µ, dened in [START_REF] Aksamit | Integral representations of martingales for progressive enlargements of ltrations[END_REF]. Proposition 4.1 The F (τ ) -semimartingale decomposition of the F-martingale µ is given by

µ((0, t], A) = µ (τ ) ((0, t], A) + t 0 A f s (τ, z) p s-(τ ) η s (dz) ds, ∀t ≥ 0 , ∀A ∈ B(R) ,
where µ (τ ) is an F (τ ) -martingale and f is given in [START_REF] Bielecki | Special semimartingales and shrinkage of ltration[END_REF]. We assume that ν τ ) has a nite mass.

In other terms, the process (T n , Z n ) n≥1 is a marked point process with F (τ ) -compensator ν (τ ) , where we have

ν (τ ) (dt, dz) = 1 + f t (τ, z) p t-(τ ) η t (dz) dt, ∀t ≥ 0 , ∀z ∈ R . (7)
In particular, we have

1 + f t (τ, z)/p t-(τ ) ≥ 0, ∀t ≥ 0, ∀z ∈ R.
Proof: From the results of initial enlargement 1 , the process µ (τ ) dened by

µ (τ ) ((0, t], A) = µ((0, t], A] - t 0 d⟨ µ((0, •], A), p(u)⟩ F s p s-(u) u=τ , ∀t ≥ 0 , ∀A ∈ B(R) ,
is an F (τ ) -martingale. In order to compute the predictable covariation, we start by computing the quadratic covariation of the processes µ and p(u), for each u ≥ 0. Obviously, we have

µ, p(u) t = t 0 R f s (u, z) µ(ds, dz), ∀t, u ≥ 0 ,
and hence, from Theorem 6.28, part 2 in [START_REF] He | Semimartingale Theory and Stochastic Calculus[END_REF],

µ, p(u)

F t = t 0 R f s (u, z) η s (dz) ds, ∀t, u ≥ 0 . It follows that µ (τ ) ((0, t], R) = µ((0, t], R) - t 0 R f s (τ, z) p s-(τ ) + 1 η s (dz) ds, ∀t ≥ 0 , is an F (τ ) -martingale and the F (τ ) -compensator of µ is ν (τ ) (dt, dz) = f t (τ, z) p t-(τ ) + 1 η t (dz) dt, ∀t ≥ 0 , ∀z ∈ R , since the process µ (τ ) = µ -ν - • 0 R f s (τ, z) p s-(τ ) η s (dz) ds ≡ µ -ν (τ )
is an F (τ ) -martingale. This completes the proof. □

Note that the weak predictable representation property for the marked point process µ (τ ) holds in F (τ ) (See Proposition 4.6 or [START_REF] Bandini | Progressively enlargement of ltrations and control problems for step processes[END_REF]). 1 One applies Theorem 2.1 in [START_REF] Jacod | Grossissement initial, hypothèse (H ′ ) et théorème de Girsanov[END_REF] which states that, under Jacod's hypothesis, for any F-martingale X = (X t ) t≥0 , the process X(τ ) = (X t (τ )) t≥0 dened by

X t (τ ) = X t - t 0 d⟨X, p(u)⟩ F s p s-(u) u=τ , ∀t ≥ 0 , is an F (τ ) -martingale. Note that p(τ ) does not vanish. Proposition 4.2 Each (P, F (τ ) )-martingale Y (τ ) = (Y t (τ )) t≥0 admits a representation of the form Y t (τ ) = Y 0 (τ ) + t 0 R ψ s (τ, z) µ (τ ) (ds, dz), ∀t ≥ 0 , (8)
for some

P(F) ⊗ B(R + ) ⊗ B(R)-measurable process ψ satisfying t 0 R |ψ s (τ, z)| ν (τ ) (ds, dz) < ∞, ∀t ≥ 0 ,
where ν (τ ) is dened in [START_REF] Brémaud | Point Processes and Queues: Martingale Dynamics[END_REF].

The progressively enlarged ltration

We denote by G = (G t ) t≥0 the progressive enlargement of F with τ , that is, the right-continuous version of G 0 where

G 0 t = s>t F s ∨ σ(τ ∧ s) , ∀t ≥ 0 . (9)
Note that τ is a G-stopping time and that, according to the hypothesis that the random variable τ is strictly positive, the σ-algebra G 0 is trivial, so that the initial value of a G-adapted process is a deterministic one. Observe that, under Jacod's hypothesis, any F-martingale is a G-semimartingale (see, e.g., Proposition 5.30, page 116 in [START_REF] Aksamit | Enlargement of ltration with nance in view[END_REF] or Theorem 3.1 in [START_REF] Jeanblanc | Progressive enlargement of ltrations with initial times[END_REF]), and thus, a special semimartingale according to Chapter VI, Theorem 4, page 367 in [START_REF] Protter | Stochastic Integration and Dierential Equations[END_REF].

We observe that the completion of the two enlargements G and

F (τ ) follows from F ∞ ⊂ G ∞ ⊂ F (τ ) ∞ ⊂ A.
We further indicate with the superscript G the processes which are G-adapted, as Y G , as we shall do now for the G-adapted process 1 1 {τ ≤t} which will be denoted H G . We recall that, under Jacod's hypothesis, any G-optional process Y G can be written as

Y G t = 1 1 {τ >t} Y 0 t + 1 1 {τ ≤t} Y 1 t (τ ), ∀t ≥ 0 , where Y 0 is F-optional and Y 1 is O(F) ⊗ B(R + )-
measurable (see Theorem 6.9 in [START_REF] Song | Optional splitting formula in a progressively enlarged ltration[END_REF]). A particular case occurs when Y G is the optional projection of a process Y (τ ) . In that case, one has

Y 0 t = 1 G t ∞ t Y t (u) p t (u) ρ(du), ∀t ≥ 0 , and Y 1 t (u) = Y t (u), ∀t, u ≥ 0 with t ≥ u ,
where the process G is dened in [START_REF] Björk | Point Processes and Jump Diusions: An Introduction with Finance Applications[END_REF]. Here, Y 0 is called the F-optional reduction of Y G . We also recall that any G-predictable process Y G t = (Y G t ) t≥0 can be written as

Y G t = 1 1 {τ ≥t} Y 0 t + 1 1 {τ <t} Y 1 t (τ ), ∀t ≥ 0 ,
where the process Y 0 is F-predictable and Y 1 is P(F) ⊗ B(R + )-measurable (see, e.g., Proposition 2.11, page 36 in [START_REF] Aksamit | Enlargement of ltration with nance in view[END_REF]). In this case, Y 0 is called the F-predictable reduction of Y G . Note that the càg process (1 1 {τ <t} ) t≥0 is G-predictable.

As it follows from the Doob-Meyer decomposition of the supermartingale H G and the fact that any G-predictable process is equal, on the set {τ ≥ t} to an F-predictable process, there exists an F-predictable increasing process Λ = (Λ t ) t≥0 such that the process

M G = (M G t ) t≥0
dened by

M G t = H G t -Λ t∧τ , ∀t ≥ 0 , (10)
is a G-martingale. It is known that, under Jacod's hypothesis, the process Λ admits the representation (we also use the fact that ρ has no atoms)

Λ t = t 0 p s (s) G s ρ(ds) = t 0 p s-(s) G s- ρ(ds), ∀t ≥ 0 ,
(see Proposition 4.4 in [START_REF] El Karoui | What happens after a default: the conditional density approach[END_REF] or Corollary 5.27 (b), page 114 in [START_REF] Aksamit | Enlargement of ltration with nance in view[END_REF]). In this respect, the process λ = (λ t ) t≥0 dened by λ t = p t-(t)/G t-, for t ≥ 0, is the intensity rate of τ with respect to the measure ρ (see Proposition 2.15, page 37 in [START_REF] Aksamit | Enlargement of ltration with nance in view[END_REF]).

The Doob-Meyer decomposition of the Azéma supermartingale can be given explicitly and its multiplicative decomposition is as follows.

Proposition 4.3 Suppose that Jacod's hypothesis holds. The Doob-Meyer decomposition of the Azéma supermartingale G is

G t = 1 - t 0 G s λ s ρ(ds) + t 0 R ∞ s f s (u, z) µ(ds, dz) ρ(du), ∀t ≥ 0 , (11)
or in a simplied form

G t = 1 - t 0 G s λ s ρ(ds) + t 0 R φ(s, z) µ(ds, dz), ∀t ≥ 0 ,
where

φ(t, z) = ∞ t f t (u, z) ρ(du), ∀t ≥ 0 , ∀z ∈ R . (12)
Proof: The Doob-Meyer decomposition of G is obtained using Itô-Ventzell formula as developed in Theorem 3.1 in [START_REF] Øksendal | The Itô-Ventzell formula and forward stochastic dierential equations driven by Poisson random measures[END_REF] to the process

G t (x) = P(τ > x | F t ) = ∞ x p t (u) ρ(du) = ∞ x p 0 (u)ρ(du) + t s=0 R ∞ u=x f s (u, z) µ(ds, dz) ρ(du), ∀t ≥ 0 ,
with a parameter x, where the forward integral (with respect to the compensated measure) in [START_REF] Øksendal | The Itô-Ventzell formula and forward stochastic dierential equations driven by Poisson random measures[END_REF] is the usual stochastic integral in our setting since we integrate predictable processes. Then (with the notation of [START_REF] Øksendal | The Itô-Ventzell formula and forward stochastic dierential equations driven by Poisson random measures[END_REF]

γ = 0, H(s, x) = ∞ x f s (u, z)ρ(du)), G t = G t (t) = 1 - t 0 λ s G s ρ(ds) + t 0 ∞ s R f s (u, z) ρ(du) µ(ds, dz), ∀t ≥ 0 .

□

In the following proposition, we give the semimartingale decomposition of the process µ dened in [START_REF] El Karoui | What happens after a default: the conditional density approach[END_REF] in the ltration G.

µ((0, t], A) = µ G ((0, t], A) + t∧τ 0 A φ s (z) G s- η s (dz) ds + t t∧τ A f s (τ, z) p s-(τ ) η s (dz) ds, ∀t ≥ 0 , ( 13
)
where µ G ((0, •], A) is a G-martingale, φ is dened in [START_REF] El Karoui | What happens after a default: the conditional density approach[END_REF], and f is dened in [START_REF] Bielecki | Special semimartingales and shrinkage of ltration[END_REF]. The predictable random measure

ν G (dt, dz) = 1 1 {τ ≥t} φ t (z) G t + 1 + 1 1 {τ <t} f t (τ, z) p t-(τ ) + 1 η t (dz) dt, ∀t ≥ 0, ∀z ∈ R , ( 14
)
is the G-compensator of the random jump measure µ.

Proof: Recall that the process G admits a Doob-Meyer decomposition as G = m -(H G ) p,F with a martingale m = (m t ) t≥0 (see 2 ) where, from ( 11) and ( 12), we get

m t = t 0 R φ s (z) µ(ds, dz) , ∀t ≥ 0 .
The G-semimartingale decomposition 3 of the F-martingale µ is given by

µ((0, t], A) = µ G ((0, t], A) + t∧τ 0 d⟨ µ(A), m⟩ F s G s- + t t∧τ d⟨ µ(A), p(u)⟩ F s p s-(u) u=τ = µ G ((0, t], A) + t∧τ 0 A φ s (z) G s- η s (dz) ds + t t∧τ A f s (τ, z) p s-(τ ) η s (dz) ds , ∀t ≥ 0, ∀A ∈ B(R) ,
where ( µ G ((0, t], A)) t≥0 is a G-martingale. It thus follows that the G-compensator of µ is given by [START_REF] Gapeev | Projections of martingales in enlargements of Brownian ltrations under Jacod's equivalence hypothesis[END_REF].

□

Remark 4.5 The compensator ν G of µ in ( 14) being increasing, the process f t (τ, z)/p t-(τ )+1, ∀t ≥ 0, ∀z ∈ R, is nonnegative, as well as the process φ t (z)/G t + 1, ∀t ≥ 0, ∀z ∈ R.

Y G t = Y G 0 + t 0 R α G s (z) µ G (ds, dz) + t 0 β 0 s dM G s , ∀t ≥ 0 , ( 15
)
2 It is known that, under the assumption that the random time τ avoids all F-stopping times, the dual optional projection of H is continuous and equal to the dual predictable projection of H, denoted by H p (see Proposition 1.48 (a), page 22 in [START_REF] Aksamit | Enlargement of ltration with nance in view[END_REF]). Therefore the martingale m which appears in the general formulae of the semimartingale decomposition (see Proposition 5.30, page 116 in [START_REF] Aksamit | Enlargement of ltration with nance in view[END_REF]) is equal to the martingale part of the Doob-Meyer decomposition of G, that is, one has G = m-H p with an F-martingale m = (m t ) t≥0 . In particular, the predictable projection of

G is p G = p m -H p = m --H p = G -.
3 One can use Remark 2 and Theorem 5.30, page 116 in [START_REF] Aksamit | Enlargement of ltration with nance in view[END_REF] to deduce that, for any F-martingale X, the process X G = (X G t ) t≥0 dened by

X G t = X t - t∧τ 0 d⟨X, m⟩ F s G s- - t t∧τ d⟨X, p(u)⟩ F s p s-(u) u=τ , ∀t ≥ 0 , is a G-martingale.
for some

P(G) ⊗ B(R)-measurable process α G satisfying t 0 R α G s (z) ν G (ds, dz) < ∞, ∀t ≥ 0 ,
where ν G is dened in [START_REF] Gapeev | Projections of martingales in enlargements of Brownian ltrations under Jacod's equivalence hypothesis[END_REF]. Here, the process α G is of the form

α G t (z) = 1 1 {τ ≥t} α 0 t (z) + 1 1 {τ <t} α t (τ, z), ∀t ≥ 0, ∀z ∈ R , (16)
where α 0 is a P(F) ⊗ B(R)-measurable process, α is a P(F) ⊗ B(R + ) ⊗ B(R)-measurable process, while β 0 is an F-predictable process.

Proof: Due to the avoidance property, the process µ((0, •], A) + H G is a G-measurable marked point process, with sequence of jumps (∪ n T n ) ∪ τ and jumps (Z n , n ≥ 1) (after ordering the jump times). The result follows from the denition of M G in (10) above. See also [START_REF] Bandini | Progressively enlargement of ltrations and control problems for step processes[END_REF]. □ Remark 4.7 Note that, if the process β G admits the representation

β G t = 1 1 {τ ≥t} β 0 t + 1 1 {τ <t} β 1 t (τ ), ∀t ≥ 0 , then the equality t 0 β G s dM G s = t 0 β 0 s dM G s , ∀t ≥ 0 ,
holds, for any choice of the P(G) ⊗ B(R)-measurable process β 1 , since M G is at after τ (i.e., M G t = M G t∧τ , for all t ≥ 0).

Optional projections of martingales

Let Y (τ ) be an F (τ ) -martingale. Then, Y (τ ) admits the integral representation given by [START_REF] Callegaro | Carthagian enlargement of ltrations[END_REF]. We study the G-optional projection Y G (a G-martingale) of the process Y (τ ) and the F-optional projection Y (an F-martingale) of Y (τ ) . The G-martingale Y G admits the integral representation given by [START_REF] Grorud | Insider trading in a continuous time market model[END_REF], with some process α G in the form of ( 16), which has a P(F) ⊗ B(R)-measurable process α 0 , a P(F) ⊗ B(R + ) ⊗ B(R)-measurable process α, as well as an F-predictable process β 0 .

Observe that any square integrable F-martingale Y admits the representation (3) with some

P(F) ⊗ B(R)-measurable process ξ satisfying E t 0 R ξ 2 s (z) η s (dz) ds < ∞, ∀t ≥ 0 ,
(see Chapter VIII, Theorem T8, page 239 in [START_REF] Brémaud | Point Processes and Queues: Martingale Dynamics[END_REF]).

Similarly, we observe that any square integrable F (τ ) -martingale Y (τ ) admits the representation (8) with some P(F) ⊗ B(R

+ ) ⊗ B(R)-measurable process ψ satisfying E t 0 R ψ 2 s (τ, z) ν (τ ) (ds, dz) < ∞, ∀t ≥ 0 , (17)
where ν (τ ) is dened in [START_REF] Brémaud | Point Processes and Queues: Martingale Dynamics[END_REF] (see Chapter VIII, Theorem T8, page 239 in [START_REF] Brémaud | Point Processes and Queues: Martingale Dynamics[END_REF]).

Finally, we observe that any square integrable G-martingale Y G admits the representation [START_REF] Grorud | Insider trading in a continuous time market model[END_REF] with some P(G) ⊗ B(R)-measurable process α G satisfying

E t 0 R α G s (z) 2 ν G (ds, dz) < ∞, ∀t ≥ 0 , (18)
and F-predictable process β 0 , where ν G is dened in [START_REF] Gapeev | Projections of martingales in enlargements of Brownian ltrations under Jacod's equivalence hypothesis[END_REF] (see Chapter VIII, Theorem T8, page 239 in [START_REF] Brémaud | Point Processes and Queues: Martingale Dynamics[END_REF]).

5.1 The projections of F (τ ) -martingales on G Proposition 5.1 Let Y (τ ) be an F (τ ) -martingale with the representation (8) above:

Y t (τ ) = Y 0 (τ ) + t 0 R ψ s (τ, z) µ (τ ) (ds, dz)
for some

P(F) ⊗ B(R + ) ⊗ B(R)-measurable process ψ, We study the G-optional projection Y G of the process Y (τ ) . Note that Y G is a G-martingale.
The G-martingale Y G admits the integral representation given by ( 15), with some process α G in the form of ( 16), which has a P(F) ⊗ B(R)-measurable process α 0 , a P(F) ⊗ B(R + ) ⊗ B(R)-measurable process α, as well as an F-predictable process

β 0 . Then Y G 0 = E[Y t (τ )].
The P(F) ⊗ B(R)-measurable process α 0 , the P(F) ⊗ B(R + ) ⊗ B(R)-measurable process α and the F-predictable process β 0 are of the form

α 0 t (z) = 1 φ t (z) + G t- (19) × ∞ t ψ t (u, z) + Y t-(u) f t (u, z) + p t-(u) -Y t-(u) φ t (z) G t- + 1 p t-(u) ρ(du), ∀t ≥ 0 , ∀z ∈ R , α t (u, z) = ψ t (u, z), ∀u ≥ t ≥ 0 , ∀z ∈ R , (20)
β 0 t = p,F Y t-(t) -Y 0 t-, ∀t ≥ 0 , (21)
where

Y 0 = (Y 0 t ) t≥0 is the F-predictable reduction of Y G given by Y 0 t-= 1 G t- ∞ t Y t-(u) p t-(u) ρ(du), ∀t ≥ 0 , (22)
and Y 0 -is its left limit.

Proof: In the rst part of the proof (the rst and the second step), we assume that the F (τ ) -martingale Y (τ ) is square integrable, so that the G-martingale Y G is square integrable too.

In the rst step, we determine α G (z), for each z ∈ R, and, in the second step, we determine β 0 . We generalize the result to any F (τ ) -martingale by localisation in the second part of the proof (third step).

We introduce the sign TP

= to indicate that the tower property for conditional expectations is applied.

First step: We assume that the F (τ ) -martingale Y (τ ) is square integrable, so that the G-martingale Y G is square integrable too. In particular, Y 0 (τ ) is square integrable and the P(F)⊗B(R + )⊗B(R)-measurable process ψ satises [START_REF] Jacod | Grossissement initial, hypothèse (H ′ ) et théorème de Girsanov[END_REF] as well as the P(G)⊗B(R)-measurable process α G satises [START_REF] Jacod | Calcul Stochastique et Problèmes de Martingales[END_REF]. Then, consider a bounded P(G) ⊗ B(R)-measurable process γ G such that γ G (z) = (γ G t (z)) t≥0 , for each z ∈ R, as well as a bounded F-predictable process θ 0 = (θ 0 t ) t≥0 , and dene the process

K G = (K G t ) t≥0 by K G t = K G 0 + t 0 R γ G s (z) µ G (ds, dz) + t 0 θ 0 s dM G s , ∀t ≥ 0 , (23)
where ν G is dened in [START_REF] Gapeev | Projections of martingales in enlargements of Brownian ltrations under Jacod's equivalence hypothesis[END_REF]. It is seen that the process K G is a square integrable G-martingale, since γ G satises the condition

E t 0 R γ G s (z) 2 ν G (ds, dz) < ∞, ∀t ≥ 0 , (24)
and the process θ 0 is F-predictable and bounded. In this case, the square integrable random variable

Y G t = E[Y t (τ ) | G t ] is the only G t -measurable random variable such that E Y t (τ ) K G t = E Y G t K G t , ∀t ≥ 0 , (25)
holds. Thus, since one has

E Y t (τ ) K G 0 = E Y G t K G 0 , ∀t ≥ 0 ,
the equality ( 25) is equivalent to the system of two following equalities

E Y t (τ ) t 0 R γ G s (z) µ G (ds, dz) = E Y G t t 0 R γ G s (z) µ G (ds, dz) , ∀t ≥ 0 , (26)
and

E Y t (τ ) t 0 θ 0 s dM G s = E Y G t t 0 θ 0 s dM G s , ∀t ≥ 0 . (27)
We now determine the processes α 0 and α from the equality [START_REF] Protter | Stochastic Integration and Dierential Equations[END_REF]. On the one hand, one has

E Y t (τ ) t 0 R γ G s (z) µ(ds, dz) -ν G (ds, dz) = E Y t (τ ) t 0 R γ G s (z) µ(ds, dz) -ν (τ ) (ds, dz) + t 0 R γ G s (z) ν (τ ) (ds, dz) -ν G (ds, dz) , ∀t ≥ 0 ,
where ν (τ ) is dened in [START_REF] Brémaud | Point Processes and Queues: Martingale Dynamics[END_REF]. Integrating by parts on the time interval [0, t] the product the two F (τ ) -martingales Y (τ ) and Υ = (Υ t ) t≥0 dened by

Υ t = t 0 R γ G s (z) µ (τ ) (ds, dz), ∀t ≥ 0 , (28)
and taking into account the fact that Υ t-dY t (τ ) and Y t-(τ )dΥ t correspond to true martingales, as we shall prove in Appendix below, one has

E Y t (τ ) t 0 R γ G s (z) µ (τ ) (ds, dz) = E t 0 R γ G s (z) ψ s (τ, z) ν (τ ) (ds, dz) , ∀t ≥ 0 .
Now, integrating by parts on the time interval [0, t] the product of the martingale Y (τ ) and the bounded variation process Γ(τ ) = (Γ t (τ )) t≥0 dened by

Γ t (τ ) = t 0 R γ G s (z) ν (τ ) (ds, dz) -ν G (ds, dz) , ∀t ≥ 0 , one obtains E Y t (τ ) Γ t (τ ) = E t 0 R Y s-(τ ) γ G s (z) ν (τ ) (ds, dz) -ν G (ds, dz) , ∀t ≥ 0 .
On the other hand, one has by integration by parts

E Y G t t 0 R γ G s (z) µ (τ ) (ds, dz) = E t 0 R γ G s (z) α G s (z) ν G (ds, dz) , ∀t ≥ 0 .
Finally, ( 26) is equivalent to, for any γ G satisfying (24), we have

E t 0 R γ G s (z) ψ s (τ, z) ν (τ ) (ds, dz) + Y s-(τ ) ν (τ ) (ds, dz) -ν G (ds, dz) = E t 0 R γ G s (z) α G s (z) ν G (ds, dz) , ∀t ≥ 0 . ( 29
)
For γ G (z) such that γ G t (z) = 1 1 {τ ≥t} γ 0 t (z), ∀t > 0, ∀z ∈ R, where γ 0 is P(F) ⊗ B(R)-measurable, using the identities ( 7) and ( 14), we have

E t 0 R γ 0 s (z) 1 1 {τ ≥s} ψ s (τ, z) + Y s-(τ ) f s (τ, z) p s-(τ ) + 1 -Y s-(τ ) φ s (z) G s- + 1 η s (dz) ds = E t 0 R γ 0 s (z) 1 1 {τ ≥s} α 0 s (z) φ s (z) G s- + 1 η s (dz) ds , ∀t ≥ 0 , (30)
and, introducing by tower property a conditioning with respect to F s and using the existence of the conditional density, setting F t (u, z) = f t (u, z)/p s-(u) + 1 and Φ t (z) = φ t (z)/G t-+ 1, ∀t, u ≥ 0, ∀z ∈ R, the left-hand side of (30) is equal to

E t 0 R γ 0 s (z) 1 1 {τ ≥s} ψ s (τ, z) + Y s-(τ ) F s (τ, z) -Y s-(τ ) Φ s (z) η s (dz) ds TP = E t 0 R γ 0 s (z) ∞ s ψ s (u, z) + Y s-(u) F s (τ, z) -Y s-(u) Φ s (z) p s-(u) ρ(du) η s (dz) ds , ∀t ≥ 0 , ( 31
)
where, in the last equality, we have used the fact that the F-predictable projection of p(u) is p -(u), the process p(u) being a martingale, for each u ≥ 0 xed. We note also that, using the fact that G -is the F-predictable projection of G (see Remark 2), the right-hand side of ( 30) is

E t 0 R γ 0 s (z) 1 1 {τ ≥s} α 0 s (z) Φ s (z) η s (dz) ds = E t 0 R γ 0 s (z) G s α 0 s (z) Φ s (z) η s (dz) ds = E t 0 R γ 0 s (z) G s-α 0 s (z) Φ s (z) η s (dz) ds , ∀t ≥ 0 . ( 32
)
It follows from (30) that the right-hand sides of ( 31) and (32) are equal, for any γ 0 , and hence,

∞ t ψ t (u, z) + Y t-(u F t (u, z) -Y t-(u) Φ t (z) p t-(u) ρ(du) = ∞ t ψ t (u, z) + Y t-(u) f t (u, z) + p t (u) -Y t-(u) φ t (z) G t- + 1 p t-(u) ρ(du) = G t-α 0 t (z) Φ t (z) = α 0 t (z) φ t (z) + G t -, ∀t ≥ 0, ∀z ∈ R ,
and expression ( 19) holds.

Using the identities ( 7) and ( 14), for γ G of the form γ G t = γ t (τ, z)1 1 {τ <t} , ∀s > 0, for γ ∈ P(F) ⊗ B(R + ) ⊗ B(R), equality (29) leads to

E t 0 R γ s (τ, z) 1 1 {τ <s} ψ s (τ, z) F s (τ, z) η s (dz) ds = E t 0 R γ s (τ, z) 1 1 {τ <s} α s (τ, z) F s (τ, z) η s (dz) ds , ∀t ≥ 0 ,
and we can choose α = ψ on the event {τ < t}, so that expression (20) holds. Second step: In the second step, we compute the value of β 0 , from the expression [START_REF] Runggaldier | Jump-diusion models[END_REF]. It

is straightforward to see that E Y G t t 0 θ 0 s dM G s = E t 0 β 0 s θ 0 s λ s 1 1 {τ >s} ρ(ds) TP = E t 0 β 0 s θ 0 s λ s G s ρ(ds) , ∀t ≥ 0 .
From the denition of M G , it follows that

E Y t (τ ) t 0 θ 0 s dM G s = E Y t (τ ) 1 1 {τ ≤t} θ 0 τ - t 0 1 1 {τ >s} θ 0 s λ s ρ(ds) TP = E t 0 Y t (s) θ 0 s p t (s) ρ(ds) - t 0 θ 0 s λ s E Y (τ ) s 1 1 {τ >s} F s ρ(ds) = E t 0 Y s-(s) p s-(s) θ 0 s ρ(ds) - t 0 θ 0 s λ s ∞ s Y s-(u) p s-(u) ρ(du) ρ(ds) = E t 0 p,F Σ s p s-(s) θ 0 s ρ(ds) - t 0 θ 0 s λ s ∞ s Y s-(u) p s-(u) ρ(du) ρ(ds) , ∀t ≥ 0 ,
where we have used in the third equality that Y (u)p(u) is an F-martingale [START_REF] Aksamit | Enlargement of ltration with nance in view[END_REF]Pro 4.33] with predictable projection Y -(u)p -(u), for each u ≥ 0, and dened Σ = (Σ t ) t≥0 by Σ t = Y t-(t), for all t ≥ 0. We are not able to give conditions so that Σ is predictable, since we do not have regularity of the process Y t-(u) with respect to u, for each u ≥ 0, this is why we have to take its predictable projection.

It follows that

β 0 t = 1 λ t G t- p,F Σ t p t-(t) -λ t ∞ t Y t-(u) p t-(u) ρ(du) (33) = p,F Σ t - 1 G t- ∞ t Y t-(u) p t-(u) ρ(du), ∀t ≥ 0 ,
where we have used the fact that λ t = p t-(t)/G t-, for t ≥ 0. The expression in (33) implies the fact means that (21) holds with [START_REF] Last | Marked Point Processes on the Real Line. The Dynamic Approach[END_REF].

Third step: The extension to F (τ ) -martingales is done using usual localisation procedure (see Third step of Proof of Proposition 5.1 in [START_REF] Gapeev | Projections of martingales in enlargements of Brownian ltrations under Jacod's equivalence hypothesis[END_REF]).

□

5.2

The projections of F (τ ) -martingales on F Proposition 5.2 Let Y (τ ) be an F (τ ) -martingale with the representation given by equality [START_REF] Callegaro | Carthagian enlargement of ltrations[END_REF].

Then, its F-optional projection Y = (Y t ) t≥0 admits the representation (3), with P(F) ⊗ B(R)measurable process ξ, given by

ξ t (z) = ∞ 0 ψ t (u, z) f t (u, z) + p t-(u) + Y t-(u) f t (u, z) ρ(du), ∀t ≥ 0 .
Proof: As before, we assume that Y (τ ) is square integrable. Then, consider a bounded P(F) ⊗ B(R)-measurable process ζ such that ζ(z) = (ζ t (z)) t≥0 , for each z ∈ R, and dene the process K = (K t ) t≥0 by

K t = K 0 + t 0 R ζ s (z) µ(ds, dz), ∀t ≥ 0 .
It is seen that the process K is a square integrable G-martingale, since the process ζ satises the condition

E t 0 R ζ 2 s (z) η s (dz) ds < ∞, ∀t ≥ 0 .
In this case, the square integrable random variable

Y t = E[Y t (τ ) | F t ] is the only F t -measurable random variable such that E Y t (τ ) K t = E Y t K t , ∀t ≥ 0 , (34)
holds. Thus, the equality (34) is equivalent to the following equality

E Y t (τ ) t 0 R ζ s (z) µ(ds, dz) = E Y t t 0 R
ζ s (z) µ(ds, dz) , ∀t ≥ 0 .

On the one hand, one has

E Y t (τ ) t 0 R ζ s (z) µ(ds, dz) = E Y t (τ ) t 0 R ζ s (z) µ (τ ) (ds, dz) + Y t (τ ) t 0 R ζ s (z) ν (τ ) (ds, dz) -η s (dz) ds , ∀t ≥ 0 ,
where ν (τ ) is dened in [START_REF] Brémaud | Point Processes and Queues: Martingale Dynamics[END_REF]. Integrating by parts on the time interval [0, t] the product of the two F (τ ) -martingales Y (τ ) and Φ(τ ) = (Φ t (τ )) t≥0 dened by

Φ t (τ ) = t 0 R ζ s (z) µ (τ ) (ds, dz), ∀t ≥ 0 ,
one has, using the square integrability assumption, that

E Y t (τ ) Φ t (τ ) = E t 0 R ζ s (z) ψ s (τ, z) ν (τ ) (ds, dz) = E t 0 R ζ s (z) ψ s (τ, z) F s (τ, z) η s (dz) ds , ∀t ≥ 0 .
By integrating by parts the product of Y (τ ) and the process ∆(τ ) = (∆ t (τ )) t≥0 of bounded variation dened by

∆ t (τ ) = t 0 R
ζ s (z) ν (τ ) (ds, dz) -η s (dz) ds , ∀t ≥ 0 , one obtains, using the equality [START_REF] Brémaud | Point Processes and Queues: Martingale Dynamics[END_REF], that

E Y t (τ ) ∆ t (τ ) = E t 0 R ζ s (z) Y s-(τ ) f s (τ, z) p s (τ ) η s (dz) ds , ∀t ≥ 0 .
Hence, we have

E Y t (τ ) t 0 R ζ s (z) µ(ds, dz) = E Y t (τ ) Φ t (τ ) + E Y t (τ ) ∆ t (τ ) , ∀t ≥ 0 .
On the other hand, one has

E Y t t 0 R ζ s (z) µ(ds, dz) = E t 0 R ζ s (z) ξ s (z) η s (dz) ds , ∀t ≥ 0 .
Finally, the expression (34) implies

E t 0 R ζ s (z) ∞ 0 ζ s (u, z) f s (u, z) + Y s-(u) f s (u, z) -1 p s-(u) ρ(du) η s (dz) ds = E t 0 R ζ s (z) ξ s (z) η s (dz) ds , ∀t ≥ 0 ,
then, we obtain the expression (5.2).

□

5.3

The projections of G-martingales on F Proposition 5.3 Let Y G be a G-martingale with the representation given by equality ( 15)

Y G t = Y G 0 + t 0 R α G s (z) µ G (ds, dz) + t 0 β 0 s dM G s , ∀t ≥ 0 ,
and the decomposition given in (4.2) above. Then, its F-optional projection Y is given by (3) above,

Y t = Y 0 + t 0 R ξ s (z) µ(ds, dz), ∀t ≥ 0
where the P(F) ⊗ B(R)-measurable process ξ is given by

ξ t (z) = α 0 t (z) φ t (z) + G t-+ Y 0 t-φ t (z) + t 0 α t (u, z) f t (u, z) + p t-(u) + Y t-(u) f t (u, z) p t-(u) ρ(du), ∀t ≥ 0, ∀z ∈ R ,
with the supermartingale G given by the equality [START_REF] Björk | Point Processes and Jump Diusions: An Introduction with Finance Applications[END_REF].

Proof: As before, for any G-adapted bounded process θ G , we consider the equality satised by Y such that

E Y t t 0 R θ G s µ(ds, dz) = E Y G t t 0 R θ G s µ(ds, dz) , ∀t ≥ 0 .
The left-hand side is equal to

E t 0 R ξ s (z) θ G s η s (dz) ds , ∀t ≥ 0 . The right-hand side is equal to E Y G t t 0 R θ G s µ G (ds, dz) + ν G s (ds, dz) -η s (dz) ds = E t 0 R α G s θ G s ν G (ds, dz) + t 0 R θ G s Y G s-ν G (ds, dz) -η s (dz) ds = E t 0 R θ G s α 0 s (z) φ s (z) G s- + 1 + Y 0 s- φ s (z) G s- η s (dz) 1 1 {τ >s} ds + E t 0 R θ G s s 0 α s (u, z) f s (u, z) + p s-(u) + Y s-(u) f s (u, z) ρ(du) η s (dz) ds , = E t 0 R θ G s α 0 s (z) φ s (z) + G s + Y 0 s-φ s (z) η s (dz) ds + E t 0 R θ G s s 0 α s (u, z) f s (u, z) + p s (u) + Y s-(u) f s (u, z) ρ(du) η s (dz) ds , ∀t ≥ 0 ,
where ν G is dened in [START_REF] Gapeev | Projections of martingales in enlargements of Brownian ltrations under Jacod's equivalence hypothesis[END_REF]. Hence, this equality being true for any θ G , the proof is complete. □ Remark 5.4 Using the same methodology, wee can extend the result in the Brownian case, established in [START_REF] Gapeev | Projections of martingales in enlargements of Brownian ltrations under Jacod's equivalence hypothesis[END_REF] under equivalence Jacod's hypothesis to the case of absolute continuity hypothesis. We do not give details,the results are the same as the ones in [START_REF] Gapeev | Projections of martingales in enlargements of Brownian ltrations under Jacod's equivalence hypothesis[END_REF].

6 Changes of probability measures and applications

In this section, as an example of application of the results from the previous section, we consider the relationships between strictly positive F (τ ) -martingales (or G-martingales) and their optional projections. We then apply the results in a nancial market framework to study the set of equivalent martingale measures in dierent ltrations.

A probability measure Q is said to be locally equivalent to P on the ltration H if there exists a strictly positive H-martingale L = (L t ) t≥0 such that dQ dP Ht = L t , ∀t ≥ 0 .

The martingale L is called the Radon-Nikodym density of Q with respect to P. The locally" terminology is needed, since as in [START_REF] Amendinger | Martingale representation theorems for initially enlarged ltrations[END_REF], we cannot dene the new probability measure Q on H ∞ , because the density process L is not necessarily an uniformly integrable martingale on H.

6.1 The projections of strictly positive F (τ ) -martingales on G Let L(τ ) be a strictly positive F (τ ) -martingale. Then, in particular, we have L 0 (τ ) > 0 (P-a.s.). Moreover, applying Proposition 4.2 with Y (τ ) = L(τ ), we can write L(τ ) in the form of

L t (τ ) = L 0 (τ ) + t 0 L s-(τ ) R * Θ s (τ, z) -1 µ (τ ) (ds, dz) ∀t ≥ 0 , (35)
where L t (τ )(Θ t (τ, z) -1) = ψ t (τ, z), ∀t ≥ 0, ∀z ∈ R. Note that, since at jumps times L Tn = L Tn-Θ Tn , ∀n ≥ 1, one has Θ > 0. Note that, if E[L 0 (τ )] = 1, then we can associate to the strictly positive F (τ ) -martingale L(τ ) the probability measure P locally equivalent to P on the ltration F (τ ) dened by d P dP F (τ ) t = L t (τ ), ∀t ≥ 0 .

Remark 6.1 As in Remark 6.1 in [START_REF] Gapeev | Projections of martingales in enlargements of Brownian ltrations under Jacod's equivalence hypothesis[END_REF], the particular choice of L 0 (τ ) = 1 (P-a.s.) is equivalent to the property P(τ > u) = P(τ > u), for each u ≥ 0.

We now consider the G-optional projection L G = (L G t ) t≥0 of the strictly positive martingale L(τ ). In this case, applying Proposition 4.6 with Y G = L G , and setting L(κ -1) = α G , so that κ > 0, we see that L G admits the representation

L G t = L G 0 + t 0 L G s- R κ G s (z) -1 µ G (ds, dz) + t 0 L G s-ξ 0 s dM G s , ∀t ≥ 0 . ( 36
)
with a strictly positive and P(G) ⊗ B(R)-measurable process κ G satisfying

t 0 R κ G s (z) -1 ν G (ds, dz) < ∞, ∀t ≥ 0 ,
where ν G is dened in [START_REF] Gapeev | Projections of martingales in enlargements of Brownian ltrations under Jacod's equivalence hypothesis[END_REF].

Here, the process κ G is of the form

κ G t (z) = 1 1 {τ ≥t} κ 0 t (z) + 1 1 {τ <t} κ t (τ, z), ∀t ≥ 0, ∀z ∈ R ,
where κ 0 is a P(F)⊗B(R)-measurable process, κ is a P(F)⊗B(R + )⊗B(R)-measurable process, while ξ 0 > -1 is an F-predictable process. Proposition 6.2 Let L(τ ) = (L t (τ )) t≥0 be a strictly positive martingale of the form (35).

Then, its G-optional projection L G satises (36) with the G-predictable processes κ G and the F-predictable process ξ 0 given by

κ G t (z) -1 = 1 1 {τ ≥t} L 0 t-(φ t (z) + G t-) (37)
× ∞ t L t-(u) Θ t (u, z) f t (u, z) + p t-(u) - φ t (z) G t- + 1 p t-(u) ρ(du) + 1 1 {τ <t} Θ t (τ, z) -1 , ∀t ≥ 0 , ∀z ∈ R , ξ 0 t = p,F (L t-(t)) L 0 t- -1, ∀t ≥ 0 , (38)
L 0 t-= 1 G t- ∞ t L t-(u) p t-(u) ρ(du), ∀t ≥ 0 ,
where L 0 = (L 0 t ) t≥0 is the F-predictable reduction of L G and L 0 -is its left-hand limit.

Proof:

Consider the F (τ ) -martingale L(τ ) given by equality (35). In this case, its Goptional projection L G has the form of (36). Then, Proposition 5.1 applies with Y (τ ) = L(τ ) and ψ(u, z) = L -(u)(Θ(u, z) -1), for all u ≥ 0 and z ∈ R, and therefore, equalities ψ =

L G t (κ G t (z) -1) = α G t (z). That is, L 0 t-(κ 0 t (z) -1) = α 0 t (z) and L t-(u)(κ t (u, z) -1) = α t (u, z
), for all u ≥ 0 and z ∈ R, and ξ 0 t L 0 t-= β 0 t hold, for all t ≥ 0 and z ∈ R, and

L t-(u) κ t (u, z) -1 = 1 φ t (z) + G t- × t L t-(u) Θ t (u, z) f t (u, z) + p t-(u) - φ t (z) G t- + 1 p t-(u) ρ(du), ∀t ≥ 0 . □ 6.2
The projections of strictly positive F (τ ) -martingales on F Let L(τ ) be a strictly positive F (τ ) -martingale of the form (35). Then, applying Proposition 5.2 with Y (τ ) = L(τ ), we obtain that the F-optional projection L of L(τ ) admits the integral representation

L t = E L 0 (τ ) + t 0 L s- R χ s (z) µ(ds, dz), ∀t ≥ 0 ,
where the P(F) ⊗ B(R)-measurable process χ is given by

χ t (z) = 1 L t- ∞ 0 L t-(u) Θ t (u, z) f t (u, z) + p t-(u) + f t (u, z) ρ(du), ∀t ≥ 0 , ∀z ∈ R .

The projections of strictly positive G-martingales on F

It follows from Proposition 5.3 that any strictly positive G-martingale L G = (L G t ) t≥0 admits the equivalent representation (36) and, being a process, it admits the decomposition

L G t = 1 1 {τ >t} L 0 t + 1 1 {τ ≤t} L 1 t (τ ), ∀t ≥ 0 ,
where the process L 0 is F-optional and the process L 1 is O(F) ⊗ B(R + )-measurable. By similar arguments, it follows that its F-optional projection L = (L t ) t≥0 admits the integral representation

L t = L G 0 + t 0 L s- R σ s (z) µ(ds, dz), ∀t ≥ 0 ,
where σ(z) = (σ t (z)) t≥0 is an F-predictable process. In order to derive σ, it suces to apply Proposition 5.3 with Y G = L G , (L -(κ(z) -1)) 0 = α 0 (z), L -(κ(z) -1) = α(z) and L -σ(z) = ξ(z), for all z ∈ R, so that Y 0 = L 0 and Y = L. The equality (L -(κ(z) -1)) 0 = L 0 -(κ 0 (z) -1), for all z ∈ R, follows from the denition of predictable reduction. Therefore, we conclude, after easy simplications, that

σ t (z) = 1 L t- L 0 t-κ 0 t (z) φ t (z) + G t--G t + t 0 L 1 t (u) κ t (u, z) -1 f t (u, z) + p t-(u) + f t (u, z) p t-(u) ρ(du) , ∀t ≥ 0 , ∀z ∈ R .

The equivalent martingale measures

Let us now consider a model of a nancial market in which the risky asset price process S = (S t ) t≥0 follows the stochastic dierential equation

dS t = S t δ t dt + S t- R h t (z) µ(dt, dz) ,
where h belongs to P(F) ⊗ B(R) and is greater than -1 (to satisfy the positivity of S). We also assume that the riskless asset has a zero interest rate and δ is F-adapted.

Change of probability in F: A change of the probability measure in F has a Radon-Nikodym density process L = (L t ) t≥0 satisfying the stochastic dierential equation given from the weak predictable representation property [24, Pro. 2.1] by

dL t = L t- R α t (z) -1 µ(dt, dz), L 0 = 1 , (39)
for a strictly positive P(F) ⊗ B(R)-measurable process α.

This Radon-Nikodym density process will correspond to an equivalent martingale measure if SL is an F-martingale. From the integration-by-parts formula, we have

S t L t = S 0 + t 0 L s-dS s + t 0 S s-dL s + [S, L] t , ∀t ≥ 0 ,
where the quadratic variation 4 is given by

S, L t = Tn≤t S Tn-L Tn-h Tn (Z n ) (α Tn (Z n ) -1) = t 0 S s-L s- R h s (z) α s (z) -1 µ(ds, dz) + t 0 S s-L s- R h s (z) α s (z) -1 η s (dz) ds, ∀t ≥ 0 , and S t L t = S 0 + t 0 L s-dS s + t 0 S s-dL s + [S, L] t , = S 0 + t 0 L s-S s-δ s + R h s (z) f s (τ, z) p s-(τ ) η s (dz) ds + t 0 S s-L s- R h s (z) α s (z) -1 η s (dz) ds + M t , ∀t ≥ 0 ,
where the process M = (M t ) t≥0 is an F-martingale.

We now dene

θ t = δ t + R h t (z) f t (τ, z) p t-(τ ) η t (dz)
and we obtain that the process SL is an F-martingale if and only if α (see (39)) satises the equality

θ t + R h t (z) α t (z) -1 η t (dz) = 0, ∀t ≥ 0 .
4 For any two semimartingales X = (X t ) t≥0 and Y = (Y t ) t≥0 , one has

[X, Y ] t = ⟨X c , Y c ⟩ t + 0≤s≤t ∆X s ∆Y s , ∀t ≥ 0 ,
where X c and Y c are continuous (local) martingale parts of X and Y , while ∆X t = X t -X t-and ∆Y t = Y t -Y t- (see Chapter VIII, Denition 8.2, page 209 in [START_REF] He | Semimartingale Theory and Stochastic Calculus[END_REF]).

(see Section 2.1 in [START_REF] Prigent | Option pricing with a general marked point process[END_REF] for similar results). This equivalent martingale measure is unique in that case.

Change of probability in G: In the ltration G, using the equality ( 14), we have

dS t = S t δ t dt + S t- R h t (z) ν G (dt, dz) -ν(dt, dz) + S t- R h t (z) µ G) (dt, dz) = S t-δ t + R h t (z) 1 1 {τ ≥t} φ t (z) G t- + 1 1 {τ <t} f t (τ, z) p t-(τ ) η t (dz) dt + S t- R h t (z) µ G (dt, dz) = S t-θ G t dt + S t- R h t (z) µ G (dt, dz) ,
where the last term is an G-martingale and

θ G t = δ t + R h t (z) 1 1 {τ ≥t} φ t (z) G t- + 1 1 {τ <t} f t (τ, z) p t-(τ ) η t (dz) .
Using the fact that any positive G-martingale has the form

dL G t = L G t- R κ G (z) -1 µ G (dt, dz) + ξ 0 t dM G t
for some strictly positive P(G) ⊗ B(R)-measurable process κ G (see Proposition 4.6 and that

S, L

G G t = t 0 R h s (z) κ G t (z) -1 η G t (dz), ∀t ≥ 0 ,
the set P(G) of (locally) equivalent martingale measures on G corresponds to the set of Radon-Nikodym density processes of the form

dL G t = L G t- R κ G t (z) -1 µ G (dt, dz) + ξ 0 s dM G t with κ G such that θ G t + R h t (z) κ G t (z) -1 ν G (dt, dz) = 0, ∀t ≥ 0 .
This change of probability is not unique, ξ 0 is to be chosen.

Change of probability in F (τ ) : In the ltration F (τ ) , using the representation (7), we have

dS t = S t δ t dt + S t- R h t (z) ν (τ ) (dt, dz) -ν(dt, dz) + S t- R h t (z) µ τ (dt, dz) = S t-δ t + R h t (z) f t (τ, z) p t-(τ ) η t (dz) dt + S t- R h t (z) µ (τ ) (dt, dz) ,
and the last term is an F (τ ) -martingale.

As it is seen in Proposition 4.2, a change of probability in F (τ ) has the Radon-Nikodym density process L(τ ) = (L t (τ )) t≥0 satisfying the stochastic dierential equation given in (35) which can be written in the form dL t (τ ) = L t-(τ ) R Θ t (τ, z) -1 µ (τ ) (dt, dz), L 0 (τ ) = ℓ(τ ) , for a strictly positive P(F (τ ) ) ⊗ B(R)-measurable process Θ and a strictly positive Borel function ℓ(u), for u ≥ 0. This Radon-Nikodym density process will correspond to an equivalent martingale measure if SL(τ ) is an F (τ ) -martingale. Using the fact that S, L(τ ) t = t 0 S s-L s-(τ ) R h s (z) Θ s (τ, z) -1 µ (τ ) (ds, dz) This change of probability is not unique, ℓ being to be chosen. Let P * be the set of G-optional projections L * ,G of L * (τ ), which satises (36) where the processes κ G and ξ 0 given by equalities (37) and (38). More precisely, one has κ G t (z) -1 = 1 1 {τ ≥t}

+ t 0 S s-L s-(τ ) R h s (z) Θ s (τ, z) -1 f s (τ,
where L * ,0 is the F-predictable reduction of L * ,G . Here, each element of P * is a (locally) equivalent martingale measure on G. Note that κ G does not depend on the choice of L * 0 (see (40)), whereas ξ 0 depends on it.

Since in (6.4), there are no constraints on ξ 0 , the set P(G) is strictly greater than P * .

Appendix

Using the same methodology as in [START_REF] Gapeev | Projections of martingales in enlargements of Brownian ltrations under Jacod's equivalence hypothesis[END_REF] we prove the martingale property of the two local martingales used in the proof of Proposition 5.1.

• We rst prove that the F (τ ) -local martingale M (τ ) = ( M t (τ )) t≥0 dened by M t (τ ) = holds, for some C p > 0 depending on p only (see, e.g., Chapter IV, Section 4, Theorem 48, page 195 in [START_REF] Protter | Stochastic Integration and Dierential Equations[END_REF]).

is a true martingale. As above, by Burkholder-Davis-Gundy's inequality, this will be the case when, for any T > 0 xed E ⟨ M (τ )⟩ F (τ ) T 1/2 < ∞ .

Note that we have

E ⟨ M (τ )⟩ F (τ ) T 1/2 = E T 0 Y 2 s (τ ) γ G s (z) 2 ν (τ ) (ds, dz) 1/2 ≤ E sup 0≤s≤T Y s (τ ) T 0 R γ G s (z) 2 ν (τ ) (ds, dz) 1/2 ≤ E sup 0≤s≤T Y s (τ ) 2 + E T 0 R γ G s (z)
2 ν (τ ) (ds, dz) .

Proposition 4 . 4

 44 The G-semimartingale decomposition of the F-martingale µ([0, •], A) is given by for any A ∈ B(R)

Proposition 4 . 6

 46 Each (P, G)-martingale Y G = (Y G t ) t≥0 can be represented as

t 0 Υψ 2 s 2 s 2 s 2 s

 02222 s-dY s (τ ), ∀t ≥ 0 , is a true martingale. This will be the case when, for any T > 0 xed, the propertyE sup 0≤t≤T M t (τ ) < ∞holds (see Chapter I, Theorem 51, page 38 in[START_REF] Protter | Stochastic Integration and Dierential Equations[END_REF]). By Burkholder-Davis-Gundy's inequality 5 , this condition is satised ifE ⟨ M (τ )⟩ F (τ ) T 1/2 < ∞ . Note that we have E ⟨ M (τ )⟩ F (τ ) (τ, z) ν (τ ) (ds, dz) (τ, z) ν (τ ) (ds, dz) (τ, z) ν (τ ) (ds, dz) ,where we have used the fact that |ab| ≤ (a 2 + b 2 ), for any a, b ∈ R. Using again Burkholder-Davis-Gundy's inequality Υ dened in[START_REF] Song | Optional splitting formula in a progressively enlarged ltration[END_REF] being a martingale, we obtain that s ) 2 ν (τ ) (ds, dz) < ∞ , for some constant C > 0. Moreover, by the assumption of square integrability of the F (τ )martingale Y (τ ), we have E (τ, z) ν (τ ) (ds, dz) < ∞ , so that the process M (τ ) is a martingale.• We now prove that the F (τ ) -local martingale M (τ ) = ( M t (τ )) t≥0 dened byM t (τ ) = t 0 Y s (τ ) dΥ s , ∀t ≥ 0 ,5 Burkholder-Davis-Gundy's inequality states that, if M is a local martingale, for any p ≥ 1, then the expressionE sup 0≤t≤T M t p ≤ C p E (⟨M ⟩ T ) p/2

2 ν

 2 It follows, using again Burkholder-Davis-Gundy's inequality, thatE sup 0≤s≤T Y s (τ ) 2 ≤ C E T 0 ψ 2 s (τ, z) ν (τ ) (ds, dz) < ∞ ,for some constant C > 0. Moreover, by the assumption of square integrability of the F (τ )martingale Υ, we have E (τ ) (ds, dz) < ∞ , so that the process M (τ ) is a martingale.

  z) p s-(τ )η s (dz) ds, ∀t ≥ 0 , and the F (τ ) -martingale property of the rst term above, we obtain that the process SL(τ ) is an F (τ ) -martingale if and only if Θ satises the equality δ t +

∞ t h t (z) f t (τ, z) p t-(τ ) η t (dz) + ∞ t h t (z) Θ t (τ, z)) -1 f t (τ, z) p t-(τ ) η t (dz) = 0, ∀t ≥ 0 ,

which can be simplied to

δ t + ∞ t h t (z) Θ t (τ, z) f t (τ, z) p t-(τ )

η t (dz) = 0, ∀t ≥ 0 .

  (u) Θ t (u, z) f t (u, z) + p t-(u) -φ t (z) G t- + 1 p t-(u) ρ(du) + 1 1 {τ <t} Θ t (τ, z) -1 , ∀t ≥ 0 , ∀z ∈ R ,

					1 t-(φ t (z) + G t-) L * ,0	(40)
	∞
	× t-ξ 0 t L * t = p,F (L * t-(t)) t-L * ,0	-1, ∀t ≥ 0 ,	(41)
	L * ,0 t-=	1 G t-	t	∞	L * t-(u) p t-(u) ρ(du), ∀t ≥ 0 ,

ILB, Labex ANR 11-LABX-0019