Projections in enlargements of filtrations under Jacod's equivalence hypothesis for marked point processes *

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Projections in enlargements of filtrations under Jacod’s equivalence hypothesis for marked point processes *

Pavel Gapeev, Monique Jeanblanc, Dongli Wu

To cite this version:

Pavel Gapeev, Monique Jeanblanc, Dongli Wu. Projections in enlargements of filtrations under Jacod’s

equivalence hypothesis for marked point processes *. 2021. �hal-03207058�

Projections in enlargements of filtrations under Jacod’s equivalence hypothesis for marked point processes ^∗

Pavel V. Gapeev

Monique Jeanblanc

Dongli Wu

Abstract

We consider the initial and progressive enlargements of a filtration generated by a marked point process (called the reference filtration) with a strictly positive random time. We assume Jacod’s equivalence hypothesis, that is, the existence of a strictly positive conditional density for the random time with respect to the reference filtration. Then, starting with the predictable integral representation of a martingale in the initially enlarged reference filtration, we derive explicit expressions for the coefficients which appear in the predictable integral representations for the optional projections of the martingale on the progressively enlarged filtration and on the reference filtration. We also provide similar results for the optional projection of a martingale in the progressively enlarged filtration on the reference filtration.

1 Introduction

In this paper, we consider the initial (resp. progressive) enlargements of a filtration F (called hereafter the reference filtration) with a strictly positive random variable τ (called hereafter the random time), denoted byF^(τ) (resp. G). We study the case in which Fis generated by a marked point process (MPP). We assume that the law of τ has no atoms and that Jacod’s equivalence hypothesis introduced in [2, 10] holds (see Section 3 for details). We prove that these hypotheses imply that the weak predictable representation property holds in the filtrationF^(τ) and (adding a

∗This research benefited from the support of ILB, Labex ANR 11-LABX-0019.

†London School of Economics, Department of Mathematics, Houghton Street, London WC2A 2AE, United King- dom; e-mail: p.v.gapeev@lse.ac.uk

‡Université Paris Saclay, CNRS, Univ Evry, Laboratoire de Mathématiques et Modélisation d’Évry 91037 Évry France; e-mail: monique.jeanblanc@univ-evry.fr

§CCB Fintech; email: wudongli.sh@ccbft.com

0Mathematics Subject Classification 2010: Primary 60G44, 60J65, 60G40. Secondary 60G35, 60H10, 91G40.

0Key words and phrases: Marked point process, compensator, conditional probability density, Jacod’s equiva- lence hypothesis, initial and progressive enlargements of filtrations, predictable (martingale) representation property, changes of probability measures.

0Date: April 23, 2021

pure jump martingale) in the filtration G. We study the relationship between the representation of martingales in the initially (resp. progressively) enlarged filtration and the various optional projections. The paper is an extension of our previous paper [9] for the case of models driven by marked point processes. We refer the reader to the monograph [1] for results on enlargements of filtrations. Our results can be useful to compare the optimal strategies of investors having different information flows, and to investigate optimal stopping problems in different filtrations.

The reason why we are working with marked point processes is that a marked point process in Fremains a marked point processes, in particular a semi-martingale, in any enlargement ofF(with possibly a different compensator).

The paper is organised as follows. In Section 2, we recall standard definitions of projections as well as other results of stochastic analysis that we use in the paper. In Section 3, we give some basic definitions and results related to the initial and progressive enlargements of a filtrationFgenerated by a marked point process (MPP) with a random timeτ, denoted byF^(τ)andG, respectively, under Jacod’s equivalence hypothesis. In Section 4, we recall that the weak predictable representation property holds in the reference filtration 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 filtration involved. In Section 5, we consider the optional projections of anF^(τ)-martingale on the filtrationsGandF. We derive explicit expressions for the coefficients in the integral representations of these optional projections in terms of the originalF^(τ)-martingale and the components in its representation as a stochastic integral and give analogous results in the case ofF-optional projections of aG-martingale.

2 Preliminary definitions and results

We denote byB(R)(resp. B(R⁺)) the Borel sets ofR(resp. ofR⁺= [0,∞)) and setR^∗=R\ {0}.

We work on a standard complete probability space (Ω,G,P), on which there exists a sequence (Tn, Zn)n≥1, where (Tn)n≥1 is a strictly increasing sequence of finite strictly positive random vari- ables with no accumulation point, and(Z_n)_n≥1a sequence of real-valued random variables. We shall say that the sequenceN = (Tn, Zn)_n≥1is amarked point process(MPP) onR(see Def. 1.1.6 in [24], Section 1.2, pages 3-4 in [21] and Chapter VIII in [5]).

We introduce the associated random measureµonΩ× B(R⁺)× B(R)which is defined, for any setA∈ B(R)and anyt≥0, by

µ(ω; (0, t], A) =X

n≥1

11_{Tn(ω)≤t}11_{Zn(ω)∈A},

which is called the jump measure of the marked point process N. Note that the process N(A) = (Nt(A))t≥0 is a counting process. We denote byF= (Ft)t≥0 the natural filtration of the MPP

which is a right-continuous filtration (see Proposition 3.39 in [14]). We callFhereafter thereference filtration, and note thatTn, for n≥1, are F-stopping times. We define the compensatorν of the jump measureµwith respect toFas the unique random measure¹onΩ× B(R⁺)× B(R)such that, for anyA∈ B(R), the process

ν(ω; (0, t], A) = Z t

0

Z

A

ν(ω;ds, dz),∀t≥0, (2.1)

isF-predictable and the processN(A) = (Nt(A))t≥0 given by

N_t(A) =µ((0, t], A)−ν((0, t], A),∀t≥0, (2.2) is anF-martingale. We shall say thatNis theF-compensatedmartingale of the marked point process N, and, by abuse of language, thatν is the compensator of N. More generally, if Kis a filtration larger thatF, we say thatν^K is theK-compensator ofN if, for anyA∈ B(R), the process

µ((0, t], A)−ν^K((0, t], A),∀t≥0, (2.3) is aK-martingale, and the processν^K((0,·], A)isK-predictable.

We assume, as in Chapter VIII, Definition D5, page 236 of [5] and [26] that the compensatorν admits the representation

ν(ω;dt, dz) =dt η_t(ω;dz),∀t≥0, (2.4) whereη(dz) is a transition kernel, so thatη(A) = (ηt(A) =R

Aηt(dz), t≥0)is the intensity of the counting processN(A).

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

Recall that, ifξis a P(F)⊗ B(R)-measurable process such that Z t

0

Z

R^∗

|ξs(z)|ηs(dz)ds <∞,∀t≥0, (2.5) the processY = (Y_t)_t≥0defined as

Yt=Y0+ Z t

0

Z

R^∗

ξs(z) µ(ds, dz)−ηs(dz)ds

,∀t≥0, (2.6)

is anF-local martingale. Under the stronger assumption E

Z t

0

Z

R^∗

|ξs(z)|ηs(dz)ds

<∞, ∀t≥0, (2.7)

1The compensatorνis given, for anyA∈ B(R), by ν(ω; (0, t], A) =

Zt 0

Z

A

X

k≥1

11{T_k−1<s≤T_k}

P(Tk∈ds, Zk∈dz| F_Tk−1)

P(Tk> s| F_Tk−1) ,∀t≥0, (see, e.g., Section 1.10 in [21], [12] or Chapter 11, Section 4 in [11]).

the processY is anF-martingale (see Chapter VIII, Corollary C4, page 235 in [5]).

Furthermore, anyF-martingaleY admits a representation as Y_t=Y₀+

Z t

0

Z

R^∗

ξ_s(z) µ(ds, dz)−η_s(dz)ds

,∀t≥0,

with ξ satisfying (2.5) (see Chapter VIII, Theorem T8, page 239 in [5] and Theorem 2.2 in [26]).

This property is referred to as theweak predictable representation property(WPRP) of the marked point processN on the filtrationF with respect to thecompensated jump measure µ−ν (see also Theorem 13.19 in [11], Th. 1.13.2 in [21] or Theorem 1.1.21 in [24]) . Such a representation is essentially unique (P×ηt(dz)×dt-a.s.).

LetX = (Xt)_t≥0 be a measurable process and Hbe a filtration satisfying the usual hypotheses of completeness and right continuity. We denote by^p,FX = (^p,FXt)t≥0 (resp. ^o,FX = (^o,FXt)t≥0) its F-predictable (resp. optional) projection when they exist (see Chapter V, Th. 5.1 (resp. 5.2) in [11]

or Section 1.3.1, page 15 in [1]).

3 Jacod’s equivalence hypothesis

In the whole paper, we work on a probability space(Ω,G,P)which supports a marked point process with a continuous on right and completed natural filtration F = (Ft)_t≥0 and a strictly positive random variableτ. Note that the inclusion F_∞⊂ G holds and, in general, this inclusion is strict.

We recall that anyF-martingale is càdlàg.

Hypothesis 3.1 We assume in the whole paper, as in [2] and [10], that Jacod’s equivalence hy- pothesis holds, that is, the regular conditional distributions ofτ given Ft are equivalent toρ, the unconditional law of the random variableτ:

P(τ ∈ · | F_t)∼P(τ∈ ·),∀t≥0 (P-a.s.).

In our model, this assumption implies (see Lemma 2.3 in [8]) that there exists a family of strictly positive processes p(u) = (pt(u))_t≥0 such that the function (ω, t, u) 7→ pt(u;ω) is O(F)⊗ B(R⁺)- measurable, and, for eachu≥0, the processp(u)is a càdlàgF-martingale. Moreover, for any Borel bounded functionf, the following equality holds

E f(τ)

Ft

= Z ∞

0

f(u)pt(u)ρ(du),∀t≥0 (P-a.s.). (3.1) The expression in (3.1) implies that the following equality holds

P(τ > s| Ft) = Z ∞

s

pt(u)ρ(du), ∀t, s≥0 (P-a.s.), so that, from the assumption of strict positivity ofτ, the equality

Z ∞

0

p_t(u)ρ(du) = 1,(P-a.s.),

We shall call the family ofF-optional processesp(u), for eachu≥0, theF-conditional density fam- ily with respect toρ(du)(or the conditional density ofτif there is no ambiguity on the filtration).

The following proposition is proved as a consequence of the WPRP in [23, Pro. 2.1].

Proposition 3.2 There exists a strictly positive and P(F)⊗ B(R+)⊗ B(R)-measurable process f such that, for anyu≥0, the strictly positive F-martingalep(u)admits the representation

pt(u) =p0(u) exp Z t

0

Z

R^∗

ln fs(u, z)

µ(ds, dz)− Z t

0

Z

R^∗

fs(u, z)−1

ηs(dz)ds

, ∀t≥0, (3.2) or, equivalently,p(u) satisfies the stochastic differential equation

dpt(u) =pt−(u) Z

R^∗

ft(u, z)−1

µ(dt, dz)−ηt(dz)dt

, p0(u) = 1. (3.3) Let us denote by H = (Ht)_t≥0 with Ht = 11_{τ≤t}, for all t ≥ 0, which is called the indicator default processin the credit risk theory, whereτdenotes the time at which a default occurs. Moreover, sinceH is a càdlàg process, we can introduce the F-supermartingaleG= (G_t)_t≥0 defined byG=

o,F(1−H), that is, theF-optional projection of1−H satisfying the property

Gt=P(τ > t| Ft), ∀t≥0 (P-a.s.), (3.4) which, according to the equality (3.1), can be represented in the form

Gt= Z ∞

t

pt(u)ρ(du),∀t≥0 (P-a.s.). (3.5) Note thatGis strictly positive and that, from the assumption of strict positivity of the random variableτ, one hasG0 = 1. TheF-supermartingale G is called the conditional survival processor theAzéma supermartingale of the random timeτ.

Hypothesis 3.3 We assume that the distribution lawρof the strictly positive random variableτ is non-atomic.

Remark 3.4 It is known that, under the assumption that the law ρ of the random variable τ is non-atomic and under Jacod’s equivalence hypothesis, the random timeτ avoids all F-stopping times (see Corollary 2.2 in [7]). This will allow us to obtain a simpler formula for the semimartin- gale decomposition. More precisely, under the avoidance hypothesis, the dual optional projection ofH is continuous and equal to the dual predictable projection ofH, denoted byH^p (see Proposi- tion 1.48 (a), page 22 in [1]). Therefore the martingalemwhich appears in the general formulae of the semimartingale decomposition (see Proposition 5.30, page 116 in [1]) is equal to the martingale part of the Doob-Meyer decomposition of G, that is, one has G = m−H^p. In particular, the predictable projection of G is ^pG= ^pm−H^p = m−−H^p = G−. The fact that ρis non-atomic implies that, for a càdlàg processX, one has

Z t

0

X_s−ρ(ds) = Z t

0

Xsρ(ds),∀t≥0.

4 Enlargement of filtrations and martingales

We will consider two enlarged filtrations: the initial enlargement ofFobtained by adding theσ-field σ(τ) at time 0 and denoted F^(τ), and the progressive enlargement of F obtained by progressively adding informationσ(τ∧t)at timet≥0, or, in other terms, the smallest filtrationGcontainingF and turning outτ 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 ofN(A)defined in (2.2) as well as theG-semimartingale decompo- sition ofH. We underline that the martingale partN^(τ)(A)of theF^(τ)-semimartingale decomposi- tion ofN(A)enjoys theF^(τ)-predictable representation property, while the pair(N^G(A), M^G)of the martingale parts of theG-semimartingale decompositions of N(A)andH enjoys the G-predictable representation property, where the integral with respect to the pair is understood componentwise as in (4.15) below.

4.1 The initially enlarged filtration

As in the introduction, let us denote byF^(τ)= (F_t^(τ))t≥0= (Ft∨σ(τ))t≥0the initial enlargement of the filtrationFwith the random timeτ. We recall that, under Jacod’s equivalence hypothesis, anyF- local martingale is anF^(τ)-special semimartingale (see, e.g., Theorem 2.1 in [13] or Proposition 5.30, page 116 in [1]). Note that, according to Proposition 3.3 in [2], the filtrationF^(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 asX, or lower casex, orϕ, or evenx⁰.

We also recall that, for anyt≥0fixed, anyF_t^(τ)-measurable random variableY_t^(τ)is of the form Yt(ω, τ(ω)), for some Ft⊗ B(R⁺)-measurable function(ω, u)7→Yt(ω, u)(see, e.g., Proposition 2.7, part (i) in [6]). In particular, anyF₀^(τ)-measurable random variable is a Borel function ofτ. Recall that anyF^(τ)-predictable process can be represented in the formY_t(ω, τ(ω)), for allt≥0, where the mapping(ω, t, u)7→Yt(ω, u)defined onΩ×R⁺×R⁺ and valued inRisP(F)⊗ B(R⁺)-measurable.

Moreover, under Jacod’s equivalence density hypothesis, anyF^(τ)-optional processY^(τ)= (Y_t^(τ))_t≥0 can be written asY_t^(τ)=Yt(τ), for allt≥0, where the processY isO(F)⊗ B(R+)-measurable (see Theorem 6.9 in [27]).

As an immediate consequence of Jacod’s equivalence hypothesis, we observe that, for eacht≥0, if theF_t^(τ)-measurable random variableYt(τ)is integrable, then the following representation holds

E Yt(τ)

Ft

= Z ∞

0

Yt(u)pt(u)ρ(du),∀t≥0, (4.1) (see, e.g., Proposition 4.18 (b), page 85 in [1]).

In the following proposition, we give the semimartingale decomposition ofN(A), defined in (2.2), inF^(τ).

Proposition 4.1 For any Borel setA, theF^(τ)-semimartingale decomposition of the F-martingale N(A)is given by

Nt(A) =N^(τ)_t (A) + Z t

0

Z

A

fs(τ, z)−1

ηs(dz)ds,∀t≥0,

where N^(τ)(A) is an F^(τ)-martingale and f is given in (3.3). In other terms, the process N = (Tn, Zn)n≥1 is a marked point process withF^(τ)-compensatorν^(τ), where we have

ν^(τ)(dt, dz) =ft(τ, z)ηt(dz)dt,∀t≥0,∀z∈R. (4.2) Proof: From the results of initial enlargement², for any A ∈ B(R), the process N^(τ)(A) = (N^(τ)_t (A))_t≥0 defined by

N^(τ)_t (A) =Nt(A)− Z t

0

dhN(A), p(u)i^F_s p_s−(u)

_u=τ

,∀t≥0,

is anF^(τ)-martingale. In order to compute the predictable covariation, we start by computing the quadratic covariation of the processesN(A)andp(u), for eachu≥0. Obviously, we have

N(A), p(u)

t= Z t

0

Z

A

p_s−(u) fs(u, z)−1

µ(ds, dz),∀t, u≥0, and hence

N(A), p(u)F t =

Z t

0

Z

A

p_s−(u) fs(u, z)−1

ηs(dz)ds,∀t≥0,∀z∈R. It follows that

N_t(A)− Z t

0

Z

A

f_s(τ, z)−1

η_s(dz)ds,∀t≥0, is anF^(τ)-martingale and theF^(τ)-compensator ofN is

ν^(τ)(dt, dz) =ft(τ, z)ηt(dz)dt,∀t≥0,∀z∈R.

This completes the proof.

Note that Jacod’s equivalence hypothesis allows us to prove the stability of weak predictable representation property in the enlargement of filtration by means of the following lemma (see [10]

or Theorem 4.37, page 94 in [1]).

2One applies Theorem 2.1 in [13] which states that, under Jacod’s hypothesis, for anyF-martingaleX= (Xt)t≥0, the processX(τ) = (Xt(τ))t≥0 defined by

Xt(τ) =Xt− Z t

0

dhX, p(u)i^F_s ps−(u)

u=τ

,∀t≥0, (4.3)

is anF^(τ)-martingale.

Lemma 4.2 Let P^∗ be the probability on F^(τ) defined by means of dP^∗

dP _F(τ)

t

= 1

pt(τ),∀t≥0.

Then,Fandτare independent underP^∗, as well asP^∗|_Ft=P^∗|_Ft, for allt≥0, andP^∗|_σ(τ)=P|_σ(τ). In particular, immersion holds underP^∗, i.e., any(P^∗,F)-martingale is a(P^∗,G)-martingale.

Proposition 4.3 Each (P,F^(τ))-martingaleY(τ) = (Yt(τ))t≥0 admits a representation of the form Y_t(τ) =Y₀(τ) +

Z t

0

Z

R^∗

ψ_s(τ, z) µ(ds, dz)−ν^(τ)(ds, dz)

,∀t≥0, (4.4)

for someP(F)⊗ B(R⁺)⊗ B(R)-measurable processψ satisfying Z t

0

Z

R^∗

|ψs(τ, z)|ν^(τ)(ds, dz)<∞,∀t≥0, (4.5) whereν^(τ)is defined in (4.2).

Proof: Note that, under the probability measure P^∗, the conditional density of the random variableτ is equal toρ. Hence, we conclude, applying Proposition 2.1 in [6] to the probability P^∗ that any (P^∗,F^(τ))-martingale Y^∗,(τ) is of the formY_t^∗,(τ) =Y_t^∗(τ), where Y^∗(u), for each u≥0, is a(P^∗,F)-martingale, hence a(P,F)-martingale, which is a stochastic integral with respect to the (P,F)(or equivalently (P^∗,F^(τ)))-compensated jump measureµ−ν. Observe that, for each u≥0, we have

Y_t^∗(u) =Y₀^∗(u) + Z t

0

Z

R^∗

ψ_s^∗(u, z) µ(ds, dz)−ηs(dz)ds

,∀t≥0, withψ^∗ beingP(F)⊗ B(R⁺)⊗ B(R)-measurable and satisfying (4.5), and thus

Y_t^∗(τ) =Y₀^∗(τ) + Z t

0

Z

R^∗

ψ^∗_s(τ, z) µ(ds, dz)−η_s(dz)ds

,∀t≥0,

and WPRP holds forF^(τ)underP^∗. Since WPRP is stable by equivalent change of probability mea- sures (see, e.g., Chapter 13, Th. 13.22 in [11]), it follows that the weak predictable representation

property holds forF^(τ) underPwith respect to µ−ν^(τ).

As a particular case, we can represent all strictly positiveF^(τ)-local martingales:

Proposition 4.4 Every strictly positiveF^(τ)-local martingaleL(τ) = (L_t(τ))_t≥0 can be represented as

Lt(τ) =L0(τ) + Z t

0

L_s−(τ) Z

R^∗

Θs(τ, z)−1

µ(ds, dz)−ν^(τ)(ds, dz)

,∀t≥0, (4.6) whereΘis strictly positive andP(F^(τ))⊗ B(R)-measurable and ν^(τ) is defined in (4.2).

4.2 The progressively enlarged filtration

We denote byG= (G_t)_t≥0 the progressive enlargement ofFwithτ, that is, Gt= \

s>t

Fs∨σ(τ∧s)

,∀t≥0. (4.7)

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 G0 is trivial, so that the initial value of a G-adapted process is a deterministic one. Observe that, under Jacod’s equivalence hypothesis, any F-martingale is a G-semimartingale (see, e.g., Proposition 5.30, page 116 in [1] or Theorem 3.1 in [16]), and thus, a special semimartingale according to Chapter VI, Theorem 4, page 367 in [25]).

We observe that the completion of the two enlargementsGand F^(τ) follows fromF_∞ ⊂ G_∞ ⊂ F∞^(τ)⊂ A, and we note thatF₀^(τ)=σ(τ).

We further indicate with the superscript Gthe processes which are G-adapted, as Y^G, except for theG-adapted processH.

We recall that anyG-predictableprocessK^G= (K_t^G)_t≥0 can be written as K_t^G= 11_{τ≥t}K_t⁰+ 11_{{τ <t}}K_t¹(τ),∀t≥0,

where the processK⁰isF-predictable andK¹isP(F)⊗B(R⁺)-measurable (see, e.g., Proposition 2.11, page 36 in [1]). Under Jacod’s equivalence density hypothesis, any G-optional process Y^G can be written as

Y_t^G= 11_{{τ >t}}Y_t⁰+ 11_{τ≤t}Y_t¹(τ),∀t≥0, (4.8) whereY⁰is F-optional andY¹isO(F)⊗ B(R+)-measurable (see Theorem 6.9 in [27]).

As it follows from the Doob-Meyer decomposition of the supermartingale H and the fact that anyG-predictable process is equal, on the set {τ ≥t} to anF-predictable process, there exists an F-predictable increasing processΛ = (Λ_t)_t≥0 such that the processM^G= (M_t^G)_t≥0 defined by

M_t^G=H_t−Λ_t∧τ,∀t≥0, (4.9)

is aG-martingale. It is known that, under Jacod’s equivalence hypothesis, the processΛadmits the representation (we use also the fact thatρhas no atoms)

Λt= Z t

0

ps(s) Gs

ρ(ds) = Z t

0

p_s−(s)

G_s− ρ(ds),∀t≥0, (4.10)

(see Proposition 4.4 in [7] or Corollary 5.27 (b), page 114 in [1]). In this respect, the process λ= (λt)_t≥0 defined by λt =p_t−(t)/G_t−, for t ≥ 0, is the intensity rate of τ with respect to the measureρ(see Proposition 2.15, page 37 in [1]).

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

Proposition 4.5 Suppose that Jacod’s equivalence hypothesis holds. The Doob-Meyer decomposi- tion of the Azéma supermartingaleGis

Gt= 1− Z t

0

Gsλsρ(ds) + Z t

0

G_s−

Z

R^∗

ϕs(z)−1

µ(ds, dz)−ηs(dz)ds

,∀t≥0, (4.11) where the functionϕ defined by

ϕ_t(z) = 1 Gt−

Z ∞

t

p_t−(u)f_t(u, z)ρ(du),∀t≥0,∀z∈R, (4.12) is strictly positive andP(F)⊗ B(R)-measurable.

The multiplicative decomposition of the Azéma supermartingaleGhas the form Gt=e^−Λt exp

Z t

0

Z

R^∗

ln ϕs(z)

µ(ds, dz)− Z t

0

Z

R^∗

ϕs(z)−1

ηs(dz)ds

,∀t≥0, (4.13) whereΛ is given by (4.10).

Proof:The Doob-Meyer decomposition ofGis obtained using Itô-Ventzell formula as developed in Theorem 3.1 in [22] to the process

Gt(s) =P(τ > s| Ft) = Z ∞

s

pt(u)ρ(du),∀t, s≥0,

with parameters, where the forward integral (with respect to the compensated measure) in [22] is the usual stochastic integral in our setting since we integrate predictable processes.

In the following proposition, we give the semimartingale decomposition of the process N(A) defined in (2.2) in the filtrationG.

Proposition 4.6 For anyA∈ B(R), theG-semimartingale decomposition of theF-martingaleN(A) is given by

Nt(A) =N^G_t(A) + Z t∧τ

0

Z

A

ϕs(z)−1

ηs(dz)ds+ Z t

t∧τ

Z

A

fs(τ, z)−1

ηs(dz)ds,∀t≥0,

whereN^G(A) is a G-martingale, ϕ is defined in (4.12), and f is defined in (3.3). The predictable random measure

ν^G(dt, dz) = 11_{τ≥t}ϕt(z) + 11_{{τ <t}}ft(τ, z)

ηt(dz)dt,∀t≥0,∀z∈R, (4.14) is theG-compensator of the random jump measureµof the marked point processN.

Proof: Recall that G admits a Doob-Meyer decomposition asG =m−H^p (see Remark 3.4.

TheG-semimartingale decomposition³ of theF-martingaleN(A)is given by N_t(A) =N^G_t(A) +

Z t∧τ

0

dhN(A), mi^F_s Gs−

+ Z t

t∧τ

dhN(A), p(u)i^F_s ps−(u)

_u=τ

=N^G_t(A) + Z t∧τ

0

Z

A

(ϕ_s(z)−1)G_s−

Gs−

η_s(dz)ds+ Z t

t∧τ

Z

A

f_s(τ, z)−1

η_s(dz)ds

=N^G_t(A) + Z t∧τ

0

Z

A

ϕs(z)−1

ηs(dz)ds+ Z t

t∧τ

Z

A

fs(τ, z)−1

ηs(dz)ds,∀t≥0,

whereN^G(A) = (N^G_t(A))_t≥0is aG-martingale. It thus follows that theG-compensator ofµis given

by (4.14).

Proposition 4.7 Every (P,G)-martingaleY^G= (Y_t^G)_t≥0 can be represented as Y_t^G=Y₀^G+

Z t

0

Z

R^∗

α^G_s(z) µ(ds, dz)−ν^G(ds, dz) +

Z t

0

β_s⁰dM_s^G,∀t≥0, (4.15) for someP(G)⊗ B(R)-measurable process α^G satisfying

Z t

0

Z

R^∗

|α^G_s(z)|ν^G(ds, dz)<∞,∀t≥0, (4.16) whereν^G is defined in (4.14). Here, the process α^G is of the form

α^G_t(z) = 11_{τ≥t}α⁰_t(z) + 11_{{τ <t}}αt(τ, z),∀t≥0,∀z∈R, (4.17) whereα⁰ isP(F)⊗ B(R)-measurable process,αis aP(F)⊗ B(R⁺)⊗ B(R)-measurable process, while β⁰ is anF-predictable process.

Proof:The weak predictable representation property holds for the filtrationGunder the prob- ability measureP^∗, due to the independence betweenF and σ(τ) under P^∗. This property means that any(P^∗,G)-martingaleY^∗,G = (Y_t^∗,G)_t≥0 admits the representation

Y_t^∗,G=Y₀^∗,G+ Z t

0

Z

R^∗

α^G_s(z) µ(ds, dz)−ηs(dz)ds +

Z t

0

β_s⁰dM_s^∗,G,∀t≥0, (4.18) with someP(G)⊗ B(R)-measurable process α^G satisfying (4.16) and being of the form (4.17), for some P(F)⊗ B(R)-measurable processα⁰, some P(F)⊗ B(R⁺)⊗ B(R)-measurable processα, and someF-predictable processβ⁰. Note that, due to immersion property the(P^∗,G)compensator ofµ is the measureηs(dz)ds. Here the processM^∗,G= (M_t^∗,G)_t≥0is the(P^∗,G)-compensated martingale

3One can use Remark 3.4 and Theorem 5.30, page 116 in [1] to deduce that, for anyF-martingaleX, the process X^G= (X_t^G)t≥0defined by

X_t^G=Xt− Z t∧τ

0

dhX, mi^F_s Gs−

− Z t

t∧τ

dhX, p(u)i^F_s ps−(u)

u=τ

,∀t≥0, is aG-martingale.

associated withH defined similar toM^G in (4.9), but under the probability measureP^∗. Since the weak predictable representation property (WPRP) is stable under an equivalent change of probabil-

ity measure (see Th. 13.22 in [11]), the result follows.

Remark 4.8 Note that, if the processβ^G admits the representation β^G_t = 11_{τ≥t}β⁰_t+ 11_{{τ <t}}β_t¹(τ),∀t≥0, then the equality

Z t

0

β_s^GdM_s^G= Z t

0

β_s⁰dM_s^G,∀t≥0, (4.19)

holds, for any choice of the P(G)⊗ B(R)-measurable process β¹, since M^G is flat after τ (i.e., M_t^G=M_t∧τ^G , for allt≥0).

As a particular case of Proposition 4.7, we obtain:

Proposition 4.9 Every strictly positive G-local martingale L^G= (L^G_t)t≥0 can be represented as L^G_t =L^G₀ +

Z t

0

L^G_s−

Z

R^∗

κ^G_s(z)−1

µ(ds, dz)−ν^G(ds, dz) +

Z t

0

L^G_s− ξ^G_s −1

dM_s^G,∀t≥0, for a strictly positive andP(G)⊗ B(R)-measurable processκ^G of the form

κ^G_t(z) = 11_{{τ >t}}κ⁰_t(z) + 11_{τ≤t}κt(τ, z), ∀t≥0,∀z∈R, and a strictly positiveF-predictable process ξ^G, whereν^G is defined in (4.14).

5 Optional projections of martingales

LetY(τ)be anF^(τ)-martingale. Then,Y(τ)admits the integral representation given by (4.4). We study theG-optional projection Y^G of the process Y(τ) and the F-optional projectionY of Y(τ).

Note thatY^Gis aG-martingale andY is anF-martingale. TheG-martingaleY^Gadmits the integral representation given by (4.15), with some processesα^G andβ⁰ that can be represented as

α^G_t(z) = 11_{τ≥t}α⁰_t(z) + 11_{{τ <t}}αt(τ, z),∀t≥0,∀z∈R, (5.1) where, as in (4.17),α⁰ isP(F)⊗ B(R)-measurable, αis P(F)⊗ B(R+)⊗ B(R)-measurable, andβ⁰ is anF-predictable process.

Observe that any square integrable F-martingale Y admits the representation (2.6) with some P(F)⊗ B(R)-measurable processξsatisfying

E Z tZ

ξ²(z)ηs(dz)ds

<∞,∀t≥0, (5.2)

(see Chapter VIII, Theorem T8, page 239 in [5]).

Similarly, we observe that anysquare integrable F^(τ)-martingaleY(τ)admits the representation (4.4) with someP(F)⊗ B(R⁺)⊗ B(R)-measurable processψsatisfying

E Z t

0

Z

R^∗

ψ²_s(τ, z)ν^(τ)(ds, dz)

<∞,∀t≥0, (5.3)

whereν^(τ)is defined in (4.2) (see Chapter VIII, Theorem T8, page 239 in [5]).

Finally, we observe that anysquare integrableG-martingaleY^Gadmits the representation (4.15) with someP(G)⊗ B(R)-measurable processα^G satisfying

E Z t

0

Z

R^∗

α^G_s(z)²

ν^G(ds, dz)

<∞,∀t≥0, (5.4)

andF-predictable processβ⁰, whereν^Gis defined in (4.14) (see Chapter VIII, Theorem T8, page 239 in [5]).

5.1 The projections of F

-martingales on G

Proposition 5.1 Let Y^(τ)=Y(τ)(with Y beingO(F)⊗ B(R+)-measurable) be anF^(τ)-martingale with the representation (4.4), whereψ is P(F)⊗ B(R+)⊗ B(R)-measurable. Then, its G-optional projectionY^G= (Y_t^G)_t≥0 is theG-martingale with representation given by the equation (4.15) with Y0 =E[Yt(τ)], where α^G admits the decomposition (5.1). TheP(F)⊗ B(R)-measurable process α⁰, theP(F)⊗ B(R⁺)⊗ B(R)-measurable processαand the F-predictable processβ⁰ are of the form

α⁰_t(z) = 1 ϕt(z)G_t−

Z ∞

t

ψt(u, z) +Y_t−(u)

ft(u, z)−Y_t−(u)ϕt(z)

p_t−(u)ρ(du),∀t≥0,∀z∈R, αt(u, z) =ψt(u, z),∀u≥t≥0,

β_t⁰= ^p,FΣt− 1 G_t−

Z ∞

t

Yt−(u)pt−(u)ρ(du), ∀t≥0, withΣt=Y_t−(t), for all t≥0.

Proof: In the first part of the proof (the first and the second step), we assume that the F^(τ)- martingaleY(τ)is square integrable, so that theG-martingaleY^G is square integrable too. In the first step, we determine α^G(z), for each z ∈ R, and, in the second step, we determine β⁰. 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- martingaleY^G is square integrable too. In particular, Y0(τ) is square integrable and the P(F)⊗ B(R⁺)⊗ B(R)-measurable processψsatisfies (5.3) as well as the P(G)⊗ B(R)-measurable process

α^Gsatisfies (5.4). Then, consider a boundedP(G)⊗ B(R)-measurable processγ^Gsuch thatγ^G(z) = (γ_t^G(z))t≥0, for eachz∈R^∗, as well as a boundedF-predictable processθ⁰= (θ_t⁰)t≥0, and define the processK^G= (K_t^G)_t≥0 by

K_t^G=K₀^G+ Z t

0

Z

R^∗

γ_s^G(z) µ(ds, dz)−ν^G(ds, dz) +

Z t

0

θ⁰_sdM_s^G,∀t≥0, (5.5) where ν^G is defined in (4.14). It is seen that the process K^G is a square integrable G-martingale, sinceγ^G satisfies the condition

E Z t

0

Z

R^∗

γ_s^G(z)²

ν^G(ds, dz)

<∞,∀t≥0, (5.6)

and the processθ⁰isF-predictable and bounded. In this case, the square integrable random variable Y_t^G=E[Y_t(τ)| Gt]is the onlyGt-measurable random variable such that

E

Y_t(τ)K_t^G

=E

Y_t^GK_t^G

,∀t≥0, (5.7)

holds. Thus, since one has

E

Yt(τ)K₀^G

=E

Y_t^GK₀^G

,∀t≥0, the equality (5.7) is equivalent to the system of two following equalities

E

Yt(τ) Z t

0

Z

R^∗

γ^G_s(z) µ(ds, dz)−ν^G(ds, dz)

(5.8)

=E

Y_t^G Z t

0

Z

R^∗

γ_s^G(z) µ(ds, dz)−ν^G(ds, dz)

,∀t≥0, and

E

Y_t(τ) Z t

0

θ_s⁰dM_s^G

=E

Y_t^G Z t

0

θ⁰_sdM_s^G

,∀t≥0. (5.9)

We now determine the processesα⁰ andαfrom the equality (5.8). On the one hand, one has E

Y_t(τ)

Z t

0

Z

R^∗

γ_s^G(z) µ(ds, dz)−ν^G(ds, dz)

=E

Yt(τ) Z t

0

Z

R^∗

γ_s^G(z) µ(ds, dz)−ν^(τ)(ds, dz) +

Z t

0

Z

R^∗

γ_s^G(z) ν^(τ)(ds, dz)−ν^G(ds, dz)

, ∀t≥0,

where ν^(τ) is defined in (4.2). Integrating by parts on the time interval [0, t]the product the two F^(τ)-martingalesY(τ)andΥ = (Υ_t)_t≥0 defined by

Υt= Z t

0

Z

R^∗

γ_s^G(z) µ(ds, dz)−ν^(τ)(ds, dz)

,∀t≥0,

and taking into account the fact thatΥ_t−dYt(τ)andY_t−(τ)dΥtcorrespond to true martingales, as we shall prove in Appendix below, one has

E

Yt(τ) Z t

0

Z

R^∗

γ_s^G(z) µ(ds, dz)−ν^(τ)(ds, dz)

=E Z t

0

Z

R^∗

γ_s^G(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 martingaleY(τ)and the bounded variation processΓ(τ) = (Γt(τ))_t≥0 defined by

Γt(τ) = Z t

0

Z

R^∗

γ_s^G(z) ν^(τ)(ds, dz)−ν^G(ds, dz)

,∀t≥0, one obtains

E

Yt(τ) Γt(τ)

=E Z t

0

Z

R^∗

Y_s−(τ)γ_s^G(z) ν^(τ)(ds, dz)−ν^G(ds, dz)

, ∀t≥0. On the other hand, one has by integration by parts

E

Y_t^G Z t

0

Z

R^∗

γ_s^G(z) µ(ds, dz)−ν^G(ds, dz)

=E Z t

0

Z

R^∗

γ^G_s(z)α^G_s(z)ν^G(ds, dz)

,∀t≥0. Finally, (5.8) is equivalent to, for anyγ^Gsatisfying (5.6), we have

E Z t

0

Z

R^∗

γ_s^G(z)

ψs(τ, z)ν^(τ)(ds, dz) +Ys−(τ) ν^(τ)(ds, dz)−ν^G(ds, dz)

#

=E Z t

0

Z

R^∗

γ_s^G(z)α^G_s(z)ν^G(ds, dz)

,∀t≥0. (5.10)

Forγ^G(z)such thatγ_s^G(z) = 11_{τ≥s}γ_s⁰(z), for alls >0, whereγ⁰ isP(F)⊗ B(R)-measurable, using the identities (4.2) and (4.14), we have

E Z t

0

Z

R^∗

γ⁰_s(z) 11_{τ≥s}

ψs(τ, z) +Ys−(τ)

fs(τ, z)−Ys−(τ)ϕs(z)

ηs(dz)ds

=E Z t

0

Z

R^∗

γ_s⁰(z) 11_{τ≥s}α_s⁰(z)ϕs(z)ηs(dz)ds

,∀t≥0, (5.11)

and, introducing by tower property a conditioning with respect toFsand using the existence of the conditional density, the left hand side of (5.11) is

E Z t

0

Z

R^∗

γ_s⁰(z) 11_{τ≥s}

ψs(τ, z) +Ys−(τ)

fs(τ, z)−Ys−(τ)ϕs(z)

ηs(dz)ds

TP= E Z t

0

Z

R^∗

γ⁰_s(z) Z ∞

s

ψs(u, z) +Y_s−(u)

fs(u, z)−Y_s−(u)ϕs(z)

ps(u)ρ(du)

ηs(dz)ds

=E Z t

0

Z

R^∗

γ_s⁰(z) Z ∞

s

ψs(u, z) +Y_s−(u)

fs(u, y)−Y_s−(u)ϕs(z)

p_s−(u)ρ(du)

ηs(dz)ds

,∀t≥0, (5.12)

where, in the last equality, we have used the fact that theF-predictable projection ofp(u)isp₋(u), the processp(u)being a martingale, for eachu≥0.

We note also that, using the fact thatG₋ is theF-predictable projection ofG(see Remark 3.4), the right-hand side of (5.11) is

E Z t

0

Z

R^∗

γ_s⁰(z) 11_{τ≥s}α⁰_s(z)ϕ_s(z)η_s(dz)ds

=E Z t

0

Z

R^∗

γ⁰_s(z)Gsα⁰_s(z)ϕs(z)ηs(dz)ds

=E Z t

0

Z

R^∗

γ⁰_s(z)G_s−α⁰_s(z)ϕs(z)ηs(dz)ds

,∀t≥0. (5.13)

It follows from (5.11) that the right-hand sides of (5.12) and (5.13) are equal for anyγ⁰, hence α⁰_s(z) = 1

ϕ_s(z)G_s−

Z ∞

s

ψs(u, z) +Y_s−(u)

fs(u, z)−Y_s−(u)ϕs(z)

p_s−(u)ρ(du),∀s≥0. Using the identities (4.2) and (4.14), for γ^G of the form γ_s^G = γs(τ, z)11_{{τ <s}}, for all s > 0, for γ∈P(F)⊗ B(R⁺)⊗ B(R), equality (5.10) leads to

E Z t

0

Z

R^∗

γs(τ, z) 11_{{τ <s}}ψs(τ, z)fs(τ, z)ηs(dz)ds

=E Z t

0

Z

R^∗

γ_s(τ, z) 11_{{τ <s}}α_s(τ, z)f_s(τ, z)η_s(dz)ds

, ∀t≥0, and we can chooseα=ψ.

Second step: In the second step, we compute the value ofβ⁰, from (5.9). It is straightforward to see that

E

Y_t^G Z t

0

θ⁰_sdM_s^G

=E Z t

0

β_s⁰θ_s⁰λ_s11_{{τ >s}}ρ(ds) TP

= E Z t

0

β_s⁰θ_s⁰λ_sG_sρ(ds)

,∀t≥0. From the definition ofM^G, it follows that

E

Yt(τ) Z t

0

θ⁰_sdM_s^G

=E

Yt(τ)

11_{τ≤t}θ_τ⁰− Z t

0

11_{{τ >s}}θ⁰_sλsρ(ds)

TP= E Z t

0

Yt(s)θ_s⁰pt(s)ρ(ds)− Z t

0

θ⁰_sλsE

Y_s^(τ)11_{{τ >s}}

Fs ρ(ds)

=E Z t

0

Y_s−(s)p_s−(s)θ_s⁰ρ(ds)− Z t

0

θ⁰_sλ_s Z ∞

s

Y_s−(u)p_s−(u)ρ(du)

ρ(ds)

=E Z t

0

p,FΣsps−(s)θ_s⁰ρ(ds)− Z t

0

θ⁰_sλs

Z ∞

s

Ys−(u)ps−(u)ρ(du)

ρ(ds)

,∀t≥0, where we have used in the third equality thatY(u)p(u)is anF-martingale with predictable projection Y₋(u)p₋(u), for eachu≥0, and definedΣ = (Σ_t)_t≥0 byΣ_t=Y_t−(t), for allt≥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)

It follows that

β_s⁰= 1 λsG_s−

p,FΣsps−(s)−λs

Z ∞

s

Ys−(u)ps−(u)ρ(du)

= ^p,FΣs− 1 G_s−

Z ∞

s

Y_s−(u)p_s−(u)ρ(du),∀s≥0, where we have used the fact thatλ_s=p_s−(s)/G_s−, fors≥0.

Third step: The extension to F^(τ)-martingales is done using usual localisation procedure (see

Third step of Proof of Proposition 5.1 in [9]).

5.2 The projections of F

-martingales on F

Proposition 5.2 Let Y(τ) be an F^(τ)-martingale with the representation given by equality (4.4).

Then, its F-optional projection Y = (Y_t)_t≥0 admits the representation (2.6), with P(F)⊗ B(R)- measurable processξ, given by

ξt(z) = Z ∞

0

ψt(u, z) +Y_t−(u)

ft(u, z)−Y_t−(u)

p_t−(u)ρ(du),∀t≥0. (5.14) Proof: As before, we assume that Y(τ) is square integrable. Then, consider a bounded P(F)⊗ B(R)-measurable process φ such that φ(z) = (φt(z))_t≥0, for each z ∈ R^∗, and define the processK= (Kt)t≥0by

K_t=K₀+ Z t

0

Z

R^∗

φ_s(z) µ(ds, dz)−η_s(dz)ds

,∀t≥0.

It is seen that the process K is a square integrable G-martingale, since the process φsatisfies the condition

E Z t

0

Z

R^∗

φ²_s(z)ηs(dz)ds

<∞, ∀t≥0.

In this case, the square integrable random variable Y_t = E[Y_t(τ)| Ft] is the only Ft-measurable random variable such that

E

Yt(τ)Kt

=E YtKt

,∀t≥0, (5.15)

holds. Thus, the equality (5.15) is equivalent to the following equality E

Yt(τ)

Z t

0

Z

R^∗

φs(z) µ(ds, dz)−ηs(dz)ds

=E

Yt

Z t

0

Z

R^∗

φs(z) µ(ds, dz)−ηs(dz)ds

,∀t≥0. (5.16)