Integral Representations of Martingales for Progressive Enlargements Of

INTEGRAL REPRESENTATIONS OF MARTINGALES FOR PROGRESSIVE ENLARGEMENTS OF FILTRATIONS ANNA AKSAMIT, MONIQUE JEANBLANC AND MAREK RUTKOWSKI Abstract. We work in the setting of the progressive enlargement G of a reference filtration F through the observation of a random time τ. We study an integral representation property for some classes of G-martingales stopped at τ. In the first part, we focus on the case where F is a Poisson filtration and we establish a predictable representation property with respect to three G-martingales. In the second part, we relax the assumption that F is a Poisson filtration and we assume that τ is an F-pseudo-stopping time. We establish integral representations with respect to some G-martingales built from F-martingales and, under additional hypotheses, we obtain a predictable representation property with respect to two G-martingales. Keywords: predictable representation property, Poisson process, random time, progressive enlargement, pseudo-stopping time Mathematics Subjects Classification (2010): 60H99 1. Introduction We are interested in the stability of the predictable representation property in a filtration enlargement setting: under the postulate that the predictable representation property holds for F and G is an enlargement of F, the goal is to find under which conditions the predictable representation property holds for G. We focus on the progressive enlargement G = (Gt)t∈R of a +reference filtration F = (Ft)t∈R through the observation of the occurrence of a random time τ +and we assume that the hypothesis (H′) is satisfied, i.e., any F-martingale is a G-semimartingale. It is worth noting that no general result on the existence of a G-semimartingale decomposition after τ of an F-martingale is available in the existing literature, while, for any τ, any F-martingale stopped at time τ is a G-semimartingale with an explicit decomposition depending on the Azéma supermartingale of τ. Moreover, most papers in this area are devoted to either the case of honest times (see, e.g., Barlow [6] and Jeulin & Yor [24, 25]) or the case where the Jacod’s absolute continuity hypothesis holds (see, e.g., Jeanblanc & Le Cam [18]). The particular case where the Jacod’s equivalence hypothesis holds is presented in Callegaro et al. [9]. More information can be found in [3]. The stability of predictable representation property was studied first by Kusuoka [26], under the assumptions that the filtration F is generated by a Brownian motion, the intensity rate of τ exists and the filtration F is immersed in its progressive enlargement G, that is, any F-martingale is a G-martingale. The immersion property is also known as the hypothesis (H) (see Brémaud & Yor [8]). Kusuoka has shown that any G-martingale can be represented as a sum of a stochastic integral with respect to the Brownian motion and a stochastic integral with respect to the compensated G-martingale M(τ) of the indicator process A := 1[[τ,∞[[. Under the hypotheses that the filtration F enjoys the predictable representation property, F is immersed in G, and τ avoids Date: May 21, 2018. 1<ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">A. AKSAMIT, M. JEANBLANC AND M. RUTKOWSKI </li></ul>stopping times, this result was extended for a positive Azéma supermartingale Z in Aksamit & Jeanblanc [3, Th. 3.13] and, for the case of an arbitrary supermartingale Z, in Coculescu et al. [10]. Barlow [6] started the study of the predictable representation property in the case where τ is honest. He solved the problem for honest times avoiding F-stopping times when all F-martingales are continuous in an unpublished manuscript [7]. The predictable representation property in a progressively enlarged filtration was then studied in Jeanblanc & Song [21] in full generality, assuming that the predictable representation property holds in the filtration F, with respect to a (possibly multidimensional) martingale M and that the hypothesis (H′) between F and G holds. Theorem 3.2 in [21] gives conditions for the validity (τ) c<ul style="display: flex;"><li style="flex:1">of the predictable representation property for G-martingales with respect to the pair (M, M </li><li style="flex:1">)</li></ul>cwhere M is the G-martingale part of the G-semimartingale M. In particular, they work under the assumption that Gτ = Gτ− (see (2.4) for the definition of these σ-fields). Comparing to their result we limit ourselves to some particular set-ups but we attempt here to derive explicit expressions for the predictable integrands, rather than merely to establish their existence and uniqueness and we do not make the assumption that Gτ = Gτ−. Our main result in this spirit, concerning a Poisson filtration, is stated in Theorem 3.1 and its further simplifications are given in Theorem 3.2 and Corollaries 3.2 and 3.3. We discuss the condition Gτ = Gτ− in Remark 3.2. Moreover, we derive different integral representations with not necessarily predictable integrands in the specific case where τ is assumed to be a pseudo-stopping time. The results of this type are collected in Propositions 4.3, 4.4 and 4.5. The paper is structured as follows. In Section 2, we introduce the setup and notation. We also review very briefly the classic results regarding the G-semimartingale decomposition of F- martingales stopped at τ. In Section 3, we assume that the filtration F is generated by a Poisson process N with compensated F-martingale M and we consider the filtration G, which is the progressive enlargement of F with an arbitrary random time. Since we work with a Poisson filtration, all F-martingales are necessarily processes of finite variation and thus any F-martingale is manifestly a G-semimartingale; in other words, the hypothesis (H′) is satisfied. In Section 3.1, we establish the predictable representation property for a class of G-martingales stopped at time τ in terms of three martingales: the compensated G-martingale M(τ) of the indicator process A, the G-martingale part M of the G-semimartingale M and the G-martingale, denoted by M∗, which takes care of the common cjumps of M(τ) and M. In Section 3.2, we illustrate the above result with a particular example cobtained with a Cox construction. In Section 4, we consider a general filtration F and we work under the postulate that τ is an F-pseudo-stopping time (see Nikeghbali & Yor [31]), so that its Azéma supermartingale is an F- optional, decreasing process. We first study in Section 4.1 the particular case of a random time τ independent of the filtration F. In Section 4.2 we consider some classes of G-martingales stopped at τ. We derive several alternative integral representations with respect to martingales built from F-martingales, under the assumption that the Azéma supermartingale of τ is strictly positive. In Section 4.3, the positivity assumption is relaxed and we obtain a more general representation result. Under additional hypotheses on the common jumps of specific F-martingales and M(τ), we (τ) cobtain a predictable representation property with respect to the two martingales M and M integrals used in what follows are implicitly postulated to be well defined without further explicit .<ul style="display: flex;"><li style="flex:1">R</li><li style="flex:1">R</li></ul>s<ul style="display: flex;"><li style="flex:1">Conventions. We use throughout the notational convention that </li><li style="flex:1">=</li><li style="flex:1">(t,s]. Moreover, all </li></ul>t<ul style="display: flex;"><li style="flex:1">INTEGRAL REPRESENTATIONS FOR PROGRESSIVE ENLARGEMENTS </li><li style="flex:1">3</li></ul>mentioning. In several (but not all) instances the existence of integrals is easy to check, since any process h from the class Mo(G, τ) (or Mp(G, τ)) is bounded (see Section 2.3). Finally, we write X = 0 when the process X is indistinguishable from the null process. 2. Preliminaries We first introduce the notation for an abstract setup in which a reference filtration F is progressively enlarged through observation of a random time. Let (Ω, G, F, P) be a probability space where F is an arbitrary filtration satisfying the usual conditions of right-continuity and P-completeness, and such that F0 is a trivial σ-field and F∞ ⊆ G. Let τ be a random time, that is, a [0, ∞]-valued random variable on (Ω, G, P). The indicator process A := 1[[τ,∞[[ is an increasing process, which is F-adapted if and only if τ is an F-stopping time. By convention, we shall put A0− := 0 whenever we consider the jump of A at 0. We denote by G the progressive enlargement of the filtration F with the random time τ, that is, the smallest right-continuous and P-completed filtration G such that the inclusion F ⊂ G holds and the process A is G-adapted so that the random time τ is a G-stopping time. In other words, the progressive enlargement is the smallest extension of the filtration F, which satisfies the usual conditions and renders τ a stopping time. 2.1. Azéma Supermartingale. For a filtration H ∈ {F, G}, we denote by Bp,H (resp., Bo,H) the dual H-predictable (resp., the dual H-optional) projection of a process B of finite variation, whereas p,HU (resp., o,HU) stands for the H-predictable (resp., H-optional) projection of a process U. The H-predictable covariation process of two H-semimartingales X and Y is denoted by hX, Y iH. Recall that hX, Y iH is defined as the dual H-predictable projection of the covariation process [X, Y ], that is, hX, Y iH := [X, Y ]p,H. The following definition is due to Azéma [5]. Definition 2.1. The càdlàg, bounded F-supermartingale Z given by <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul>Zt := P(τ > t | Ft) = o,F 1[[0,τ[[ = 1 − o,FAt tis called the Azéma supermartingale of the random time τ. Note that Z∞ := limt→∞ Zt = P(τ = ∞ | F∞). By convention, we set Z0− := 1. It is known that the processes Z and Z− do not vanish before τ. Specifically, the random set {Z− = 0} is disjoint from [[0, τ]] so that the set {Z = 0} is disjoint from [[0, τ[[ (see, e.g., [32]). The Azéma supermartingale Z is of class (D) and thus it admits a (unique) Doob-Meyer decomposition Z = µ−Ap where µ is a positive BMO F-martingale and Ap is an F-predictable, increasing process. Specifically, Ap is the dual F-predictable projection of the increasing process A, that is, Ap = Ap,F, and the positive F-martingale µ is given by the equality µt := E(A∞p + 1{τ=∞} | Ft). Then Ap0 = E(A0 | F0) = P(τ = 0 | F0) = 1 − Z0 as in [3, Proposition 1.24] and we obtain µ0 = 1. It is well known [23, p. 64]) that the process M(τ) given by ZdAps t∧τ (2.1) Mt(τ) := At − Zs− 0is a uniformly integrable G-martingale. We shall also use the dual F-optional projection Ao := Ao,F of A and the positive BMO F-martingale m given by mt := E(Ao∞ + 1{τ=∞} | Ft). Then A0o = E(A0 | F0) and m0 = 1. It is also well known that Z = m − Ao (see, e.g., [3, Proposition 1.46]). <ul style="display: flex;"><li style="flex:1">4</li><li style="flex:1">A. AKSAMIT, M. JEANBLANC AND M. RUTKOWSKI </li></ul>Remark 2.1. It is well known (see, e.g., Lemma 8.13 in Nikeghbali [30] or [3, Lemma 1.48]) that if either: (a) τ avoids all F-stopping times, that is, P(τ = σ < ∞) = 0 for any F-stopping time σ or (b) all F-martingales are continuous so that all F-optional processes are F-predictable, then the equalities Ao = Ap and µ = m are valid. These properties motivated other authors to focus on models in which either the avoidance or the continuity properties are satisfied. 2.2. Semimartingale Decompositions for Progressive Enlargements. For the reader’s con- venience, we recall some useful results for the progressive enlargement of a filtration F and an arbitrary random time τ. As was already mentioned, no general result giving tractable conditions for the stability of space of semimartingales under enlargement of filtration is known (one can see Jeulin [23, th. 2.6]) nor furnishing a G-semimartingale decomposition after τ of an F-martingale is available in the existing literature, although some partial results were established for particular classes of random times, such as: the honest times (see Barlow [6] and Jeulin & Yor [25]), the random times satisfying the density hypothesis (see Jeanblanc & Le Cam [18]), as well as for some random times obtained through various extensions of the so-called Cox construction of a random time (see, e.g., [16, 19, 20, 28, 29, 31]) and recently for a class of thin random times (see Aksamit et al. [2]). In this work, we will use semimartingale decompositions for F-martingales stopped at τ and thus we first quote some classic results regarding this case. For the result stated in Proposition 2.1, the reader is referred to Jeulin [23, Proposition 4.16] and Jeulin & Yor [24, Théorème 1, pp. 87–88]). Proposition 2.1. For any F-local martingale X, the process <ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><ul style="display: flex;"><li style="flex:1">ꢂ</li><li style="flex:1">ꢃ</li></ul><ul style="display: flex;"><li style="flex:1">t∧τ </li><li style="flex:1">t∧τ </li></ul><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">1</li></ul>dhX, miFs = Xt∧τ −1{Z dhX, µiFs + dJs bXt := Xt∧τ (2.2) −s−<1} Zs− Zs− <ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul>where J := A∆Xτ p,F, is a G-local martingale (stopped at τ). By comparing the two decompositions in (2.2), we obtain the following well known result, which will be used in the proof of the main result of this paper (see Theorem 3.1). Corollary 2.1. The following equality holds for any F-local martingale X ZProof. For completeness, we give the proof of the corollary. We note that the processes Z<ul style="display: flex;"><li style="flex:1">t∧τ </li><li style="flex:1">t∧τ </li></ul><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">1</li></ul>(2.3) <ul style="display: flex;"><li style="flex:1">dJs = </li><li style="flex:1">dhX, m − µiFs . </li></ul><ul style="display: flex;"><li style="flex:1">Zs− </li><li style="flex:1">Zs− </li></ul><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><ul style="display: flex;"><li style="flex:1">ꢂ</li><li style="flex:1">ꢃ</li></ul><ul style="display: flex;"><li style="flex:1">·∧τ </li><li style="flex:1">·∧τ </li></ul><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">1</li></ul>dhX, miFs and 1{Z dhX, µiFs + dJs s−<1} Zs− Zs− <ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul>are G-predictable. Hence equations (2.2) yield two equivalent forms of the Doob-Meyer decomposition of the special G-semimartingale X·∧τ . Due to the uniqueness of the Doob-Meyer decomposition, we obtain ZZZ Z<ul style="display: flex;"><li style="flex:1">ꢂ</li><li style="flex:1">ꢃ</li></ul>t∧τ t∧τ t∧τ t∧τ <ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">1</li></ul>0 = =1{Z dhX, µiFs + dJs −dhX, miFs s−<1} ZZs− s− 000 0Z<ul style="display: flex;"><li style="flex:1">ꢂ</li><li style="flex:1">ꢃ</li><li style="flex:1">ꢂ</li><li style="flex:1">ꢃ</li></ul>t∧τ 1dhX, µ − miFs + dJs −1{Z dhX, µiFs + dJs s−=1} Zs− 0<ul style="display: flex;"><li style="flex:1">ꢂ</li><li style="flex:1">ꢃ</li></ul>1 =dhX, µ − miFs + dJs Zs− where the last equality follows from Lemme 4(b) in [24]. Hence equality (2.3) is valid. ✷<ul style="display: flex;"><li style="flex:1">INTEGRAL REPRESENTATIONS FOR PROGRESSIVE ENLARGEMENTS </li><li style="flex:1">5</li></ul>2.3. Problem Formulation. For a filtration H ∈ {F, G} and a random time τ, we introduce two σ-fields Hτ and Hτ− defined as ( ꢄHτ = σ Yτ 1{τ<∞}, Y is an H-optional process ꢄ(2.4) Hτ− = σ Yτ 1{τ<∞}, Y is an H-predictable process . Obviously, Hτ− ⊂ Hτ and τ ∈ Hτ−. We recall that, in a progressive enlargement setting, for any G-predictable process Y , there exists an F-predictable process y such that Yt1{t≤τ} = yt1{t≤τ} (see, e.g., [3, Proposition 2.11]) and thus we have that Gτ− = Fτ−. In contrast, the σ-fields Gτ and Fτ may differ, in general (for counterexamples, see Barlow [6, Page 319] or [3, Example 5.13]). Note that since the random time τ is a G-stopping time, the definitions of Gτ and Gτ− given above coincide with the usual ones, specifically, Gτ = {G ∈ G∞ : G ∩ {τ ≤ t} ∈ Gt, ∀t} whereas Gτ− is the smallest σ-field which contains G0 and all the sets of the form G ∩ {t < τ} where t > 0 and G ∈ Gt. Let us introduce the following notation for the classes of G-martingales studied in this work ꢄMo(G, τ) := Yth := E(hτ | Gt): h ∈ Ho(F) where Ho(F) := {h : h a bounded F-optional process with limit at + ∞}. We denote by Hp(F) the set of all processes h in Ho(F) that are F-predictable and we set ꢄMp(G, τ) := Yth := E(hτ | Gt): h ∈ Hp(F) . Remark 2.2. (a) Due to the equality Gτ− = Fτ−, the set Mp(G, τ) represents all bounded G- <ul style="display: flex;"><li style="flex:1">martingales Y for which Yτ is equal to the value at time τ of a G-predictable process, i.e., Yτ ∈ Gτ− </li><li style="flex:1">.</li></ul>(b) Since the equality Gτ = Fτ is not valid, in general, Mo(G, τ) may differ from the class of all bounded G-martingales stopped at τ. Note that Mo(G, τ) is equal to the class of all bounded G-martingales stopped at τ if τ is an F-stopping time. If Gτ = Gτ− = Fτ , then we have equality between the two classes Mp(G, τ) and Mo(G, τ). In general, one has only Gτ− ⊂ Gτ where the inclusion may be strict. For instance, if τ is the first jump of a compound Poisson process

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