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 = (G_t)_t∈R of a

+

reference filtration F = (F_t)_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 Az´ema 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 Br´emaud & 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

• 2
• A. AKSAMIT, M. JEANBLANC AND M. RUTKOWSKI

stopping times, this result was extended for a positive Az´ema 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

• of the predictable representation property for G-martingales with respect to the pair (M, M
• )

c

where M is the G-martingale part of the G-semimartingale M. In particular, they work under the assumption that G_τ = G_τ− (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 G_τ = G_τ−. 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 G_τ = G_τ− 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

c

jumps of M^(τ) and M. In Section 3.2, we illustrate the above result with a particular example

c

obtained 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 Az´ema 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 Az´ema 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)

c

obtain 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
.

• R
• R

s

• Conventions. We use throughout the notational convention that
• =
• _(t,s]. Moreover, all

t

• INTEGRAL REPRESENTATIONS FOR PROGRESSIVE ENLARGEMENTS
• 3

mentioning. In several (but not all) instances the existence of integrals is easy to check, since any process h from the class M^o(G, τ) (or M^p(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. Az´ema Supermartingale. For a filtration H ∈ {F, G}, we denote by B^p,H (resp., B^o,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 i^H. Recall that hX, Y i^H is defined as the dual H-predictable projection of the covariation process [X, Y ], that is, hX, Y i^H := [X, Y ]^p,H. The following definition is due to Az´ema [5].

Definition 2.1. The c`adl`ag, bounded F-supermartingale Z given by

• ꢀ
• ꢁ

Z_t := P(τ > t | F_t) = ^o,F 1_[[0,τ[[ = 1 − ^o,FA_t

t

is called the Az´ema supermartingale of the random time τ. Note that Z_∞ := lim_t→∞ Z_t = 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 Az´ema supermartingale Z is of class (D) and thus it admits a (unique) Doob-Meyer decomposition Z = µ−A^p where µ is a positive BMO F-martingale and A^p is an F-predictable, increasing process. Specifically, A^p is the dual F-predictable projection of the increasing process A, that is, A^p = A^p,F, and the positive F-martingale µ is given by the equality µ_t := E(A_∞^p + 1_{τ=∞} | F_t). Then A^p₀ = 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

Z

dA^p_s

t∧τ

(2.1)

M_t^(τ) := A_t −

Z_s−

0

is a uniformly integrable G-martingale. We shall also use the dual F-optional projection A^o := A^o,F of A and the positive BMO F-martingale m given by m_t := E(A^o_∞ + 1_{τ=∞} | F_t). Then A₀^o = E(A₀ | F₀) and m₀ = 1. It is also well known that Z = m − A^o (see, e.g., [3, Proposition 1.46]).

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• A. AKSAMIT, M. JEANBLANC AND M. RUTKOWSKI

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 A^o = A^p 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, Th´eor`eme 1, pp. 87–88]).

Proposition 2.1. For any F-local martingale X, the process

• Z
• Z

• ꢂ
• ꢃ

• t∧τ
• t∧τ

• 1
• 1

dhX, mi^F_s = X_t∧τ

−

1_{Z

dhX, µi^F_s + dJ_s

b

X_t := X_t∧τ

(2.2)

−

_s−<1}

Z_s−

Z

s−

• 0
• 0

• ꢀ
• ꢁ

where J := A∆X_τ ^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

Z

Proof. For completeness, we give the proof of the corollary. We note that the processes

Z

• t∧τ
• t∧τ

• 1
• 1

(2.3)

• dJ_s =
• dhX, m − µi^F_s .

• Z_s−
• Z_s−

• 0
• 0

• Z
• Z

• ꢂ
• ꢃ

• ·∧τ
• ·∧τ

• 1
• 1

dhX, mi^F_s

and

1_{Z

dhX, µi^F_s + dJ_s

_s−<1}

Z_s−

Z

s−

• 0
• 0

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

• ꢂ
• ꢃ

t∧τ t∧τ t∧τ t∧τ

• 1
• 1

0 =

=

1_{Z

dhX, µi^F_s + dJ_s

−

dhX, mi^F_s

_s−<1}

Z

Z_s−

s−

000
0

Z

• ꢂ
• ꢃ
• ꢂ
• ꢃ

t∧τ

1

dhX, µ − mi^F_s + dJ_s

−

1_{Z

dhX, µi^F_s + dJ_s

_s−=1}

Z_s−

0

• ꢂ
• ꢃ

1
=

dhX, µ − mi^F_s + dJ_s

Z_s−

where the last equality follows from Lemme 4(b) in [24]. Hence equality (2.3) is valid.

✷

• INTEGRAL REPRESENTATIONS FOR PROGRESSIVE ENLARGEMENTS
• 5

2.3. Problem Formulation. For a filtration H ∈ {F, G} and a random time τ, we introduce two σ-fields H_τ and H_τ− defined as

(
ꢄ

H_τ = σ Y_τ 1_{τ<∞}, Y is an H-optional process

ꢄ

(2.4)

H_τ− = σ Y_τ 1_{τ<∞}, Y is an H-predictable process .

Obviously, H_τ− ⊂ H_τ and τ ∈ H_τ−. We recall that, in a progressive enlargement setting, for any G-predictable process Y , there exists an F-predictable process y such that Y_t1_{t≤τ} = y_t1_{t≤τ} (see, e.g., [3, Proposition 2.11]) and thus we have that G_τ− = F_τ−. In contrast, the σ-fields G_τ and F_τ 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 G_τ and G_τ− given above coincide with the usual ones, specifically,

G_τ = {G ∈ G_∞ : G ∩ {τ ≤ t} ∈ G_t, ∀t}

whereas G_τ− is the smallest σ-field which contains G₀ and all the sets of the form G ∩ {t < τ} where t > 0 and G ∈ G_t.
Let us introduce the following notation for the classes of G-martingales studied in this work

ꢄ

M^o(G, τ) := Y_t^h := E(h_τ | G_t): h ∈ H^o(F) where H^o(F) := {h : h a bounded F-optional process with limit at + ∞}. We denote by H^p(F) the set of all processes h in H^o(F) that are F-predictable and we set

ꢄ

M^p(G, τ) := Y_t^h := E(h_τ | G_t): h ∈ H^p(F) .
Remark 2.2. (a) Due to the equality G_τ− = F_τ−, the set M^p(G, τ) represents all bounded G-

• martingales Y for which Y_τ is equal to the value at time τ of a G-predictable process, i.e., Y_τ ∈ G_τ−
• .

(b) Since the equality G_τ = F_τ is not valid, in general, M^o(G, τ) may differ from the class of all bounded G-martingales stopped at τ. Note that M^o(G, τ) is equal to the class of all bounded G-martingales stopped at τ if τ is an F-stopping time. If G_τ = G_τ− = F_τ , then we have equality between the two classes M^p(G, τ) and M^o(G, τ). In general, one has only G_τ− ⊂ G_τ where the inclusion may be strict. For instance, if τ is the first jump of a compound Poisson process