PROGRESSIVE ENLARGEMENTS OF FILTRATIONS AND SEMIMARTINGALE DECOMPOSITIONS

Transcription

1 PROGRESSIVE ENLARGEMENTS OF FILTRATIONS AND SEMIMARTINGALE DECOMPOSITIONS Libo Li and Marek Rutkowski School of Mathematics and Statistics University of Sydney NSW 26, Australia July 1, 211 Abstract We deal with decompositions of F-martingales with respect to the progressively enlarged filtration G and we extend there several related results from the existing literature. Results on G-semimartingale decompositions of F-local martingales are crucial for applications in financial mathematics, especially in the context of modeling credit risk, where τ represents the moment of occurrence of some credit event e.g., the default event. The research of L. Li and M. Rutkowski was supported under Australian Research Council s Discovery Projects funding scheme DP

2 2 Progressive Enlargements and Semimartingale Decompositions Contents 1 Introduction Applications to Financial Mathematics Preliminaries Conditional Distribution of a Random Time Multiplicative Construction of a Random Time Properties of Conditional Distributions Hypotheses H and HP Complete Separability Separability Extended Density Hypothesis Enlargements of Filtrations Case of an Honest Time General Case Conditional Expectations Conditional Expectations under Hypothesis HP Conditional Expectations under the Extended Density Hypothesis Properties of G-Local Martingales G-Local Martingales under Hypothesis HP G-Local Martingales under the Extended Density Hypothesis Compensator of H Compensator of H under Complete Separability Compensator of H under the Extended Density Hypothesis Semimartingale Decompositions Case of Continuous Martingales Case of Discontinuous Martingales Semimartingale Decomposition under Hypothesis HP Semimartingale Decomposition under the Extended Density Hypothesis Case of Multiplicative Construction Applications to Financial Mathematics Martingale Measures Hypothesis H in a General Set-up Martingale Measures via Hypotheses H and HP Information Theory

3 L. Li and M. Rutkowski 3 1 Introduction We work on a probability space Ω, G, P endowed with a filtration F satisfying the usual conditions and we assume that τ is any random time with values in R + defined on this space. Definition 1.1. By an enlargement of F associated with τ or, briefly, an enlargement of F we mean any filtration K = K t t R+ in Ω, G, P, satisfying the usual conditions, and such that: i the inclusion F K holds, meaning that F t K t for all t R +, and ii τ is a K-stopping time. The following two particular enlargements of F were extensively studied in the literature see, e.g., Dellacherie and Meyer [12], Jeulin [24, 25, 26], Jeulin and Yor [27, 28], and Yor [35, 36]; Nikeghbali [31] see Section 8 therein provides a short overview of the most pertinent results, but mostly under the assumptions that all F-martingales are continuous and/or the random time τ avoids all F-stopping times. Definition 1.2. The initial enlargement of F is the filtration G = Gt t R+ Gt = s>t στ F s for all t R +. given by the equality The initial enlargement does not seem to be well suited for the analysis of a random time since it postulates that στ G, meaning that all the information about τ is already available at time. It appears that the following notion of the progressive enlargement is more suitable for formulating and solving problems associated with an additional information conveyed by observations of occurrence of a random time τ. Definition 1.3. The progressive enlargement of F is the minimal enlargement, that is, the smallest filtration G = G t t R+, satisfying the usual conditions, such that F G and τ is a G-stopping time. More explicitly, G t = s>t G o s where we denote G o t = στ t F t for all t R +. Let H be the filtration generated by the process H t = 1 {τ t}. It is clear that F H G G. In fact, the inclusion F H K necessarily holds for any enlargement of F. In what follows, we will mainly work with the progressive enlargement G. However, we would like first to clarify the relationships between various filtrations encountered in the existing literature. Note that some definitions are only applicable to the case of an honest time see Definition 3.1. We are now in the position to describe very briefly the related results from the existing literature. For any two filtrations F K on a probability space Ω, G, P, we say that the hypothesis H holds for F and K under P if any P, F-semimartingale is also a P, K-semimartingale. By the classic result due to Jacod [18], the hypothesis H is satisfied in the case of the initial enlargement of F provided that τ is an initial time. Recall that τ is called an initial time if there exists a measure η on R +, B R + such that the F-conditional distributions of τ are absolutely continuous with respect to η, that is, F du,t ηdu. This property is commonly referred to as the density hypothesis. The hypothesis H should be contrasted with the stronger hypothesis H for F and K under P, also known as the immersion property, which stipulates that any P, F-martingale is also a P, K- martingale see Brémaud and Yor [7]. In the case of the progressive enlargement G of a filtration F through a random time τ defined on Ω, G, P, the abovementioned hypothesis H for F and G under P is well known to be equivalent to the hypothesis H introduced in Definition 2.4 see, for instance, Elliott et al. [15] and thus no confusion may arise. However, the main hypothesis examined in the present work are the hypothesis HP and extended density hypothesis introduced in Definition 2.4 and 2.7 respectively. the hypothesis HP is clearly weaker than the hypothesis H and it is satisfied, in particular, when a random time is constructed using the multiplicative approach. Also, under mild technical assumption, it is equivalent to the either complete or partial separability of the F-conditional distribution of τ see Definitions 2.5 and 2.6. On the other hand, the extended density hypothesis is a reformulation/extension of the concept of density hypothesis: there exists an F-adapted increasing process D such that F du,t dd u see Definition 2.7. We do so to avoid the technical assumption on strict positivity of F-conditional

4 4 Progressive Enlargements and Semimartingale Decompositions distribution of the random time τ. It is worth to point out that most results obtained for initial times can be extended to this new setting without any serious difficulty. The problem of the semimartingale decomposition of a P, F-martingale with respect to a progressive enlargement G was examined in several papers during the last thirty years. In particular, the following properties are well known see Yor [35] and Dellacherie and Meyer [12]: a a P, F-martingale may fail to be a semimartingale with respect to a progressive enlargement G, b any P, F-semimartingale stopped at τ is a P, G-semimartingale, c any P, F-semimartingale is a P, G-semimartingale if τ is an honest time. Having the above points in mind, if a process U is a P, F-local martingale. It is natural to ask whether U is a P, G-semimartingale and to compute its canonical semimartingale decomposition with respect to G. In the path-breaking paper by Jeulin and Yor [27] see also Barlow [3, 4] and Jeulin and Yor [28], the authors derived the P, G-semimartingale decomposition of the stopped process U τ t for any random time τ, as well as the P, G-semimartingale decomposition of U under an additional assumption that τ is an honest time. The latter result was recently extended to the case of initial times in papers by Nikeghbali and Yor [32], Jeanblanc and Le Cam [2] and El Karoui et al. [15]. Of course, we are not in a position to discuss these results in detail here, although some of them will be stated in what follows. The paper is organized as follows. In Section 2, we outline briefly the multiplicative approach to constructing of a random time with a predetermined Azéma supermartingale that was developed in Li and Rutkowski [29]. we also examine in this section some properties of F-conditional distributions of a random time under HP hypothesis and extended density hypothesis. In Section 3, we revisit and clarify the relationship between filtrations in the existing literature, while paying special attention to the progressive enlargement of F with a random time. We provide an alternative characterization of the filtration G. This alternative characterization is used in the subsequent subsections in computations of conditional expectation of G-adapted processes under hypothesis HP and extended density hypothesis. With these results, we are then able to give sufficient conditions for a class of G-adapted processes to a P, G-martingale. In Section 4, we compute the semimartingale decompositions of P, F-martingales with respect to the progressively enlarged filtration G and extend there several related results from the existing literature. It should be stressed that is not assumed here that the pair τ, P was obtained using the multiplicative system approach summarized in Section 2. Results on P, G-semimartingale decompositions of F-local martingales are crucial for applications in financial mathematics, especially modeling of credit risk where τ represents the moment of occurrence of some credit event e.g., a default. For all unexplained concepts from the general theory of stochastic processes we refer to, for instance, the monographs by Dellacherie [11] and Jacod and Shiryaev [19], as well as the survey paper by Nikeghbali [31]. 1.1 Applications to Financial Mathematics In view of financial mathematics and applications, we shall provide two examples. As usual, one is given a random time τ on a probability space Ω, G, P endowed with a filtration F satisfying the usual conditions. In view of model construction, it is natural to demand that any given model should be arbitrage free. In the recent work of Coculescu et al. [9], the existence of an equivalent probability measure for which the hypothesis H holds was shown to be a sufficient condition for a model with enlarged filtration to be arbitrage-free, provided that the model based on the filtration F is arbitrage free. In subsection 5.1, we sketch the most pertinent results from [9] and then show that under mild technical assumptions, if τ, P satisfies the hypothesis HP then there exists an equivalent probability measure for which hypothesis H holds. That is, if the market trading with information F is

5 L. Li and M. Rutkowski 5 arbitrage-free, then so is the market that trades with information described by G. Next, we consider utility maximisation under asymmetric information. We give a brief sketch on how the results obtain on G-semimartingale decomposition in Proposition 4.2 can be applied to insiders information problems studied in the existing literature. Given a P, F-semimartingale S, one argues as follows. Heuristically, different investors makes different investment decisions based on the information they have, which is modeled by different filtrations. The filtration F can be interpreted as information that is publicly available to all investors. An investor is said to be an insider, if he has access to a larger filtration K that contains the reference filtration F. In particular, We refer to [1, 16] for studies on asymmetric information in financial markets and [8, 5, 13] on anticipative approach to the insider problem. Broadly speaking, given a filtration K F, let K be the set of K-adapted x-superadmissible trading strategies, that is any trading strategies such that the investors wealth process V t x is strictly positive for all t. In the case of F-adapted x-superadmissible strategies, it is described by the following condition V t x := x + φ s ds s >, t. where x is the initial wealth. The anticipative or the forward integral approach under certain technical assumptions allows one to write down an investor s wealth process or hedging portfolio as a forward integral of their K-adapted x-superadmissible trading strategy φ against the F-adapted process S, even if the process S is not a K-semimartingale. The mathematical formulation of the utility maximisation problem of insiders are given in [8]. We shall introduce it below and hint on how our results can be applied. First in [8] the author aims to maximise insider utility and the optimization problem is formulated as follows, U K x := max E P U x + φ t d S t 1 φ K where the integral is understood as a forward stochastic integrals and the function U is any utility functions. In the case where the hypothesis H holds between F and K, the forward stochastic integral can be replaced by a standard stochastic integral. With the results developed in this paper, one can work with the filtration K = G been the progressive enlargement of the reference filtration F with a positive random variable τ satisfying the extended density hypothesis. Although by doing so, we loss the generality of the anticipative approach, as the process S may not even have to be a semimartingale in the larger filtration K, but what we hope to do is to avoid technical assumptions of Malliavin calculus and by keeping computations to F and G adapted processes only, one may be able to provide more explicit computations. Ankirchner and Imkeller have shown in [1] that if F and K are both finite utility filtrations for the process S see Definition 5.1 then the difference in expected logarithmic utility between investors working with filtrations F and K where F K depends only on the information drift of K relative to F see Definition 5.2. Therefore, as an application, we give in Section 5.2 under extended density hypothesis and certain integrability conditions, an explicit representations for: i the solution to optimization problem 1 with Ux = logx and K = G ii the information drift of G with respect to F. We show that the progressive enlargement G of a finite utility filtration F for the process S with a random time is once again a finite utility filtration for the process S.,T ]

6 6 Progressive Enlargements and Semimartingale Decompositions 2 Preliminaries In this section, we deal with the most pertinent properties of F-conditional distributions of a random time. First, we summarize in Section 2.2 the multiplicative approach to a random time with a given in advance Azéma supermartingale G, as developed in Li and Rutkowski [29]. The first step is to produce an F-conditional distribution of τ consistent with G using the concept of the multiplicative system. Next, one constructs a random time with a predetermined F-conditional distribution on a suitable extension of the underlying probability space. The construction provided in Section 2.2 motivates us to study in Section 2.3, various properties of F-conditional distributions that will play an essential role in the study of enlarged filtrations. 2.1 Conditional Distribution of a Random Time For an arbitrary random time τ, defined on some filtered probability space Ω, F, F, P, we define the P, F-submartingale F τ and the P, F-supermartingale G τ = 1 F τ by setting F τ t = P τ t F t, G τ t = P τ > t F t. Note that the processes F τ and G τ are positive and bounded by 1. Definition 2.1. The F-conditional distribution of τ is the random field Fu,t τ u,t R+ given by F τ u,t = P τ u F t, u, t R +. The F-conditional survival distribution of τ is the random field G τ u,t u,t R+ given by G τ u,t = P τ > u F t = 1 F τ u,t, u, t R +. More generally, it is convenient to introduce also the following definition. Definition 2.2. A random field F u,t u,t R+ on a filtered probability space Ω, F, F, P is said to be an F-conditional distribution of a random time if it satisfies: i for every u R + and t R +, we have F u,t 1, P-a.s., ii for every u R +, the process F u,t t R+ is a P, F-martingale, iii for every t R +, the process F u,t u R+ is right-continuous, increasing and F,t = 1. A random field G u,t u,t R+ is said to be an F-conditional survival distribution of a random time whenever F u,t = 1 G u,t is an F-conditional distribution of a random time. The following result from Li and Rutkowski [29] shows that if a random field F u,t satisfies Definition 2.2 then there exists a random time τ on some extended probability space Ω, F, F, P such that Fu,t τ = F u,t. Lemma 2.1. For any given F-conditional distribution F u,t, there exists a random time τ on an extension of the usual probability space such that P τ u F t = F u,t for all u, t R +. Moreover, the equality P Ft = P Ft holds for all t R +. Usually, a random field F u,t is not given in advance, but one rather deals with a predetermined submartingale F. Hence the following assumption. Assumption 2.1. We are given a submartingale F = F t t R+, defined on an underlying filtered probability space Ω, F, F, P, and such that F t 1 for every t R + and F = Multiplicative Construction of a Random Time For a given in advance P, F-submartingale F, Li and Rutkowski [29] provided also a method of constructing a random field F u,t consistent with F, in the sense that F t = F t,t. We will now describe briefly the method proposed in [29]. We start by recalling the concept of a multiplicative system associated with a submartingale see Meyer [3].

7 L. Li and M. Rutkowski 7 is a multi- Definition 2.3. Given a positive submartingale F = F t t R+, we say that C u,t u,t R+ plicative system associated with F if the following conditions are satisfied: i if u s t then C u,s C s,t = C u,t and C u,t = 1 if u t, ii for any fixed u R +, the process C u,t t R+ is a predictable and decreasing, iii for any fixed t R +, the process C u,t u R+ is a right-continuous and increasing, iv the following equality holds, for all t R +, E P C t, F F t = F t. 2 For the proof of the next result, see Corollary 2.1 in Li and Rutkowski [29]. Lemma 2.2. Let C u,t u,t R+ be a multiplicative system associated with F. Then: i for any fixed u, the process C u,t F t t [u, ] is a P, F-martingale, ii the following relationship holds, for every u < t, C u,t p F t = C u,t F t. 3 The next lemma, borrowed from Li and Rutkowski [29], shows that for any positive submartingale F satisfying Assumption 2.1, one can construct an F-conditional distribution F u,t such that F t,t = F t. Lemma 2.3. Let F t t R+ field F u,t u,t R+ by setting be a submartingale satisfying Assumption 2.1. We define the random F u,t = { E P F u F t, C u,t F t, t [, u, t [u, ], 4 where C u,t is any multiplicative system associated with F. Then F u,t is an F-conditional distribution and F t,t = F t. We are in the position to state the main result from Li and Rutkowski [29]. It is worth noting that in order to establish Theorem 2.1, it suffices to combine Lemmas 2.1 and 2.3. Theorem 2.1. Assume that we are given a submartingale F t t R+ satisfying Assumption 2.1. Then there exists a random time τ on the extension Ω, F, F, P of the filtered probability space Ω, F, F, P such that the F-conditional distribution of τ under P equals { F u,t = P E P F u F t, t [, u], τ u F t = 5 C u,t F t, t u, ]. In particular, the equality P τ > t F t = G t is valid for every t [, ]. Moreover, the equality P Ft = P Ft is satisfied for all t R +. Although the multiplicative system associated with the submartingale F may not be unique see Meyer [3], the F-conditional survival distribution constructed from any multiplicative systems C u,t associated with F is always unique in this framework. Let us also observe that in Theorem 2.1 we only deal with the random field C u,t u R+, t u. Corollary 2.1. The random time τ constructed in Theorem 2.1 is with probability 1: i a finite random time if and only if the process F is continuous at infinity, that is, the equality F = F holds. ii a strictly positive random time if and only if the process F is continuous at time zero, that is, F = F =. Under the following stronger assumption the uniqueness of the multiplicative system associated with F holds and an explicit representation is known.

8 8 Progressive Enlargements and Semimartingale Decompositions Assumption 2.2. The submartingale F = F t t R+ defined on a filtered probability space Ω, F, F, P is such that F = 1 and the inequalities < F t 1 and < F t 1 are valid for every t R +. Let the process B be given by B = A G, where G = G + M A is the Doob-Meyer decomposition of the bounded P, F-supermartingale G, so that also G = M B. Let us observe that B is the F-predictable, increasing process that generates the supermartingale 1 F. It is also known that A is the P, F-dual predictable projection of the non-f-adapted increasing process H t = 1 {τ t}. Proposition 2.1. Assume that F satisfies Assumption 2.2. Then the unique multiplicative system associated with F satisfies, for every u t, C u,t = C E t,t, ] p F s 1 db s = C,u E u 6, ] p F s 1 db s where E t U stands for the stochastic exponential of the process U, that is, the unique solution to the stochastic differential equation de t U = E t U p F t 1 db t with E U = 1. The following corollary provides sufficient conditions for the strict positivity of the random field F u,t for all u, t > this property will also be referred to as the non-degeneracy of F u,t. Corollary 2.2. Consider the F-conditional distribution F u,t = P τ u F t given by 5. Under Assumption 2.2, the random field F u,t is strictly positive, except perhaps for F, = F. 2.3 Properties of Conditional Distributions We will now examine further properties of F-conditional distributions of τ. It shall become evident that the P, G-semimartingale decomposition of P, F-martingales hinges on the properties of the conditional distribution of τ Hypotheses H and HP we recall the classic hypothesis H see, for instance, Brémaud and Yor [7] or Elliott et al. [14] and the weaker hypothesis HP that was introduced in papers by Jeanblanc and Song [21] and Li and Rutkowski [29]. Definition 2.4. A pair τ, P is said to satisfy the hypothesis H whenever for all u s t F u,s = F u,t. 7 A pair τ, P is said to satisfy the hypothesis HP whenever for all u s t F u,s F s,t = F s,s F u,t. 8 The hypotheses of Definition 2.4 can be seen either as properties of a pair τ, P or, equivalently, as properties of the associated F-conditional distribution F u,t. It is clear that H implies HP. It is also known that if the pair τ, P is constructed as in Theorem 2.1 then the hypothesis HP holds see Proposition 3.1 in Li and Rutkowski [29]. By contrast, the hypothesis H is not necessarily satisfied by the pair τ, P obtained in Theorem 2.1.

9 L. Li and M. Rutkowski Complete Separability Definition 2.5. We say that an F-conditional distribution F u,t is completely separable if there exists a positive P, F-martingale X and a positive, F-adapted, increasing process Y such that F u,t = Y u X t for every u, t R + such that u t. It is easily seen that the complete separability of F u,t implies that the hypothesis HP holds. Indeed, we have that F u,s F s,t = Y u X s Y s X t = Y s X s Y u X t = F s,s F u,t for all u < s < t. Recall that in the case of the multiplicative approach to an admissible construction of a random time τ the F-conditional distribution F u,t is unique and satisfies the equality F u,t = C u,t F t = Q u,t for all u t where, for any fixed u R +, the process Q u,t t u is a uniformly integrable P, F-martingale. Using Corollary 2.1, we thus obtain the following result. Proposition 2.2. Let the F-conditional distribution F u,t = Pτ u F t be given by formula 5. Under Assumption 2.2, we have that, for all u t, F t E t, ] F p F s 1 db s u,t = E u = Y u X t, ] p F s 1 db s where the strictly positive, F-adapted, increasing process Y is given by Y u = and the strictly positive P, F-martingale X equals Hence the random field F u,t is completely separable. ] 1 [E u p F s 1 db s, ] X t = F t E t p F s 1 db s. 9, ] Separability It appears that the property of complete separability is too restrictive, since it does not cover all interesting cases. For this reason, we introduce also the weaker condition of separability that, in fact, will be shown to be equivalent to the hypothesis HP if a random time is strictly positive. Definition 2.6. We say that an F-conditional distribution F u,t is separable at v if there exist a positive P, F-martingale X v = Xt v, t R + and a positive, F-adapted, increasing process Y v = Yu v, u [v, such that the equality F u,t = Yu v Xt v holds for every v u t. Also, we say that an F-conditional distribution F u,t is separable if it is separable at all v >. It is clear from Definitions 2.5 and 2.6 that an F-conditional distribution F u,t is completely separable whenever it is separable at. Lemma 2.4. The following implications hold for any F-conditional distribution F u,t : i if F u,t is separable at v then it is also separable at all s v, ii if F u,t is separable at v then the proportionality property 8 holds for all v u < s < t. It appears that the separability of F u,t implies the hypothesis HP when F =, that is, when the random time τ is strictly positive. Proposition 2.3. If the F-conditional distribution of τ is separable and F = then the hypothesis HP holds.

10 1 Progressive Enlargements and Semimartingale Decompositions It seems natural to conjecture that in the non-degenerate case when F u,t > for all u, t > the converse implication also holds, that is, if the hypothesis HP is valid then F u,t is separable. This is indeed the case, as the following result shows. Proposition 2.4. Suppose that a non-degenerate F-conditional distribution F u,t satisfies the hypothesis HP. Then the random field F u,t is separable Extended Density Hypothesis Definition 2.7. A pair of τ, P is said to satisfy the extended density hypothesis if there exists a family of positive random field m s,t, which are P, F-martingales for t s and an F-adapted, increasing process D such that the F-conditional distribution F u,t admits the following representation F u,t = m s,t dd s, u t. [,u] The above definition can be seem as a natural extension to the density hypothesis. The results obtained under extended density hypothesis are similar to those under density hypothesis. Nevertheless, it is introduced as it allows us to circumvent an awkward non-degeneracy condition of the F-conditional distribution of a random time, which is needed in the proof of Lemma 3.5. Remark 2.1. If the F-conditional distribution of τ is constructed through multiplicative systems approach then the F-conditional distribution F u,t has the following integral representation F t C s,t F t C s,t F t C s,t F t C s,t F u,t = F t p db s = F t F p db s + s F p db s = s F p db s s F s u,t] [L t,t] where the F t -measurable random time L t equals [L t,u] [L t,u] L t = sup { u t : C u,t = } = sup { u t : C u,t = }. 1 If the Azéma supermartingale is strictly positive, then L t = for all t. This feature of the multiplicative construction can be seen as an alternative motivation for the extended density hypothesis. 3 Enlargements of Filtrations We start by analyzing the basic properties of various enlargements of F associated with a random time τ. Recall that the notion of an enlargement of F was introduced in Definition Case of an Honest Time We examine first the case of an honest time, which was extensively studied in the literature see, e.g., Jeulin [24, 25, 26] and Jeulin and Yor [27, 28]. Definition 3.1. A positive random variable τ defined on a filtered probability space Ω, G, F, P is called an honest time if, for every t >, there exists an F t -measurable random variable τ t such that τ is equal to τ t on the event {τ t}, that is, τ1 {τ t} = τ t 1 {τ t}. The following properties of honest times are well known see, in particular, remark on page 76 in Dellacherie and Meyer [12]: a an honest time τ is an F -measurable random variable, b the event {τ t} may be replaced by the event {τ < t} in Definition 3.1, c a finite honest time can equivalently be defined as the end of an F-optional set. To check that an honest time satisfies the hypothesis HP, it suffices to use the following result.

11 L. Li and M. Rutkowski 11 Lemma 3.1. [Yor [35]] The following properties are equivalent: i τ is an honest time, ii for every u < s, there exists an event A us F s such that {τ u} = A us {τ s}. Corollary 3.1. An honest time τ satisfies the hypothesis HP. Proof. Let u < s < t. Using Lemma 3.1, we obtain F u,s F s,t = Pτ u F s Pτ s F t = PA us {τ s} F s Pτ s F t = 1 Aus Pτ s F s Pτ s F t = Pτ s F s PA us {τ s} F t = Pτ s F s Pτ u F t = F s,s F u,t, which shows that equality 8 is satisfied, so that the hypothesis HP holds. The following families of σ-fields were introduced and examined in the context of honest times. Definition 3.2. We define the families G = G t t R+, G = G t t R+ and G = G t t R+ of σ-fields by setting, for all t R +, G t = {A G Ãt F t such that A {τ > t} = Ãt {τ > t}}, G t = { A G Ât F t such that A {τ t} = Ât {τ t} }, G t = { A G Ãt, Ât F t such that A = Ãt {τ > t} Ât {τ t} } = G t G t. Note that the family G is uniquely characterized by the following equalities, for all t R +, G t {τ > t} = F t {τ > t}, G t {τ t} = F t {τ t}, 11 meaning, in particular, that G does not depend on the choice of the σ-field G it only depends on F and τ. The following properties of families G, G and G are well known see Yor [35]: a for any random time τ, the family G is a filtration satisfying the usual conditions, b the families G and G are not necessarily increasing, c if the family G is increasing then τ is an honest time, d the family G is a filtration satisfying the usual conditions if and only if τ is an honest time. It is also clear that the inclusions G G and G G are valid and thus G is an enlargement of F for any random time τ, whereas G is an enlargement of F only when τ is an honest time. The choice of the filtration G is convenient when dealing with the decomposition of F-martingales with respect to enlarged filtration when we only examine processes stopped at τ not necessarily an honest time. In fact, one can use for this purpose any enlargement K that is admissible before τ, in the sense of Definition 3.3 see, e.g., Guo and Zhang [17]. It is apparent the filtrations G for any random time and G for an honest time are admissible before τ. Definition 3.3. An enlargement K of a filtration F is said to be admissible before τ if the equality K t {τ > t} = F t {τ > t} holds for every t R +. When one deals with processes that are not stopped at τ then the choice of G is no longer suitable, since the information conveyed by G after τ is too large, in general. The smaller filtration G can be used note that G G when τ is an honest time, but, obviously, it cannot be applied to a general random time. As we shall see in what follows, the filtration G coincides in fact with the progressive enlargement G when τ is an honest time see Jeulin [24] or part ii in Lemma 3.2. This feature explains why G was widely adopted as the natural choice of an enlargement of F in the literature dealing with an honest time.

12 12 Progressive Enlargements and Semimartingale Decompositions 3.2 General Case In the case of a general random time, we find it convenient to introduce the following notion, which hinges on a natural modification of the definition of the family G. Definition 3.4. The family Ĝ = Ĝt t R+ is defined by setting, for every t, Ĝ t = {A G Ãt F t and Âτ,t G t such that A = Ãt {τ > t} Âτ,t {τ t}}. We note that, for all t R +, Ĝ t {τ > t} = F t {τ > t}, Ĝ t {τ t} = G t {τ t}. 12 It is easily seen that the σ-field Ĝt is uniquely characterized by conditions 12. The next result shows that the family Ĝ coincides in fact with the progressive enlargement G. Lemma 3.2. i If τ is any random time then G = Ĝ. ii If τ is an honest time then G = G = Ĝ. Proof. Recall that G t = s>t στ s F s and G t = s>t στ F s see Definitions 1.2 and 1.3. To show that Ĝt = G t, it suffices to check that conditions 12 are satisfied by G t. The following relationship for all t R + is immediate, F t {τ > t} G t {τ > t} G t {τ > t} = F t {τ > t}. This shows G t {τ > t} = F t {τ > t}, while on the other hand, G t {τ t} = s>t στ s F s {τ t} = s>t στ F s {τ t} = G t {τ t} since στ s {τ t} = στ {τ t} for every s > t. For part ii, we start by noting that the inclusions G G Ĝ are obviously satisfied. From part i, we know that G = Ĝ for any random time and thus G = G = Ĝ for an honest time. It is easy to see that the filtration Ĝ and hence G is admissible before τ. When dealing with a semimartingale decomposition of an F-martingale after τ we will use the following concept. Definition 3.5. We say that an enlargement K is admissible after τ if the equality K t {τ t} = G t {τ t} holds for every t R +. It is clear that the filtration Ĝ and thus G is admissible after τ for any random time. Note also that if an enlargement K is admissible before and after τ then simply K = G. Lemma 3.3. For any integrable, G-measurable random variable X and any enlargement K = K t t admissible after τ we have that, for any t R +, Proof. It suffices to show that E P 1{τ t} X Kt = lim s t E P 1{τ t} X στ Fs. 13 E P 1{τ t} X Kt = EP 1{τ t} X G t. 14 The second equality in 13 will then follow from Corollary 2.4 in [34] since G t = s>t στ F s. To establish 14, we will first check that, for every A K t, E P 1A 1 {τ t} X = E P 1A 1 {τ t} E P X G t.

13 L. Li and M. Rutkowski 13 Since, by assumption, K t {τ t} = Gt {τ t}, there exists an event B Gt {τ t} = B {τ t}. Consequently, such that A E P 1A 1 {τ t} X = E P 1B 1 {τ t} X = E P 1B 1 {τ t} E P X Gt = E P 1A 1 {τ t} E P X Gt. Hence E P 1 {τ t} X K t = E P 1{τ t} E P X Gt K t = 1{τ t} E P X Gt, since the random variable 1 {τ t} E P X Gt is K t -measurable. 3.3 Conditional Expectations For a fixed T >, we consider the map ÛT : R + Ω R and we use the notation u, ω Ûu,T ω. We postulate that ÛT is a B R + F T -measurable map, so that U τ,t is a στ F T -measurable random variable. We will need the following auxiliary lemma. Lemma 3.4. The following inclusions hold for every t R + : i {τ > t} {G t > }, P-a.s., ii {τ t} {F t > }, P-a.s. Proof. Let us denote A = {F t = 1} = {Pτ t F t = 1}. Since A F t PA = F t dp = Pτ t F t dp = 1 {τ t} dp = PA {τ t}. A A Hence A = {F t = 1} = {G t = } {τ t}, P-a.s., and thus {τ > t} {G t > }, P-a.s. For part ii, let us denote B = {G t = 1} = {Pτ > t F t = 1}. Since B F t PB = G t dp = Pτ > t F t dp = 1 {τ>t} dp = PB {τ > t}. B B Hence B = {G t = 1} = {F t = } {τ > t}, P-a.s., and thus {τ t} {F t > }, P-a.s. Remark 3.1. Part i in Lemma 3.4 can also be demonstrated as follows. Let τ = inf {t R + : G t = or G t = }. Since G is a supermartingale, it is equal to zero after τ and thus Pτ < τ = E P 1 {τ<τ} = E P G τ 1 {τ< } =. A B This in turn implies that {τ > t} {G t > }, P-a.s. Remark 3.2. Let us set, by convention, / =. 1 {τ>t} G 1 t and 1 {τ t} Ft 1 are well defined, P-a.s. Then, in view Lemma 3.4, the quantities In the rest of the paper, we shall formulate results assuming that the F-conditional distribution of the given in advance random time τ, satisfies either i hypothesis HP or ii the extended density hypothesis, which are introduced in Definitions 2.4 and 2.7 respectively. In addition, the special case of complete separability will also be mentioned Conditional Expectations under Hypothesis HP The following lemma, which extends formula 2 to any t [, T ], corresponds to Theorem 3.1 in El Karoui et al. [15] where the case of an initial time was studied. Let us recall that the concept of an initial time was introduced in the study of the hypothesis H under the initial enlargement of F by some random variable see Jacod [18] and Section 4 below. However, since it formally corresponds to the co-called density hypothesis see Jacod [18] and Hypothesis 2.1 in El Karoui et al. [15], the concept of an initial time can be studied in the context of any that is, not necessarily initial enlargement of F.

14 14 Progressive Enlargements and Semimartingale Decompositions Lemma 3.5. Let U,T : R + Ω R be a B R + F T -measurable map. Assume that the pair τ, P is such that the hypothesis HP holds and the random variable U τ,t is P-integrable. i For every t [, T, we have that E P U τ,t G t = 1 {τ>t} Ũ t,t + 1 {τ t} Û τ,t,t 15 where Ũ t,t = G t 1 E P 1{τ>t} U τ,t F t = Gt 1 E P t, ] U u,t df u,t Ft 16 and, for all u t < T, Û u,t,t = 1 G t 1 E P F t,t U u,t F t. 17 ii If, in addition, F u,t is completely separable so that F u,t = Y u X t for u T then 16 yields E P 1{T τ>t} U τ,t F t = EP X T t,t ] U u,t dy u Ft 18 and 17 becomes Û u,t,t = X t 1 E P X T U u,t F t. 19 Proof. The derivation of 16 is rather standard. Note that the hypothesis HP is not needed here and we may take t [, T ]. It suffices to take U u,t = gu1 A for a Borel measurable map g : R + R and an event A F T such that the random variable U τ,t = gτ1 A is P-integrable. Using part i in Lemma 3.4 and the classic formula for conditional expectation with respect to G t, we obtain, on the event {τ > t}, E P 1 {τ>t} U τ,t G t = E P 1{τ>t} U τ,t F t = G t 1 E P 1 A E P 1{τ>t} gτ F T F t Pτ > t F t = G t 1 E P 1 A gu df u,t Ft = G t 1 E P gu1 A df u,t Ft = G t 1 E P t, ] t, ] U u,t df u,t Ft, which yields 16. Let us take any t [, T. To establish 17, we need to compute E P U τ,t G t on the event {τ t}. By an application of Lemma 3.3, t, ] E P 1{τ t} U τ,t G t = lim s t E P 1{τ t} U τ,t στ F s. We first compute for t < s < T, the conditional expectation E P 1{τ t} U τ,t στ Fs. Recall that the hypothesis HP means that the equality F u,s F s,t = F s,s F u,t holds for all u < s < T, which implies that F s,t df u,s = F s,s df u,t for any fixed t < s < T and u [, t]. Hence for any bounded, στ F s -measurable random variable H τ,s we obtain E P 1{τ t} H τ,s U τ,t = EP E P 1{τ t} H τ,s U τ,t F T = E P = E P = E P [,t] [,t] H u,s F s,s 1 F s,t U u,t df u,s = E P H u,s Û u,s,t df u,s [,t] = E P 1 {τ t} H τ,s Û τ,s,t [, ] 1 {u t} H u,s U u,t df u,t H u,s F s,s 1 E P F s,t U u,t F s df u,s

15 L. Li and M. Rutkowski 15 since {τ t} {τ s} {F s,s > }, P-a.s. see part ii in Lemma 3.4. This in turn yields E P 1{τ t} U τ,t Gt = lim s t E P 1{τ t} U τ,t στ Fs = lim s t 1 {τ t} Û τ,s,t = lim s t 1 {τ t} 1 G s 1 E P F s,t U u,t F s u=τ = 1 {τ t} 1 G t 1 E P F t,t U u,t F t u=τ = 1 {τ t} Û τ,t,t where the penultimate equality holds by right continuity of the filtration F together with the right continuity of the processes G u and F u,t in u. This completes the proof of formula 17 and thus equality 15 is established for t [, T ]. For part ii, we observe that if, in addition, the random field F u,t is completely separable then formulae 18 and 19 follow easily from equations 16 and 17, respectively. Remark 3.3. Let us observe that for t =, we first obtain on the event {τ = } E P 1{τ=} H τ,s U τ,t = EP 1 {τ=} H τ,s Û τ,s,t = E P 1 {τ=} H,s Û,s,T where, on the event {τ = } {τ s} {F s > }, Û,s,T = F s 1 E P F s,t U,T F s. In the second step, we get, on the event {τ = } {F > }, where see 17 E P 1{τ=} U τ,t G = lim s 1 {τ=} Û,s,T = 1 {τ=} Û,,T Û,,T = F 1 E P F,T U,T F = E PPτ = F T U,T F Pτ = F where F,T = Pτ = F T. We conclude that 1 E P U τ,t G = 1 {τ>} Pτ > F E P {τ=} Pτ = F E P Remark 3.4. For t = T, we have that, ] [] U u,t dpτ u F T F U u,t dpτ u F T F. E P U τ,t G T = 1 {τ>t } Ũ T,T + 1 {τ T } U τ,t, 2 where formula 16 in Lemma 3.5 yields Ũ T,T = E P 1{τ>T } U τ,t F T = G T 1 Pτ > T F T T, ] U u,t df u,t. 21 Remark 3.5. Assume that the pair τ, P is such that the hypothesis H holds. Then formula 17 simplifies as follows Û u,t,t = 1 G t 1 E P F t,t U u,t F t = F t 1 E P F t,t U u,t F t = E P U u,t F t. In particular, if Ûτ,T = gτ, we have that Ûu,t,T = gu and Ũ t,t = G t 1 E P 1{τ>t} gτ F t = Gt 1 E P t, ] gu df u Ft.

16 16 Progressive Enlargements and Semimartingale Decompositions Conditional Expectations under the Extended Density Hypothesis Under the extended density hypothesis, Lemma 3.5 can be reformulated as follows. Lemma 3.6. Let U,T : R + Ω R be a B R + F T -measurable map. Let Assumption 2.7 be satisfied and the random variable U τ,t be P-integrable. i For every t [, T, we have that E P U τ,t G t = 1 {τ>t} Ũ t,t + 1 {τ t} Û τ,t,t 22 where Ũ t,t = G t 1 E P 1{τ>t} U τ,t F t = Gt 1 E P U u,t df u,t Ft and, for all u t < T, t, ] 23 Û u,t,t = m u,t 1 E P m u,t U u,t F t. 24 Proof. One only needs to modify to proof of Lemma 3.5 on the event {τ t}. For any bounded, στ F s -measurable random variable H τ,s we obtain E P 1{τ t} H τ,s U τ,t = EP E P 1{τ t} H τ,s U τ,t F T = E P = E P = E P [,t] [,t] and by taking limit This concludes the proof. H u,s m u,t U u,t dd u = E P H u,s Û u,s,t df u,t [,t] = E P 1 {τ t} H τ,s Û τ,s,t [, ] 1 {u t} H u,s U u,t df u,t H u,s m u,s 1 E P m u,t U u,t F s m u,s dd u lim E P 1 {τ t} Û τ,t Ft = limm u,s 1 E P m u,t U u,t F s u=τ s t s t = m u,t 1 E P m u,t U u,t F t u=τ. 3.4 Properties of G-Local Martingales We consider the map Û : R+ R + Ω R and we use the notation u, t, ω Ûu,tω. We say that Û is an F-optional map if it is B R + OF-measurable where OF is the F-optional σ-field in R + Ω. In that case, the map Û,t is B R + F t -measurable and the process Ût,t t R+ is F-optional, in the usual sense. We will sometimes need an additional assumption that the process Ût,t t R+ is F-predictable G-Local Martingales under Hypothesis HP We consider an arbitrary random time τ associated with G, in the sense that G is the Azéma supermartingale of τ. The following result is an important step towards a P, G-semimartingale decomposition of a P, F-local martingale, which will be established in the subsequent section. Proposition 3.1. Assume that the hypothesis HP holds and < F u,t 1 for every < u t. Let Ū be a P-integrable process such that Ū t = 1 {τ>t} Ũ t + 1 {τ t} Û τ,t, 25

17 L. Li and M. Rutkowski 17 where Ũ is an F-adapted, P-integrable process and Û is an F-optional map such that the random variable Ûτ,t is P-integrable for all t R + and the process Ût,t t R+ is F-predictable. Let the following conditions hold: i the process W t t is a P, F-local martingale where W t = ŨtG t + Û u,u df u, 26 ii for every u > and s >, the process F s,t U u,t t u s is a P, F-local martingale. Then the process Ūt t is a P, G-local martingale. Proof. We merely sketch the proof since it proceeds along the similar lines as the proofs of Propositions 5.1 and 5.6 in El Karoui et al. [15]. In particular, we skip the details regarding suitable localization and measurability arguments. In view of the following decomposition, Ū t = Ūt1 {τ>t} + Ūτ 1 {τ t} + Ūt Ūτ 1 {τ t} = Ũt1 {τ>t} + Ûτ,τ 1 {τ t} + Ûτ,t Ûτ,τ 1 {τ t}. It is sufficient to examine the following two subcases corresponding to conditions i and ii respectively: i the case of a process Ū stopped at τ, ii the case of a process Ū starting at τ with Ūτ =. Case i. Let us first assume that a G-adapted process Ū is stopped at τ so that Ū t = 1 {τ>t} Ũ t + 1 {τ t} Û τ, 27 where Ũt t R+ is an F-adapted process and Ût := Ût,t t R+ is an F-predictable process. We start by noting that, for every s < t, E P 1 {τ t} Û τ G s = E P 1 {s<τ t} Û τ G s + E P 1 {τ s} Û τ G s = 1 {τ>s} G s 1 E P Û u dh u Fs + 1 {τ s} Û τ = 1 {τ>s} G s 1 E P = 1 {τ>s} G s 1 E P s,t] s,t] s,t] Û u da u Fs + 1 {τ s} Û τ Û u df u Fs + 1 {τ s} Û τ since the dual P, F-predictable projection of H satisfies H p = A and F t = F M t + A t. Hence, for every s < t, E P Ūt G s = 1 {τ>s} G s 1 E P ŨtG t F s + 1 {τ>s} G s 1 E P Û u df u Fs + 1 {τ s} Û τ = 1 {τ>s} G s 1 E P W t W s F s + 1 {τ>s} G s 1 E P ŨsG s F s + 1 {τ s} Û τ = 1 {τ>s} Ũ s + 1 {τ s} Û τ = Ūs where we used the assumption that the process W given by 26 is a P, F-martingale. Case ii. We now consider a G-adapted process Ū starting at τ so that Ūt = 1 {τ t} Û τ,t where Û t,t =. We need to show that the equality E P 1 {τ t} Û τ,t G s = 1 {τ s} Û τ,s holds for every s < t. From part i in Lemma 3.5, we obtain, for every s < t, E P 1 {τ t} Û τ,t G s = 1 {τ>s} G s 1 E P Û u,t df u,t Fs s,t] s,t] + 1 {τ s} 1 G s 1 E P F s,t U u,t F s u=τ = I 1 + I 2.

18 18 Progressive Enlargements and Semimartingale Decompositions We first assume that s >. Recall that we assume that the hypothesis HP holds and < F u,t 1 for every < u t. Hence, for < s u t, we can write df u,t = F s,t df u,u Fs,u 1 where the process Du s := F u,u Fs,u 1 is increasing and F-adapted. Consequently, I 1 = 1 {τ>s} G s 1 E P E P F s,t Û u,t F u ddu s F s = 1 {τ>s} G s 1 E P s,t] s,t] F s,u Û u,u ddu s F s = where we first used assumption ii and next the equality Ûu,u =. It remains to examine the case s =. We denote U τ,t + = max U τ,t, and Uτ,t = max U τ,t,. Therefore, for all t >, I 1 = 1 {τ>} G 1 E P Û u,t df u,t F = 1 {τ>} G 1 E P lim 1 {s τ t} Û τ,t Ft F = 1 {τ>} G 1 E P lim E P 1 {s τ t} Û τ,t + s F t F s E P E P lim E P ɛ By the monotone convergence theorem for conditional expectations, we obtain I 1 = 1 {τ>} G 1 lim E P Û u,t df u,t F s s,t] = 1 {τ>} G 1 lim E P E P F s,t Û u,t F u ddu s F = s s,t] 1 {s τ t} Ûτ,t F t F. where we used again assumption ii and the equality Ûu,u =. Furthermore, using again assumption ii, we obtain I 2 = 1 {τ s} 1 G s 1 E P F s,t Û u,t F s u=τ = 1 {τ s} 1 G s 1 F s,s Ûu,s u=τ = 1 {τ s} Û τ,s. We conclude that 1 {τ t} Û τ,t is a P, G-martingale for t and this completes the proof of the proposition. The following corollary to Proposition 3.1 deals with the special case when the process U given by 25 is continuous at τ. It is clear that under the assumptions of Corollary 3.2 the process Ût,t t R+ is F-predictable. Corollary 3.2. Under the assumptions of Proposition 3.1 we postulate, in addition, that the equality Ũ t = Ût,t holds for every t R +. Then the process Ū is continuous at τ and condition i in Proposition 3.1 can be replaced by the following condition: i the process W t t is a P, F-local martingale where W t = ŨtG t + Ũ u df u. 28 Assume that F u,t is completely separable. Then the computations for case ii in the proof of Proposition 3.1 can be simplified. Specifically, using part ii in Lemma 3.5, we obtain, for every s < t, E P 1 {τ t} Û τ,t G s = 1 {τ>s} G s 1 E P X t Û u,t dy u Fs + 1 {τ s} X s 1 E P X t Û u,t F s u=τ = 1 {τ>s} G s 1 E P = 1 {τ>s} G s 1 E P s,t] s,t] s,t] E P W u,t F u dy u Fs + 1 {τ s} X s 1 E P W u,t F s u=τ W u,u dy u Fs + 1 {τ s} X s 1 W u,s u=τ = 1 {τ s} Û τ,s

19 L. Li and M. Rutkowski 19 where we used condition ii in the penultimate equality and the equality W u,u = X u Û u,u = in the last one. This leads to the following corollary in which the assumptions of Proposition 3.1 are maintained. Corollary 3.3. Assume that the F-conditional distribution of τ is completely separable, so that F u,t = Y u X t where Y is a positive increasing process and X a strictly positive P, F-martingale. Then condition ii in Proposition 3.1 can be replaced by the following condition: ii For every u, the process X t Û u,t t u is a P, F-martingale G-Local Martingales under the Extended Density Hypothesis In Proposition 3.1, one has to assume the F-conditional distribution F u,t is non-degenerate, since the random measure D s u := F u,u F s,u 1 is not always well defined if the F-conditional distribution is degenerate. Thus in order to remove such technicality, one can replace it with Assumption 2.7. Proposition 3.2. Suppose the extended density hypothesis holds with density m u,t and F-adapted increasing process D. Then condition ii in Proposition 3.1 can be replaced by the following condition: ii* for every u >, the process m u,t U u,t t u is a P, F-local martingale. Proof. The only adjustment needed is in case ii. Let the process Ū start at τ so that Ūt = 1 {τ t} Û τ,t, where Ût,t =. From Lemma 3.6, E P 1 {τ t} Û τ,t G s = 1 {τ>s} G s 1 E P s,t] s,t] Û u,t df u,t Fs + 1 {τ s} m u,s 1 E P m u,t U u,t F s u=τ = I 1 + I 2. We again consider the integrals separably. The first integral I 1 is given by I 1 = 1 {τ>s} G s 1 E P Û u,t df u,t Fs = 1 {τ>s} G s 1 E P m u,t Û u,t dd u Fs = 1 {τ>s} G s 1 E P s,t] E P mu,t Û u,t F u ddu Fs = where we first used assumption ii and next the equality Ûu,u =. The integral I 2 simplifies to I 2 = 1 {τ s} m u,s 1 E P m u,t Û u,t F s u=τ = 1 {τ s} m u,s 1 m u,s Ûu,s u=τ = 1 {τ s} Û τ,s. We conclude that 1 {τ t} Û τ,t is a P, G-martingale for t and this completes the proof of the proposition. s,t] 3.5 Compensator of H Our next goal is to compute the P, G-dual predictable projection of the process H t = 1 {τ t} where G is the progressive enlargement of F through τ. To this end, we will first compute the P, F-dual predictable projection of H. Assume the Doob-Meyer decomposition of G is known and is given by G t = G + M t A t. It is well known that the P, F-dual predictable projection of H coincides with the F-predictable process of finite variation A. To find the P, G-dual predictable projection that is, the G-compensator of H, it is enough to apply the following well known theorem, due to Jeulin and Yor [27] see also Guo and Zeng [17].

20 2 Progressive Enlargements and Semimartingale Decompositions Theorem 3.1 Jeulin-Yor. Let τ be a random time with the associated Azéma supermartingale G. Then the process 1 H t da s,t τ] G s is a P, G-martingale Compensator of H under Complete Separability In this subsection, we work under the assumption that the F-conditional distribution of τ under P is completely separable see Definition 2.5, that is P τ u F t = Y u X t. We assume, in addition, that the increasing process Y is P, F-predictable. It is worth noting that both assumptions are satisfied in the multiplicative construction setting provided that G t < 1 for all t >. By applying the integration by parts formula to F and using the assumption that Y is a predictable process, we obtain F t = Y t X t = Y X + Y s dx s + X s dy s. Hence, by the uniqueness of the Doob-Meyer decomposition, we conclude that da s = X s dy s. Therefore, from the Jeulin-Yor formula, we deduce that the process X s H t dy s 1 X s Y s is a P, G-martingale.,τ t] Compensator of H under the Extended Density Hypothesis In this subsection, we examine the case where the random time τ satisfies the extended density hypothesis see Definition 2.7, that is P τ u F t = m s,t dd s Proposition 3.3. Assume that the extended density hypothesis holds and the increasing process D is predictable. Then the P, F-dual predictable projection of H is given by H p t = p m u dd u. Proof. We compute the Doob-Meyer decomposition of F t = P τ t F t. From the extended density hypothesis, it is easy to see that F t,t = m,t D + m u,t m u,u dd u + m u dd u. where m u := m u,u. The process D is predictable and increasing and thus [,u] p m u dd u = p m u dd u.