Progressive enlargement of filtrations with initial times ∗

Progressive enlargement of filtrations with initial times ^∗

Monique Jeanblanc

, Yann Le Cam

Mathematic department, Evry university, France

†

Institut Europlace de Finance,

French Treasury January 4, 2009

Abstract

The preservation of the semi-martingale property in progressive enlargement of filtra- tions has been studied by many authors. Most of them focus on progressive enlargement with a honest time, allowing for semi-martingale invariance and simple decomposition for- mulas. However, times allowing for semi-martingale invariance in initial enlargements pre- serve as well this property in progressive enlargements. This paper is devoted to the related canonical decomposition of the martingales in the reference filtration as semi-martingales in the enlarged filtration. Examples are given in credit risk modelling.

1 Introduction

The stability of the class of semi-martingales with respect to filtration shrinking or enlargement of filtration has been a field of research during the last decades. In the caseF⊂G, it is known that any G-semi-martingale which is F-adapted is an F-semi-martingale (Stricker’s theorem [24]). This situation is what is called filtration shrinking. See also the recent work of P. Protter [22] for the specific case of local-martingales.

The situation of an enlargement of filtration is more complex, and the stability of the semi- martingale property does not always hold. In this framework, forF⊂G, it is usual to say that the hypothesis (H⁰) holds betweenFandGif anyF-semi-martingale is aG-semi-martingale.

In what follows, we denote by M(F) (resp. M(G)) the set of F-martingales (resp. G- martingales). We start by recalling some well-known facts about the initial and progressive enlargements of filtrations.

• The initial enlargement of a reference filtration F by a random time τ (a non-negative random variable) is the filtrationG^(τ)defined byG_t^(τ)=∩>0(Ft+∨σ(τ)).In this framework, no general theorem guarantees that the hypothesis (H⁰) holds. However, it is well known that if the conditional laws of the random time τ (with respect to the reference filtration) are ab- solutely continuous with respect to a probability measure η, then the hypothesis (H⁰) holds (then, one can reduce attention to the case where η is the law of τ). This result is known as Jacod’s theorem (see for example the paper of J. Jacod [8] or Chapter VI in the book of P.

Protter [21]). Random times satisfying this property will be referred to as initial times in the sequel of this article (see Section 2).

∗This research benefited from the support of the Chaire Risque de Cr´edit, F´ed´eration Bancaire Fran¸caise.

•The progressive enlargement of a reference filtrationFby a random timeτ is the smallest right-continuous filtration that contains F and makes τ a stopping time. This filtration G is defined byGt=∩>0G_t+⁰ whereG_t⁰=Ft∨σ(τ∧t). Remark that, for fixedt, theσ-algebraG_t⁰ is generated by the set of random variables of the formFth(t∧τ),withha Borel function, and FtanFt-measurable random variable. It follows that the filtrationGcoincides withFbeforeτ and withG^(τ) after.

The study of the hypothesis (H⁰) in progressive enlargements can be split in two different time intervals, before and after the occurrence ofτ:

• On the set{t < τ}, the hypothesis (H⁰) always holds: precisely for anyF-martingaleX, the stopped processX^τ is aG-semi-martingale (a nice and short argument of M. Yor may be found in [26]).

Moreover, the canonical decomposition of theG-semi-martingaleX^τ writes X_t^τ =µt+

Z _t∧τ

0

dhX, Mi_s+dBs

Gs− , withµ∈ M(G)

where G_t :=P(τ > t|F_t) is the conditional survival process¹ (also called the Az´ema F- super-martingale), andM denotes the martingale part of the Doob-Meyer decomposition² of the super-martingaleG. The process B is the F-predictable dual projection of theG -adapted process (εu)_u=(Hu∆Xτ)_u, whereHt= 11τ≤t. A proof of this decomposition can be found in the books of T. Jeulin [11] and [15] or in the papers of T. Jeulin and M. Yor [13], [12] or [14].

If the random time τ avoids the F-stopping times, that is if P(τ =T) = 0 for any F- stopping timeT (this assumption is often referred to as (A)), then ∆Xτ= 0 and B= 0.

Under this condition, a proof of the above decomposition can be found in Chapter VI of the book of P. Protter [21].

• For the general case of non-stopped semi-martingales, semi-martingale invariance deeply depends on the properties of the random time. A natural extension of the proof leading to the last result - based on the structure of theG-predictableσ-fieldP(G) and its links with P(F) - lies on the study of times allowing any G-predictable processK to be written as K =K¹1_[0,τ]+K²1_]τ,∞[, where K¹, K²

are F-predictable processes (see M.T. Barlow [1], M. Yor [26], T. Jeulin [11] or C. Dellacherie and P-A. Meyer [4]). Such times are called honest times³: precisely the time τ is said “honest”if, for anyt >0, it is equal to anFt-measurable random variable on{τ < t}.

If the time is honest, the sequence ofσ-algebras

Gt={A∈ F,∀t,∃At, Bt∈ Ft, A= (At∩ {τ > t})∪(Bt∩ {τ ≤t})}

is increasing (by honesty of the time) and forms a filtration. In that framework, the hypothesis (H⁰) holds, and ifX ∈ M(F) :

Xt=µt+ Z _t∧τ

0

dhX, Mi_s+dBs

G_s− −1_{τ≤t}

Z _t

τ

dhX, Mi_s+dBs

1−G_s− , withµ∈ M(G).

1It is well known (see Jeulin [11]) thatGdoes not reach zero beforeτ. Indeed, ifT := inf{t >0, Gt= 0 or Gt−= 0},Gtis null afterT (a non-negative super-martingale sticks at zero) andP(T < τ) =E(GT1{T <∞}) = 0

2Remark thatM ∈BM O(cf. discussions in the next section). The spaceBM Ois defined as the subspace of H² composed of the local martingales N such thatE((N∞−NT−)²|FT)≤kfor any F-stopping timeT. kNk²_{BM O}denotes the smallestkif it exists (i.e.,N∈BM O) or∞.

3Honest times coincide with the end ofF-predictable set in [0,∞]×Ω, and finite honest times coincide with the end ofF-optional sets (non-finite honest times may be not the end of anF-optional set, even ifF∞-measurable, see [11]).

The theory of progressive enlargement with an honest time presents nonetheless some major drawbacks in some application fields, such as credit derivatives modelling. Within the approach based on the enlargement of a reference filtration by the progressive knowledge of a credit event (see R. Elliott, M. Jeanblanc and M. Yor in [6] or M. Jeanblanc and Y. Le Cam in [10]), the hypothesis (H⁰) is fundamental. Indeed, the absence of arbitrage in finance is closely linked to the property of semi-martingale satisfied by the assets (see F. Delbaen and W. Schachermayer in [3]), and it is necessary that the assets of the reference market (i.e.,F-semi-martingales) remain G-semi-martingales.

The most important argument which makes impossible the application of the “honest”theory in credit modelling is the belonging of the honest time to F∞: unfortunately the credit event (a change in the ranking of an obligation or an unpaid coupon for example) can neither be directly read on the market price of the asset of the reference filtration nor on its future and modelling it through an F∞-measurable random variable is not consistent with reality. The widespread model of Cox construction of the credit event - in which τ = inf{t: Λt≥Θ} with Λ anF-adapted increasing process and Θ a random variable independent of F∞ - strengthens this point (see Section 5).

The main goal in this paper is to present the progressive enlargement of a reference filtration F with an “initial”time τ, focusing on the canonical decomposition of the semi-martingale in the new filtration. The paper is organized as follows. The first section presents the definition of initial times and the features that make them a natural tool for the progressive enlargement of filtration in many applications. In the second section, we prove that ifX is anF-martingale, thenX is aG-semi-martingale with canonical decomposition

Xt=µt+ Z _t∧τ

0

dhX, Gi_u+dBu

Gu− −

Z _t

t∧τ

d X, α^θ

u

α^θ_u−

θ=τ

, withµ∈ M(G), (1) whereα^θ is the density of the conditional laws ofτ with respect to the law ofτ, defined below by (2). We generalize our study to the case of enlargement with multiple times in Section 4.

Section 5 gives examples of initial times that can be used in credit modelling, linked to Cox construction. Because of the important applications in this field, the random time τ will be called default time or credit event.

We consider a filtered probability space (Ω,F,F,P),where the filtrationFsatisfies the usual conditions (F0contains the null sets ofPandFis right continuous: ∀t≥0,Ft=Ft+:=∩s>tFs).

We do not assume thatF =F∞. Recall that under this condition, (i) everyF-martingale admits a c`adl`ag version, and

(ii) theF-predictable projection of any martingale (Mt, t≥0) is (Mt−, t≥0)⁴. We denote byP(F) the predictableσ-algebra onR⁺×Ω.

2 Initial Times

As recalled in the introduction, the notion of initial times has been introduced by J. Jacod in [8] who proved that for initial enlargement with “initial” times, the hypothesis (H⁰) holds. We chose the nameinitial time as a reference to this property.

4Recall that theF-predictable projection of a bounded measurable processX (not necessarilyF-adapted) is theF-predictable processX^pthat satisfiesX_T^p=E(XT|FT−) on the set{T <∞}, for anyF-predictable time T.

It will be useful to introduce the notation Ω =b R⁺ ×Ω and Fbt for the right-continuous completion ofB(R⁺)⊗ Ft.TheF-optionalb σ-field O(bF) (resp. theF-predictableb σ-field P(F))b will therefore refer to theσ-field onR⁺×Ω generated by the c`ad (resp. c`ag)b F-adapted processes.b Recall that⁵:

P(F) =b B R⁺

⊗ P(F).

For any positive random timeτ,and for anyt, we writeQt(ω, dT) the regularFt-conditional distribution ofτ (that exists since the random variableτ is real-valued), and

G^T_t (ω) :=P(τ > T|Ft) (ω) =Qt(ω,]T,∞[).

Definition 2.1 (Initial times) A positive random time τ is called an initial time if there exists a measure η on B(R⁺)such that a.s. for eacht≥0, Qt(ω, dT)η(dT).

The density processes. By Doob’s theorem of disintegration of measures, the definition of an initial time is equivalent to the existence of a family of positiveF-adapted processes (α^u_t, t≥0), such that:

• for anyt≥0, one hasα^u_t (ω)η(du) =Qt(ω, du), i.e., G^T_t =

Z _∞

T

α^u_t η(du), (2)

• for anyt≥0 the mapping (u, ω)7→α^u_t(ω) isB(R⁺)⊗ Ftmeasurable.

Times satisfying (2) have been introduced to model the credit events by Y. Jiao [16]. In [5], the authors have studied these time from a ”shrinkage” point of view: characterize the G-martingales in terms ofF-martingales.

Existence of a “good version”of the density. The existence of a good version of theprocesses (α^u_t, t≥0) derives from the analysis developed in C. Striker and M. Yor in [25], and is carried out in [8]. These authors establish the existence of anF-optional map (u, ω, t)b 7−→α^u_t (ω) such that:

• it is c`adl`ag int(i.e., for almost any (u, ω), t7−→α^u_t(ω) is c `adl`ag),

• for anyu≥0, α^u is anF-martingale⁶.

Remark 2.1 We will consider this version of the density function from now on. From the martingale property for eachu≥0, α^u “sticks to zero”: ifT^u= inf{t≥0, α^u_t−= 0orα^u_t = 0}, α^u>0andα^u₋>0 on[0, T^u[ andα^u= 0 on[T^u,∞[.

Applying the second point of the following Lemma 3.1 to the F-predictable processb K_t^u = 1_{T^u_<t},

P(T^τ < t) =E(K_t^τ) =E Z _∞

0

α^u_t−1{T^u<t}η(du)

= 0.

It follows that the random variable T^τ is almost surely infinite.

5Remark that the optional σ-field generated by the c`ad augmentation of a filtration is in general strictly bigger than the optionalσ-field generated by the filtration. In the opposite, the predictableσ-fields are the same. AsFbt⊃ B(R⁺)⊗ Ft,it follows thatO(bF)⊃ B(R⁺)⊗ O(F) andP(bF) =B(R⁺)⊗ P(F).

6For anyt∈Q⁺,there existsα^u_t (ω),B R⁺

⊗ Ft-measurable, such thatα^u_t(ω)η(du) =Qt(ω, du).Fors≤t, s∈Q⁺,there existsα^u_s(ω) with the same properties. By definition,{(u, ω) :α^u_s6=E(α^u_t|Fs)}hasη⊗P-measure 0. By Fubini,∀u≥0,(α^ur)_r∈Q+ is anF-martingale. The construction of the c`adl`ag versionα^uis classical and derives from the martingale property ofα^uand the right-continuity of the filtrationF.

Note that, iff(u) :=E(α^u_t) =α^u₀, then P(τ∈du) =f(u)η(du). We shall consider, without loss of generality, the case wheref(u) = 1.

Doob-Meyer decomposition of the survival process. In such a framework, we can write the survival processGt:=G^t_tas

Gt=P(τ > t|Ft) = Z _∞

t

α^u_tη(du) = Z _∞

0

α^u_u∧tη(du)− Z _t

0

α^u_uη(du) :=Mt−Aet. We denote byAtheF-predictable increasing process: At=R_t

0α^u_u−η(du).We have the

Proposition 2.1 M is an F-martingale and A is the compensator of G. If η has no atoms, At=Aet.

Proof. Let 0≤t≤T.From the positivity of the martingale densities and Fubini theorem:

E(MT|Ft) =E Z _∞

0

α^u_u∧Tη(du) Ft

= Z _∞

0

E(α^u_u∧T| Ft)η(du) = Z _∞

0

α^u_u∧tη(du) =Mt, where the third equality comes from the fact that for anyu≥0,E(α^u_u∧T| Ft) =α^u_u∧tP-a.s. It follows that M is anF-martingale.

The processAeisF-adapted and increasing (from the positivity of the densities). If η does not have any atoms, Aeis continuous henceF-predictable.

Ifη has atoms,Aemay be not predictable, for example if the process u7−→α^u_u jumps at an atom t ofη and if the size of the jump is not Ft−-measurable⁷. In such a case, it is necessary to compensate the finite variation processA. Sincee ηis deterministic, for proving thatAis the compensator ofA, it is enough to check that for any non-negativee F-predictable processK, one hasE(K.A) =e E(K.A).From the positivity of the processes, we have:

E

K.Ae

=E Z _∞

0

Kuα_u^uη(du)

= Z _∞

0

E(Kuα^u_u)η(du).

For anyu≥0,sinceK∈ P(F), one hasKu∈ Fu−, hence

E(Kuα^u_u) =E(KuE(α^u_u|Fu−)) =E Kuα^u_u−

,

(where we have used that,α^u being a martingale,E(α^u_t|Ft−) =α^u_t− for any t≥0).

Remark thatA is also theF-predictable dual projection ofHt= 1τ≤t.

Quadratic variations. Once the choice of the good version of the density processes has been made, it is possible to study the measurability of the quadratic covariation of α with some F-martingales.

LetX be a local martingale. For any u,the covariance process ([α^u, X]t, t≥0) is c`adl`ag.

Lett≥0 andTn be a partition of [0, t]. For anyu≥0, n≥0,the Riemann sum Sⁿ(u, ω) =X

Tn

α^u_ti+1(ω)−α^u_ti(ω)

Xti+1(ω)−Xti(ω)

7Recall that a process isF-predictable if and only if it jumps atF-predictable times and its jump at any F-stopping timeT belongs toFT−.

isB(R⁺)⊗ Ft -measurable. As for anyu≥0Sⁿ(u, .) converges in probability when the mesh of the partition goes to zero (existence of the bracket), there exists a version [α^u, X]t(ω) which is B(R⁺)⊗ Ft-measurable (see the first proposition of [25] for a simple example of explicit construction). It follows that (u, t, ω)7→[α^u, X]t(ω) isF-optional.b

This F-optional c`adl`ag process has paths of finite variation, hence itsb bF-predictable com- pensatorhα^u, Xiexists as soon as its variation is locally integrable. In such a case, (u, t, ω)7→

hα^u, Xi_t(ω) isbF-predictable. Different cases may be considered:

• If the local martingaleX islocally in BM O,for anyu≥0hα^u, Xiexists with no condition onα^u,from Fefferman’s inequality. Indeed any local martingaleα^uis locally in the space⁸ H¹,hence there exists a constantksuch that

E Z _∞

0

d

h

X^Tn,(α^u)^Tn i

s

≤kk(α^u)^TnkH¹kX^TnkBM O, and [X, α^u] is locally of integrable variation;

• If the local martingaleX is locally bounded, the semi-martingaleXα^u is special⁹ for any u≥0, hencehα^u, Xiexists with no condition on α^u;

• If the local martingale X is locally square integrable, the angle bracket hα^u, Xi exists if α^u islocally square integrable;

• For a local martingaleX withno regularity condition, the angle brackethX, α^uiexists as soon as for anyu≥0, α^u is locally bounded.

Whereas these properties are quite general and do not depend onX, J. Jacod proved in [8]

the following very interesting (and complex) result, central in the analysis:

Proposition 2.2 (Jacod, 1978) If a local martingale X is given, there exists a subset of R⁺:R⁺_X,satisfying η R⁺_X

= 1 such that for anyu∈R⁺_X,hα^u, Xiis defined on {α^u₋>0}.

This fundamental result derives from the following property: On each increasing stochastic interval [0, T_n^u] withT_n^u = inf{t ≥ 0, α^u_t− ≤ 1/n}, the stopped process [α^u, X]^Tnu has locally integrable variation and its compensator hα^u, Xi^Tnu is defined. As{t : α^u_t− >0} =∪_n[0, T_n^u], hα^u, Xiis defined by embedding. Beware that two very different cases may happen:

1. α^u jumps to zero: In this case the sequence T_n^u becomes constant, equal to T^u (and {t : α_t−^u >0}=∪n[0, T_n^u] = [0, T^u]).

2. α^u reaches zero continuously: In this case the sequence T_n^u increases strictly toT^u (and {t : α_t−^u >0}=∪n[0, T_n^u] = [0, T^u[).

8For any integerp,the spaceH^pis defined as the set of the local martingalesNsuch thatkNkH^p<∞,with kNk^p_Hp=E[N, N]^p/2∞

9Indeed, ifTnis a sequence that bounds the local martingaleX by a sequencexn,and ifYⁿ=X^Tnα^u, sup

s∈[0,t]

|Y_sⁿ| ≤ sup

s∈[0,t]

|X_s^Tn| sup

s∈[0,t]

|α^u_s| ≤xn sup

s∈[0,t]

|α^u_s| ∈L¹_loc sinceα^uis a local martingale. It follows thatYⁿ, henceXα^uis special.

Let α^u be (the good version of) the family of density, with no added assumption on the regularity of the paths. The martingale part of the supermartingale G is in BM O, hence hM, Xiexists for anyX local martingale. As the processA is predictable with paths of finite variation, [X, A] is a local martingale, hencehA, Xi= 0.It follows, fromG=M+A, thathX, Gi exists for any local martingale X and hX, Gi=hX, Mi. When the angle bracket between X andαexists (see discussion above), we have:

hX, Gi_t = hX, Mi_t= Z _∞

0

hX, α^u_u∧.−∆α^u_u1u≤.i_tη(du)

= Z _∞

0

(hX, α^ui_u∧t−∆hX, α^ui_u1u≤t)η(du). (3)

Initial times have the very interesting feature (for credit modelling for example) to allow the existence of a probability under which the reference filtration and the random time are independent. Precisely, as proved in A. Grorud and M. Pontier [7] we have the:

Proposition 2.3 If τ is an initial time with EP(1/α^τ_T)< ∞,∀T there exists a probability Q equivalent to Punder which τ andF∞ are independent.

This result (obtained by choosing, fort < T, dQ/dP|_Gt =EP(1/α^τ_T|Gt)/EP(1/α^τ_T)) leads to a straightforward proof of Jacod’s theorem when the integrability assumption holds, since the hypothesis (H⁰) is stable by a change of equivalent probability.

UnderQ, immersion property holds, i.e., any (F,Q)-martingale remains a (G,Q)-martingale.

There exists - at our knowledge - no such result in a “honest”expansion. Characterizations of immersion itself are also very tractable in the framework of initial times, as it will be seen in the following Corollary 3.1.

3 Invariance of Semi-martingales

Letτbe an initial time. From Jacod’s theorem, we know that the hypothesis (H⁰) holds within the initial expansionF⊂G^(τ).IfGdenotes the progressive expansion ofFbyτ,F⊂G⊂G^(τ) and Stricker’s theorem ensures that the hypothesis (H⁰) holds betweenF andG.

We now present the theG-semi-martingale decomposition of anF-martingaleX.

It has been proved in the previous section that the compensator ofGwritesAt=R_t

0α^u_u−η(du), and that it is also theF-predictable dual projection ofHt= 1τ≤t.We start with a simple lemma that will be central in the following proofs.

Lemma 3.1 Let (K_t^u)_t≥0 be a measurably indexed family of F-predictable non-negative (or bounded) processes, i.e., such that the map (ω, t, u) → K_t^u(ω) is P(F)⊗ B(R⁺)-measurable (equivalently, with the notation on the product space, K∈ P(F)). Then:b

1. TheF-optional projection of the process t7→K_t^τ is the processt7→R_∞

0 K_t^uα^u_t η(du) ; 2. TheF-predictable projection of the process t7→K_t^τ is the process t7→R_∞

0 K_t^uα^u_t−η(du).

Proof. In the proof, we shall use, as a shortcut, t 7→ α^u_t ∈ O(F) for (u, ω, t)b 7→ α^u_t(ω) is O(F)-measurable, and similar notation forb P(F).

By the monotone class theorem, it is enough to prove properties 1. and 2. for K_t^u = k(u)KtwhereKisF-predictable and non-negative andk≥0 is a non-negative Borel function.

As t 7→ α_t^u belongs to O(bF) (resp. t 7→ α^u_t− ∈ P(F)), Fubini’s theorem implies thatb t 7→

R_∞

0 k(u)α^u_t η(du)∈ O(F) (resp. t 7→R_∞

0 k(u)α^u_t−η(du)∈ P(F)). Since K is predictable, it follows that

t7→

Z _∞

0

K_t^uα^u_t η(du)∈ O(F) (resp. t7→

Z _∞

0

K_t^uα^u_t−η(du)∈ P(F) ).

Moreover,t7→R_∞

0 k(u)α^u_t η(du) is the c`ad version of anF-martingale P−a.s., ∀t≥0, E(k(τ)| Ft) =

Z _∞

0

k(u)α^u_tη(du). (4) It follows that:

•For any finiteF-stopping timeT,we have from (4):

E(K_T^τ| FT) =KTE(k(τ)| FT) =KT

Z _∞

0

k(u)α^u_Tη(du) = Z _∞

0

K_T^uα^u_Tη(du) P−a.s., hence, the process (R_∞

0 K_t^uα^u_tη(du), t≥0) is theF-optional projection ofK^τ.

•For any finiteF-predictable timeT,and increasing sequence of stopping timesTn↑T, we have from (4):

E(k(τ)| FTn) = Z _∞

0

k(u)α_T^u_nη(du) P−a.s., and letting ntend to∞:

E(k(τ)| FT−) = Z _∞

0

k(u)α^u_T−η(du) P−a.s.

and from KT ∈ FT−:

E(K_T^τ| FT−) = KTE(k(τ)| FT−) =KT

Z _∞

0

k(u)α_T^u₋η(du)

= Z _∞

0

K_T^uα^u_Tη(du) P−a.s., hence, the process (R_∞

0 K_t^uα^u_t−η(du), t≥0) is theF-predictable projection ofK^τ. Remark 3.1 The first point remains valid if K is O(F)⊗ B(R⁺)-measurable. If it is O(bF) measurable, the scheme of the proof does not hold anymore (see the first footnote of the last section).

Remark 3.2 Note that, ifτ avoids theF-stopping times and if immersion property holds, then, for any F-predictable (bounded) processX

E(Xτ11τ >t| Ft) = Z _∞

t

Xuα^u_tη(du).

Indeed, for any boundedFt-measurable random variableFt, and any boundedF-predictable pro- cess X :

E(Xτ11τ >tFt) = E Z _∞

t

XuFtdHu

=E Z _∞

t

XuFtdAu

= E

Z _∞

t

Xuα^u_uη(du)Ft

=E Z _∞

t

Xuα^u_t η(du)Ft

,

where the last equality comes from the characterization of immersion presented below in Corol- lary 3.1.

LetX be aF-local martingale. We shall prove that there exist (i)J ∈ P(F) with finite variation and

(ii)K= (Ku(θ), u≥0)∈ P(F) such that for anyb θ≥0,the paths of the processK(θ) have finite variations, that satisfy:

Yt=Xt− Z _t∧τ

0

dJu− Z _t

t∧τ

dKu(θ)

θ=τ

is a G-martingale.

Remark that both integrals are Stieljes integrals. For anyω and θ≥0, the processt 7−→

R_t

0dKu(θ) isF-predictable with finite variation. It follows that it isG-predictable and that t7−→

Z _t

0

dKu(θ)

θ=τ∧t

isG-predictable with finite variation¹⁰.

Remark also that ifK, H∈ P(bF),we have for anyt≥0, Z _t

0

Hu(θ)dKu(θ)

θ=τ

= Z _t

0

Hu(τ)dKu(τ),P−a.s.

Such a result is clear for Ku(θ) = k(θ)Ku, Hu(θ) =h(θ)Hu and derives from the monotone class theorem.

Before stating and proving the main result of this article, we start with two remarks about the implications of such a decomposition:

1.Before default:

If such a representation holds, it is necessary that dJ_u=dhX, Gi_u+dBu

Gu−

(Recall that the F-predictable process B refers to the dualF-predictable projection of theG- adapted process εu= ∆XτHu).

Indeed from Y ∈ M(G) and since τ is a G-stopping time, the stopped process Y^τ must be a G-martingale (by the optional sampling theorem), hence X_t^τ−R_t∧τ

0 dJu ∈ M(G). The result follows from Jeulin’s formula and uniqueness of the canonical decomposition of a special semi-martingale.

10If u 7−→ k(u) is c`ag, and if for any ω, t 7−→ Kt is c`ag Ft-adapted, it is Gt-adapted and for any ω, t7−→k(τ∧t)Ktis c`ag andGt-adapted.

2.After default:

Letsbe fixed, Fsbe a bounded non-negative Fs -measurable random variable andhbe a bounded non-negative Borel function. Then the random variable Fsh(τ)1τ≤sisGs-measurable and if the above decomposition holds, the martingale property ofY implies that

E(Fsh(τ)1τ≤s(Yt−Ys)) = 0, hence that:

E(Fsh(τ)1τ≤s(Xt−Xs)) =E

Fsh(τ)1τ≤s

Z _t

s

dKu(θ)

θ=τ

. (5)

• We can write:

E(Fsh(τ)1τ≤s(Xt−Xs)) = E

Fs(Xt−Xs) Z _s

0

h(θ)α^θ_tη(dθ)

= Z _s

0

h(θ)E Fs Xtα_t^θ−Xsα^θ_s η(dθ)

where the first equality comes from a conditioning w.r.t. Ft and the second from the martingale property ofα^θ for any θ≥0.For anyθ≥0, the integration by parts formula implies

Xtα^θ_t−Xsα^θ_s= Z _t

s

Xu−dα^θ_u+ Z _t

s

α^θ_u−dXu+ Z _t

s

d X, α^θ

u P−a.s.

and, since the two first integrals areF-martingales (X andα^θ areF-martingales):

E Fs Xtα^θ_t−Xsα^θ_s

=E

Fs

Z _t

s

d X, α^θ

u

. When the angle bracket always exists (see discussion above), we conclude

E

Fs

Z _t

s

d X, α^θ

u

=E

Fs

Z _t

s

d X, α^θ

u

,

but in the particular case where X is only a martingale (with no added condition), a special care must be brought. Asα^θ = 0 on [T^θ,+∞[,

X, α^θ

u = X, α^θ

u∧T^θ, and it follows that:

E

Fs

Z _t

s

d X, α^θ

u

=E

Fs

Z _t

s

1_{u≤Tθ}d X, α^θ

u

.

Depending on the wayα^θ reaches zero, the set{u≤T^θ}may be decomposed in:

{u≤T^θ}={α^θ_u−>0} ∪ {u=T^θ,∆α^θ_u= 0}, hence

E

Fs

Z _t

s

1_{u≤T^θ_}d X, α^θ

u

= E

Fs

Z _t

s

1_{α^θ

u−>0}d X, α^θ

u

+E

Fs

Z _t

s

1_{∆α^θ_u=0,u=T^θ_}d X, α^θ

u

.

From the definition of X, α^θ

on each [0, T_n^θ],

X, α^θ_Tn^θ

u −

X, α^θ_Tn^θ

u ∈ M(F),and E

Fs

Z _t

s

1_{u≤Tn^θ_}d X, α^θ

u

=E

Fs

Z _t

s

1_{u≤Tn^θ_}d X, α^θ

u

and since a.s. 1_{u≤Tn^θ_}↑1_{u≤T^θ_}≤1,Lebesgue’s theorem implies:

E

Fs

Z _t

s

1_{α^θ

u−>0}d X, α^θ

u

=E

Fs

Z _t

s

1_{α^θ

u−>0}d X, α^θ

u

. Moreover,

Z _t

s

1_{∆αθ

u=0,u=T^θ}d X, α^θ

u = 1_{s≤Tθ≤t,∆α^θ

T θ=0}∆ X, α^θ

T^θ

= 1_{s≤Tθ≤t,∆α^θ

T θ=0}∆X_Tθ∆α^θ_Tθ= 0, hence:

E

Fs

Z _t

s

d X, α^θ

u

=E

Fs

Z _t

s

1_{αθ u−>0}d

X, α^θ

u

.

It follows¹¹ that (and the indicator function may be removed when the bracket always exists):

E(Fsh(τ)1τ≤s(Xt−Xs)) = Z _s

0

h(θ)E

Fs

Z _t

s

1_{α^θ

v−>0}d X, α^θ

v

η(dθ). (6)

• For the right hand member:

E

Fsh(τ)1τ≤s

Z _t

s

dKv(τ)

= E(Fsh(τ)1τ≤sKt(τ))−E(Fsh(τ)1τ≤sKs(τ))

= E

Fs

Z _s

0

h(θ) Kt(θ)α^θ_t−Ks(θ)α^θ_s η(dθ)

by an application of Lemma 3.1 to the F-predictable processes indexed by u : t 7→

h(u)1u≤sKt(u) (s≤t) and t7→h(u)1u≤tKt(u) (we use thatFs is Ft -measurable). For anyθ≥0, using integration by parts formula,

Kt(θ)α^θ_t−Ks(θ)α^θ_s− Z _t

s

α^θ_u−dK_u^θ∈ M(F),

sinceα^θis a martingale andK(θ) isF-predictable. It follows, from Fubini’s theorem, that:

E

Fsh(τ)1τ≤s

Z _t

s

dKv(τ)

= Z _s

0

h(θ)E Fs Kt(θ)α^θ_t−Ks(θ)α^θ_s η(dθ)

= Z _s

0

h(θ)E

F_s Z _t

s

α^θ_u−dK_u^θ

η(dθ). (7)

11The measurability of the functionθ7−→E(FsR_t

sdhX, α^θiu) is insured by Fubini’s theorem, since for anyθand almost anyω,(θ, ω)7−→Fs(hX, α^θit− hX, α^θis) is measurable. Existence of integrals likeR_s

0

R_t

sdhX, α^θivη(dθ) is insured by the existence of a measurable version for anyωof (θ, v)7→ hX, α^θiv.As pointed out by an Associate Editor of the journal, the question of the null sets associated to eachθis more tricky for Ito’s integrals (the interested reader may refer to the work of A. Sznitman [23], where the question is addressed for non-finite variation integrals).