Random times with given survival probability and their F-martingale decomposition formula∗

Random times with given survival probability and their F-martingale decomposition formula

Monique Jeanblanc,^† Shiqi Song^‡ 22 juin 2010

Abstract The problem concerns credit risk modelling. We are given a filtered probability space(Ω,F,P), whereF= (F_t)_t≥0 is a filtration, anF-adapted continuous increasing processΛ and a positiveP-Flocal martingaleNsuch thatZ_t:=N_te^−Λt ≤1, t≥0, and we try to construct model of default time, i.e., a probability measure Q and a random time τ on an extension of (Ω,F,P), such that the survival probability satisfies Q[τ > t|F_t] = Z_t, t ≥ 0. In this paper, we show that there can exist various differents models with a same survival probability and that the increasing family of martingales, combined with the stochastic differential equation, constitutes a natural way to construct these models. Our models will be equipped with an enlarged filtration G = (G_t)_t≥0 with G_t = F_t∨σ(τ ∧t). We show that all P-F martingales remainsG-semimartingale and give an explicite semimartingale decomposition formula. At the end we show how this decomposition formula is intimately linked with stochastic differential equation.

1 Introduction

1.1 Bref description

Our research was motivated by credit risk modelling : for so-called intensity based models, the starting point is a given filtration F = (F_t)_t≥0 and an F-adapted increasing continuous process Λ. Then, one constructs a random time τ, called the default time, such that (H_t− Λ_t∧τ, t ≥ 0) is a G-martingale, where H_t = 11_τ≤t, and G = (G_t)_t≥0 with G_t = F_t∨σ(τ ∧t).

The "canonical" construction is the Cox model, whereτ := inf{t : Λ_t≥Θ}, whereΘis a r.v.

independent ofF_∞, with unit exponential law. In Cox model, the survival probability is given byQ[τ > t|F_t] =e^−Λt.

Our aim here is to take into account a second parameter N, a positive local F-martingale, and to consider models with a survival probability of the formN_te^−Λt, t≥0(the multiplicative decomposition of the supermartingaleQ[τ > t|F_t]), which is a more general form that a survival probability may have than that one appearing in the Cox model. We ask then two questions.

The first one is whether there exist a probability Q and a random time τ which solve the

∗This work has benefited from financial support by Fédération Bancaire Française.

†Département de Mathématiques, Université d’Evry and Institut Europlace de Finance ; email : monique.jeanblanc@univ-evry.fr

‡Département de Mathématiques, Université d’Evry

equation :

Q[τ > t|F_t] =N_te^−Λt, t≥0.

Note that, if the solution (Q, τ) exists, the process (H_t−Λ_t∧τ, t ≥0) is a localG-martingale.

This model existence problem has been studied and solved (under adequate conditions) in Gapeev et al. [2] and Jeanblanc and Song [4] where for each given survival probabilityN e^−Λ, one particular model has been constructed (see below, Theorem 3.1). In this paper we consider the set of all solution models. We will prove that there may exist various different models with a same conditional survival probability.

Our second question concerns the impact of the factorN on our models. We know that, for Cox model, theP-Fmartingales remainsQ-Gmartingales, the well-known immersion property.

This property, however, is not preserved in general. A study has been initiated in [4] where it is proved that, for their model, aP-F martingaleX is a Q-G semimartingale whose drift is written out as a function of theP-Fpredictable brackethN, Xi, just as in the classical example of the progressive enlargement of filtration by a honest time (see Barlow, Jeulin and Yor [1, 5]).

(Note, however, that the default time in [4] is not a F-honest time.) In particular the model satisfies the (H’) property of enlargement of filtration. In this paper we continue that study for more general models. We will prove again the (H’) property and provide an explicite Q-G semimartingale decomposition formula for theP-Fmartingales. We notice that, in general, the semimartingale decomposition formula can depend on factors other thanhN, Xi. We get a new type formulae.

The paper is organized as follows. We give below the precise definition of our problem. In Section 2 we introduce the notion of iM_Z and show the equivalent relation between the iM_Z and the solutions of our problem. In section 3 we explain how to construct iM_Z. The stochastic differential equation is the key point in these constrcutions. In section 4 we prove the (H’) property and establish the semimartingale decomposition formula. Once again the results are obtained by an adequate use of the stochastic differential equation which defines the iM_Z. We show in addition that any solution of our problem possessing the same semimartingale decomposition formula must satisfy the same stochastic differential equation.

1.2 The problem and conventions

Now let us formulate precisely the problem. We begin with the notion of extension. Let (Ω,A,F,P) and ( ˆΩ,A,ˆ ˆF,Q) be two filtered probability spaces, where F = (F_t)_t≥0 (resp. Fˆ = ( ˆF_t)_t≥0) is a filtration of sub-σ-fields inA(resp. inA). Letˆ πbe a measurable map fromΩˆ into Ω. We say that ( ˆΩ,A,ˆ F,ˆ Q, π) is an extension of (Ω,F,P), if Fˆ_t =π⁻¹(F_t) for all 0≤t≤ ∞, andP= Q|_Fˆ∞◦π⁻¹. Note that if the extension holds, we shall simply identifyF,PwithˆF,Q|_Fˆ∞ and equally we shall consider a random variable Y on (Ω,F_∞) as the random variable Y ◦π on ( ˆΩ,A). With this identification a process onˆ Ω is a P-F-martingale if and only if it is a Q-F-martingale.ˆ

Notice that in this paper callingaa positive number means thata≥0and callingf(t), t∈R, an increasing function meansf(s)≤f(t) for s≤t.

Throughout this paper, we fix (Ω,A,F,P) a filtered probability space satisfying the usual conditions. Let Λ be an F-adapted continuous increasing real process with Λ₀ = 0 and N a càdlàg positive P-Flocal martingale. We postulate that 0≤N_te^−Λt ≤1for all 0≤t <∞.

The problem we consider is :

Problem-A.Construct an extension( ˆΩ,A,ˆ F,ˆ Q, π)of(Ω,F,P)and a random timeτ on( ˆΩ,A)ˆ

such thatQ[τ > t|F_t] =N_te^−Λt for all0≤t <∞.

Note that in the above expression we identifyFˆ withF and we consider the random variables on Ωas random variable onΩ. Normally a solution of the problem should be indicated by anˆ expression ( ˆΩ,A,ˆ F,ˆ Q, π, τ). But we shall also write a solution simply by ( ˆΩ,Q, τ) or even by Qalone, if no confusion is possible.

Our solution to the problem-A will be constructed on the product space [0,∞]×Ω. This space will exclusively be equipped with the product σ-field B[0,∞]⊗ F_∞, with the map π : π(s, ω) =ω and the mapτ :τ(s, ω) =s, with the filtrationFˆ=π⁻¹(F)which will immediately be identified withF. In such a setting, for([0,∞]×Ω,B[0,∞]⊗ F_∞,F,ˆ Q, π)to be an extension of (Ω,F,P), or for ([0,∞]×Ω,Q, τ) to be a solution of the problem-A, the only element we have to determine is the probabilityQ. Therefore, in these cases, we shall simply say thatQis an extension, or thatQis a solution.

When( ˆΩ,A,ˆ F,ˆ Q, π, τ)is a solution of the problem-A, we equip systematically the spaceΩˆ with the progressively enlarged filtration G = (G_t)_t≥0 made from F with the random time τ, i.e.G_t=F_t∨σ(τ ∧t), t≥0. This fix the framework for our discussion on the enlargement of filtration problem.

We shall notice that, for our solution of the problem-A, the filtration Fˆ on the product space[0,∞]×Ωis not necessarily complete. In fact, we shall not really need this completeness, because the completeness will be only used to deal with limits calculus or stopping times. But, if the only limits are the limits of theF_∞measurable random variables or if the stopping times areFstopping times, they can be worked on the initial space(Ω,F,P)which satisfies the usual condition.

In this paper, if a relation between two random variables is written without mention, it is understood to be an almost sure relation with respect to the underground probability. However, sometimes, it is important to distinguish the almost sure relation from the true one. It will be then specially mentioned.

2 Increasing family of positive and bounded martingales

We shall construct the solutions of the problem-A on the product space [0,∞]×Ω. Any probability Q on this product space can be disintegrated into the probability P and the conditional law Q[τ ∈ du|F_∞], provided Q is an extension of (Ω,F,P). A natural idea is to regard Q[τ ∈ du|F_∞] as the terminal term of the probability measure valued martingale M_t = Q[τ ∈ du|F_t],0 ≤ t ≤ ∞ and to hope that, when Q is a solution of the problem-A, M can be computed from the survival probability Z = N e^−Λ. On the one hand there is the relation, foru≥t,

Q[τ > u|F_t] =Q[Z_u|F_t] =Q[Z_∞|F_t] +Q[

Z _∞

u

Z_sdΛ_s|F_t] This yields effectively

Q[τ ∈du, t≤τ ≤ ∞|F_t] =Q[Z_udΛ_u11_{t≤u<∞}+Z_∞δ_∞(du)|F_t]

On the other hand, one realizes rapidly that no one-to-one map exists between Z and Q[τ ∈ du,0 ≤ τ ≤ t|F_t]. (For a given Z, there may exist uncountable many solutions of the problem.)

Our approach consists in two steps. First of all we find (in this section) a necessary and suffisent condition on the space (Ω,F,P), which guarantees the existence of a solution of the problem-A. Then, this condition being handleable, methods are discovered (in next section) to produce variable solutions of the problem.

2.1 Family iM associated with a pair (Q, τ)

Here, we consider pairs (Q, τ) where the probability Qis an extension of(Ω,F,P) and τ is a random variable defined on the same space as Q. We ask how to read such a pair from the space(Ω,F,P). The answer is synthesized in the following notion.

An increasing family of positive martingales bounded by1(in short iM) is a family of processes (M^u: 0< u <∞) satisfying the following conditions :

1. Each M^u = (M_t^u)_u≤t≤∞ is a càdlàg P-F martingale on [u,∞]. (In particular M_∞^u = lim_t→∞M_t^u.)

2. For any u, the martingaleM^u is everywhere positive and bounded by 1

3. For each fixed 0< t ≤ ∞,u ∈(0, t) → M_t^u is everywhere a right continuous increasing map (in particular, for0< u < t <∞,M_t^u = lim_²→0M_t+²^u ≤lim_²→0M_t+²^t =M_t^tP-almost surely).

The theorem below gives the link between iM and (Q, τ). To read it we recall that we identify the elements on(Ω,F,P) with elements on its extension.

Theorem 2.1 1. If ( ˆΩ,A,ˆ F,ˆ Q, π) is an extension of (Ω,F,P), if τ is a random time on ( ˆΩ,A), then there exists a unique family iMˆ = (M^u : 0 < u < ∞) such that, for 0< u <∞, u≤t≤ ∞,

M_t^u=Q[τ ≤u|F_t].

We shall say that iM is associated with (Q, τ).

2. Let (M^u : 0 < u < ∞) be an iM. Then, there is a unique probability measure Q on ([0,∞]×Ω,B[0,∞]⊗ F_∞) which extends (Ω,F,P) and satisfies Q[τ ≤ u|F_t] = M_t^u for 0< u <∞, u≤t≤ ∞. We shall say that Qis associated with iM.

Proof : Consider the first assertion. For each0 < u <∞, let (G^u_t, t ≥0)be a càdlàg version of the P-F martingale Q[τ ≤ u|F_t]. We insist that the random variable G^u_t is chosen really F_t-measurable (not merely "almostF_t-measurable"). It is clear that, foru < v,0≤t≤ ∞, one has0 ≤G^u_t ≤G^v_t ≤1 P-almost surely (recalling that Q|_F∞ =P). For 0< u < ∞,0 ≤t≤ ∞ set

M_t^u = inf{(G^w_t ∧1)⁺:w∈Q₊, u < w}.

We have immediately the following properties :

– For0< u <∞, the process M^u= (M_t^u)_0≤t≤∞ isF-optional.

– For0≤t≤ ∞,u∈(0,∞)→M_t^u is everywhere increasing and right continuous.

– For0≤t≤ ∞,0< u < v <∞,0≤M_t^u≤M_t^v ≤1 everywhere.

Let0< u <∞, and T be an F-stopping time. We can write

Q[τ ≤u] =Q[G^u_T]≤Q[M_T^u] = inf_v∈Q+,u<vQ[G^v_T] = inf_v∈Q+,u<vQ[τ ≤v] =Q[τ ≤u]

This shows thatM_T^u=G^u_T,P-almost surely. Consequently,G^u andM^u areP-indistinguishable and, therefore,M^u itself is a càdlàgP-F uniformly integrable martingale on[0,∞). We have

M_∞^u =G^u_∞= lim

t→∞G^u_t = lim

t→∞M_t^u

P-almost surely. The family of processes(M^u : 0 < u < ∞) defines therefore a iM satisfying the first assertion.

Consider the second assertion. Let M_∞⁰ = lim

u→0M_∞^u, M_∞^∞= lim

u→∞M_∞^u

The mapu∈(0,∞)→M_∞^u −M_∞⁰ being increasing and right continuous, we denote byd_uM_∞^u the associated random measure on(0,∞). Define a probability measure on([0,∞]×Ω,B[0,∞]⊗

F_∞) by

Q[F] :=P[

Z

[0,∞]

F(u,·)(M_∞⁰ δ₀(du) +d_uM_∞^u + (1−M_∞^∞)δ_∞(du))]

whereF(t, ω)∈ B[0,∞]⊗ F_∞, F(t, ω)≥0. We compute. Firstly forA∈ F_∞ : Q[A] =Q[A∩ {0≤τ ≤ ∞}] =P[I_A

Z

[0,∞]

(M_∞⁰ δ₀(du) +d_uM_∞^u + (1−M_∞^∞)δ_∞(du))] =P[A]

Secondly let 0< u <∞, u≤t≤ ∞, A∈ F_t. We have Q[A∩ {τ ≤u}] =P[I_A

Z

[0,u]

(M_∞⁰δ₀(ds) +d_sM_∞^s )] =P[I_AM_∞^u] =P[I_AM_t^u] =Q[I_AM_t^u]

These results prove that Q is an extension of(Ω,F,P) and Q[τ ≤u|F_t] = M_t^u for 0< u <

∞, u≤t≤ ∞. The uniqueness ofQis also immediate.

Remark :We can ask whether, for a given iM, its associated probabilityQcan be constructed on the measurable space (Ω,F_∞) instead of the space ([0,∞]×Ω,B[0,∞]⊗ F_∞). This is equivalent to ask if there exists a random variableLon (Ω,F_∞) such that M_t^u =Q[L≤u|F_t].

But if it is the case, we must have M_∞^u = 11_{L≤u}. This is a fairly restrictive condition on the family iM. We understand therefore, to have a perfect correspondance between the families iM and the probability-time pair(Q, τ), we have to accept to work on enlarged spaces like the space[0,∞]×Ω.

2.2 Families iM_Z and solutions to the problem-A

LetZ_t:=N_te^−Λt,0≤t <∞. We introduce the following definition :

An increasing family of positive martingales bounded precisely by 1−Z (abbre- viation iM_Z) is an increasing family of positive martingales (M^u : 0 < u <∞) satisfying the initial value condition :M_u^u = 1−Z_u P-almost surely for0< u <∞.

Notice that, for u < t < ∞, M_t^u ≤ M_t^t = 1−Z_t P-almost surely. The theorem below establishes that a solution of the problem-Aexists if and only if an iM_Zexists. It is an immediate consequence of the Theorem 2.1.

Theorem 2.2 1. If ( ˆΩ,A,ˆ F,ˆ Q, π, τ) is a solution of the problem-A, the family iM associa- ted with(Q, τ) is an iM_Z.

2. Let (M^u : 0 < u < ∞) be an iM_Z and Q be the associated probability measure on ([0,∞]×Ω,B[0,∞]⊗ F_∞). Then, Qgives a solution of the problem-A.

3 Constructions of iM

Now, to solve the problem-A, it is enough to find iM_Z. That is what we solve in this section.

Let us begin with some basic properties of the survival probability Z_t =N_te^−Λt,0 ≤t < ∞.

First of all Z_∞ = lim_t→∞Z_t exists always. The process Z is a P-F supermartingale whose canonical decomposition is :

dZ_s=e^−ΛsdN_s−Z_sdΛ_s where the random variableR_∞

0 Z_sdΛ_sis integrable and the martingalee^−Λ.N is in BMO. When Z is a potential, i.e., whenZ_∞= 0, we have the relation

Z_t=P[

Z _∞

t

Z_sdΛ_s|F_t]

SinceN_∞ exists and finite, Z will be a potential wheneverΛ_∞=∞.

3.1 A basic iM_Z

We introduce the hypothesis : Hypothesis Hy(N,Λ)

1. For all0< t <∞,0≤Z_t<1,0≤Z_t−<1 (strictly smaller than 1).

2. P-almost surely,R_b

a dΛt

1−Zt <∞ for any 0< a < b <∞.

The following theorem is borrowed from [4].

Theorem 3.1 Assume Hy(N,Λ). The family B^u_t = (1−Z_t) exp

µ

− Z _t

u

Z_sdΛ_s 1−Z_s

¶

, 0< u <∞, u≤t≤ ∞, defines an iM_Z.

Proof : It is enough to show that B^u is a P-F local martingale. But this is an immediate consequence of Itô’s formula.

3.2 More iM_Z when 1−Z >0

We assume in this subsection the hypothesisHy(N,Λ) andZ₀= 1, as well as the following one :

Hypothesis Hy(C): AllP-Fmartingales are continuous.

Note that under this hypothesis, for0< u <∞, the stochastic integralsR_t

u e^−Λs

1−ZsdN_s, u≤t <∞, exist as a continuousP-Flocal martingale.

3.2.1 The generating equation We need the following lemma.

Lemma 3.1 Let0< u <∞. LetM be aP-Flocal martingale on[u,∞)such thatM_u ≤1−Z_u. Then, M_t ≤ (1−Z_t) on t ∈ [u,∞) if and only if the local time at zero L⁰(M −(1−Z)) of M −(1−Z) on [u,∞) is identically null. Here the local time is taken right continuous in a→L^a_t(M−(1−Z)).

Proof : a) Sufficient condition. If L⁰(M −(1−Z))≡0, for anyF stopping time T ≥u such that everything in the following calculus is integrable, we compute with Tanaka’s formula :

P[(M_T −(1−Z_T))⁺] =P[(M_u−(1−Z_u))⁺] + P[

Z _T

u

I_{{Ms−(1−Zs)>0}}d(M_s−(1−Z_s))] +1

2P[L⁰_T(M−(1−Z))]≤0

becauseM−(1−Z)is aP-Fsupermartingale. This proves the suffisance of the condition.

Necessary condition. If (M_t−(1−Z_t))≤0on t∈[u,∞), Tanaka’s formula implies imme- diately L⁰⁺(M−(1−Z))≡0.

Recall that an iM_Z must satisfy the conditions :0≤M^u ≤1−Z,M^u ≤M^v ifu < v, and M_u^u = 1−Z_u. We notice that the stochastic differential equation is a natural tool to deal with these conditions.

Equation (\) Let Y be a P-F local martingale and f be a bounded Lipschitz function with f(0) = 0. For any 0< u <∞ we consider the equation

(\_u) (

dM_t=M_t

³

−_1−Z^e−Λt_tdN_t+f(M_t−(1−Z_t))dY_t

´

, u≤t <∞ M_u =x

wherex can be anyF_u-measurable random variable.

Remark 3.1 Our method remains valid iff(M_t−(1−Z_t))is replaced by some more general function f(M_t−(1−Z_t), M_t, t, ω) such that

|f(M_t−(1−Z_t), M_t, t, ω)| ≤K|M_t−(1−Z_t)|

or if some extra termg(M_t−(1−Z_t), M_t, t, ω)is added. But we shall not involve this generality in this paper. Instead we prefer to well explain the relation between the dynamic defined by the equation(\)and the decomposition formula in enlargement of filtration problem. A simpler function f(M_t^u−(1−Z_t)) will exhibit this relation better.

Theorem 3.2 Let 0 < u < ∞. The equation (\_u) has a unique solution M for each given initial valuex. If M_u ≤1−Z_u, M is bounded by (1−Z) on [u,∞].

Proof For the existence and the uniqueness of the solution of the equation (\_u), we refer to Protter [6]. To see that the solution is bounded by(1−Z) on[u,∞), we introduce the process

∆ =M−(1−Z). According to Lemma 3.1, we prove that the local timeL⁰(∆) is identically null. To do this, we calculateh∆i using the fact that, from Itô’s calculus

d∆_t=−∆_t e^−Λt

1−Z_tdN_t+M_tf(∆_t)dY_t−Z_tdΛ_t

Therefore, dh∆i_t = ∆²_t

µ e^−Λt 1−Z_t

¶₂

dhNi_t+M_t²f(∆_t)²dhYi_t−2∆_t e^−Λt

1−Z_tM_tf(∆_t)dhN, Yi_t (1)

≤ 2∆²_t

µ e^−Λt 1−Z_t

¶₂

dhNi_t+ 2M_t²f(∆_t)²dhYi_t≤2∆²_t

µ e^−Λt 1−Z_t

¶₂

dhNi_t+ 2M_t²K²∆²_tdhYi_t From this we can write

Z _t

0

I_{0<∆s<²} 1

∆²_sdh∆i_s<∞, 0< ²,0< t <∞.

According to Revuz-Yor [7],L⁰(∆)≡0. The theorem is proved.

Theorem 3.3 Let 0 < u < ∞. Let M, L be two solutions of the equation (\_u) with initial conditionsM_u =x < y =L_u. Then, M_t≤L_t for all u≤t≤ ∞.

Proof LetL⁰(M−L)denote the local time at zero ofM−L. We have, denoting∆ =M−(1−Z), and∆^L=L−(1−Z)

d(M_t−L_t) = (M_t−L_t)(− e^−Λt

1−Z_tdN_t) +¡

M_tf(∆_t)−L_tf(∆^L_t)¢ dY_t So, using the same computation as in (1), we obtain

dhM−Li_t≤2(M_t−L_t)²

µ e^−Λt 1−Z_t

¶₂

dhNi_t+ 2¡

M_tf(∆_t)−L_tf(∆^L_t)¢₂ dhYi_t

Then, using the fact that

M_tf(∆_t)−L_tf(∆^L_t) = (M_t−L_t)f(∆_t) +L_t(f(∆_t)−f(∆^L_t)) we obtain

dhM−Li_t ≤ 2(M_t−L_t)²

µ e^−Λt 1−Z_t

¶₂

dhNi_t+ 4(M_t−L_t)²f²(∆_t)dhYi_t+ 4L²_t(f(∆_t)−f(∆^L_t))²dhYi_t

≤ 2(M_t−L_t)²

µ e^−Λt 1−Z_t

¶₂

dhNi_t+ 4(M_t−L_t)²f²(∆_t)dhYi_t+ 4L²_tK²(M_t−L_t)²dhYi_t It yields that

Z _t

0

I_{{0<Ms−Ls<²}} 1

(M_s−L_s)²dhM−Li_s<∞, 0< ² <∞,0< t <∞ L⁰(M −L) therefore is identically null. The asserted property is proved.

Corollary 3.1 Let0< u <∞. Let M be a solution of the equation (\_u) with initial conditions M_u^u =x≥0. Then, M_t^u ≥0 for all u≤t≤ ∞.

Proof We notice that0 is the solution of equation(\_u) with initial condition 0.

As a consequence of Theorem 3.2 and Corollary 3.1, the local martingale M is bounded between 0 and 1. It is therefore a trueP-F uniformly integrable martingale.

Corollary 3.2 For0< u <∞, letL^u denote the solution of the equation (\_u) with the initial condition L^u_u = 1−Z_u. Then, for u < v ≤b <∞, L^u_b ≤L^v_b.

Proof This comparison relation is because that L^u, L^v satisfy the same equation\_v on[v,∞) andL^u_v ≤(1−Z_v) =L^v_v.

3.2.2 The iM_Z associated with the Equation(\)

For0< u <∞, letL^ube as in the preceding corollary. We know already thatL^uis a bounded P-Fmartingales,L^u_b ≤L^v_b ifu < v, and0≤L^u_b ≤(1−Z_b)for b≥u. SetL^u_∞= lim_b→∞L^u_b and

M_u^u = (1−Z_u)

M_b^u = inf{(L^v_b)⁺∧(1−Z_b) :v∈Q₊, u < v≤b}, u < b≤ ∞

Theorem 3.4 Each M^u is P-indistinguishable to L^u and (M^u : 0 < u < ∞) is an iM_Z. We shall say that this iM_Z is associated with the equation(\).

Proof We need only to prove the martingale property of M^u. Let 0 < u < ∞. Let T be a F-stopping time such that T ≥ u and everything concerned in the following computation is integrable. We have

P[M_T^u−L^u_T] = P[I_{u<T}infv∈Q,u<v≤T L^v_T −I_{u<T}L^u_T]

= inf_v∈Q,u<vP[I_{u<T}L^v∧T_T ]−P[I_{u<T}L^u_T]

= inf_v∈Q,u<vP[I_{u<T}(1−Z_v∧T)]−P[I_{u<T}(1−Z_u)]

= 0

This shows thatM^u, L^u areP-indistinguishable on[u,∞]and in particularM^u is a continuous P-F uniformly integrable martingale.

3.3 An iM_Z in case of possible zero of 1−Z

In this section we study the case where 1−Z may take the value zero. Consequently the analysis made in the preceding section does not work anymore, because, for example, _1−Z^e−Λ.N can be undefined. New method is needed to construct iM_Z.

LetH={s: 1−Z_s= 0} and define, for0< t <∞, the random times g_t := sup{0≤s < t:s∈H},

d_t := inf{s > t:s∈H}.

We introduce the hypothesis :

Hypothesis Hy(H) The setHis not empty and is closed. The measuredΛhas a decomposition dΛ_s=dV_s+dA_s whereV, A are continuous increasing processes such thatdV charges onlyH whiledA charges its complementaryH^c. Moreover, we suppose

I_{gt≤u<t}

Z _t

u

Z_s

1−Z_sdA_s=I_{u<t≤du} Z _t

u

Z_s

1−Z_sdA_s<∞ for any0< u < t <∞.

We suppose in this sectionHy(H) andZ₀ = 1.

Lemma 3.2 Let E_t^u = exp

³

−R_t

u Zs

1−ZsdA_s

´

. We have the identity, for0< u≤t≤ ∞ I_{gt≤u}E_t^u(1−Z_t) = (1−Z_u)−

Z _t

u

I_{gs≤u}E^u_se^−ΛsdN_s

Proof It is rather straightforward to prove that, for 0< u≤t≤d_u d_t(E_t^u(1−Z_t)) =E_t^u¡

Z_tdV_t−e^−ΛtdN_t¢

=−E_t^ue^−ΛtdN_t We use the balayage formula (see [7]) to calculate, foru < t <∞

I_{gt≤u}E_t^u(1−Z_t) = I_{gt≤u}

µ

(1−Z_u)− Z _t

u

E_s^ue^−ΛsdN_s

¶

= (1−Z_u)− Z _t

u

I_{gs≤u}E_s^ue^−ΛsdN_s

The lemma is proved.

Set

M_t^u =I_{gt≤u}E_t^u(1−Z_t), 0< u <∞, u≤t≤ ∞.

We see immediately that the processM^u is càdlàg, positive and bounded by(1−Z_t)on[u,∞].

According to Lemma 3.2, M^u is a P-F local martingale. So it is an P-F uniformly integrable martingale. The initial value ofM^u isM_u^u = (1−Z_u)whilst the terminal value is

I_{g≤u}exp{−

Z _∞

u

Z_s

1−Z_sdA_s}(1−Z_∞).

whereg:= lim_t→∞g_t. It yields that, for 0< t≤ ∞, the mapu∈(0, t)→M_t^u is increasing and right continuous.

We define for 0< t≤ ∞,

M_t⁰ = lim

u↓0M_t^u We have

P[M_t⁰] = lim

u↓0Q[M_t^u] = lim

u↓0Q[M_u^u] = lim

u↓0Q[1−Z_u] = 0, therefore, M⁰ ≡0. We have on the other hand

M_∞^∞=I_{g<∞}I_B(1−Z_∞) =I_B(1−Z_∞) where

B=

½

0≤u<∞inf Z _∞

u

Z_s

1−Z_sdA_s<∞

¾

We conclude

Theorem 3.5 The family (M^u: 0< u <∞) is an iM_Z. Notice that, if 1−Z_u is not identically null,R_∞

u Zs

1−ZsdA_s is not identically infinite.

Now we have an iM_Z. This implies automatically a solution of the problem-A. However for this iM_Z the solution has an interesting form we believe useful to write down explicitely. We defineQa probability measure on[0,∞]×Ωby the following relations : forA∈ F_∞,0< u <∞,

Q[A{τ ≤u}] = P[I_Aexp{−R_∞

u Zs

1−ZsdA_s}(1−Z_∞)], Q[A{τ =∞}] = P[I_A(I_BZ_∞+I_B^c)].

This definition implies immediately the equality Q[A] = P[A], A∈ F_∞, i.e. Q is an extension ofP. Now, for 0< u <∞ andA∈ F_∞,

Q[A{g∨τ ≤u}] =P[I_AI_{g≤u}exp{−

Z _∞

u

Z_s

1−Z_sdA_s}(1−Z_∞)] =P[I_AM_∞^u] In particular, if A∈ F_u,

Q[A{g∨τ ≤u}] =P[I_AM_u^u)] =P[I_A(1−Z_u)]

i.e.Q[g∨τ > u|F_u] =Z_u.We conclude

Theorem 3.6 (Q, g∨τ) is a solution of the problem-A.

4 Enlargement of filtration problem

In the preceding sections we have constructed families iM_Z which generates solutions of the problem-A. In this section we shall study the enlargement of filtration problem for these solutions. Let us recall the setting. For a solution( ˆΩ,A,ˆ F,ˆ Q, π, τ) of the problem-A, we equip it with the progressively enlarged filtration G= (G_t)_t≥0 where G_t =F_t∨σ(τ ∧t), t ≥0. We want to know if aP-F local martingaleX remains a Q-Gsemimartingale and if it is the case, what is itsQ-Gsemimartingale decomposition. The problem can be completely solved for the period before the timeτ by [5]. We have the lemma.

Lemma 4.1 Suppose Hy(B). Suppose Q[0 < τ < ∞] = 1. Let X be a P-F local martingale.

We denote byB theQ-Fpredictable dual projection of the jump process∆X_τ11_{τ≤t},0≤t <∞.

Let hN, Xi be the P-F-predictable dual projection of [N, X](hN, Xi exsits always). Then, X_·∧τ−

Z _·∧τ

0

1

Z_s−(e^−ΛsdhN, Xi_s+dB_s) is aQ-Glocal martingale.

Remark 4.1 that, if Q[τ = T] = 0 for all F-stopping time T, B ≡ 0. More generally the process B is linked with Λ by the formula : dB_s = H_s^XZ_sdΛ_s, where H^X is a F-predictable process such thatH_τ^X =Q[∆_τX|F_τ−].

Therefore the real problem for us is about X_t∨τ −X_τ for the period after the time τ. Let us mention some examples where the problem for the period after τ have been studied with success. They are the case of honest time (see Barlow, Jenlin and Yor [1, 5]) and the case of initial time (see Jeanblanc and Le Cam [3]). The situation studied here is different. We work in the framework fixed by a solution of the problem-A. Fortunately, the familes iM_Z of our solutions satisfy good stochoastic differential equations. This makes applicable a classical idea from Yor [8]. The problem will be thus solved. To complete our program, we remark that the reasoning process can be nicely reversed. We shall prove that, if for a given family iM_Z a good semimartingale decomposition formula holds for the model constructed with it, then the iM_Z must be the solutions of a certain stochastic differential equation.

4.1 The case of the basic iM_Z

The following theorem has been proved in [4] :

Theorem 4.1 Suppose Hy(N,Λ). Consider the iM_Z defined by the family (Theorem 3.1) B_t^u= (1−Z_t) exp

µ

− Z _t

u

Z_sdΛ_s 1−Z_s

¶

0< u <∞, u≤t≤ ∞,

Let Q be the probability measure on [0,∞]×Ω associated with the family (B^u : 0 < u < ∞).

Then, for anyP-F local martingaleX, X_·−

Z _·

0

I_{s≤τ}e^−Λs dhN, Xi_s Z_s⁻ +

Z _·

0

I_{{τ <s}}e^−Λs dhN, Xi_s 1−Z_s⁻ is aQ-Glocal martingale.

4.2 From the equation (\) to the decomposition formula in enlargement of filtration

In this subsection we suppose Hy(N,Λ) and Hy(C) and Z₀ = 1, Z_∞ = 0. We consider a generating equation(\) (see subsection 3.2.1) : 0< u <∞,

(\_u) (

dM_t=M_t

³

−_1−Z^e−Λt

tdN_t+f(M_t−(1−Z_t))dY_t

´

, u≤t <∞ M_u =x

withf being a continuously differentiable Lipschitz function andY being aP-Flocal martingale.

Let(M^u,0< u <∞) be the iM_Z family associated to the above equation (\). We suppose in addition the following hypothesis:

Hy(Mc) : For each0< t <∞, the mapu→M_t^u is continuous on(0, t].

We can write by monotone convergence theorem : P[ lim

v→∞M_∞^v ] = lim

v→∞P[M_∞^v ] = lim

v→∞P[M_v^v] = lim

v→∞P[(1−Z_v)] = 1.

But M_t^v ≤ 1−Z_t ≤ 1 for all v ≤ t ≤ ∞. So the above equality yields M_∞^∞ = 1. It is also true that M_∞⁰ = 0. We denote by d_uM_t^u the random measure on (0, t) induced by the map u∈(0, t)→M_t^u, for 0< t≤ ∞. The following lemma is straightforward.

Lemma 4.2 For0< u <∞, for h a bounded Borel function on [0,∞], the process : t∈[u,∞]→

Z _u

0

h(v)d_vM_t^v

is a boundedP-F martingale on[u,∞].

LetQbe the probability on the product space[0,∞]×Ωassociated with the iM_Z. Note that since M_∞^∞= 1 and M_∞⁰ = 0,Q[0< τ <∞] = 1. And also the Hy(Mc) implies Q[τ =T] = 0 for anyF stopping timeT. We need another lemma.

Lemma 4.3 Let V_s(ω),0 ≤ s < ∞, ω ∈ Ω, be a function such that, for fixed s, V_s is F_∞ measurable and for fixedω, s→V_s(ω) is càdlàg and increasing. We denote bydV_s the induced measure on[0,∞). LetF_s(t, ω),0≤s <∞,0≤t≤ ∞, ω ∈Ω, be a positive function measurable with respect toB[0,∞]⊗ B[0,∞]⊗ F_∞. Suppose Q[R_∞

0 dV_s]<∞. Then, Q[

Z _∞

0

F_sdV_s] =Q[

Z _∞

0

µZ _∞

0

F_s(v,·)d_vM_∞^v

¶ dV_s]

Proof. By monotone class theorem, we need only to check the relation for a function of formF_s(t, ω) =h(s)H(t, ω). Recall thatQis an extension of(Ω,F,P) so that it is identified as Pon F_∞. We compute

Q[

Z _∞

0

F_sdV_s] = Q[H Z _∞

0

h(s)dV_s]

= Q[

Z _∞

0

H(v,·)d_vM_∞^v Z _∞

0

h(s)dV_s]

= Q[

Z _∞

0

µZ _∞

0

F_s(v,·)d_vM_∞^v

¶ dV_s] The lemma is proved.

Theorem 4.2 Let X be a P-Flocal martingale. Then, the process X_·−X₀−R_·

011_{s≤τ}^e−Λ_Z ^s

s dhN, Xi_s

−R_·

011_{{τ <s}}(−_1−Z^e−Λs_s)dhN, Xi_s

−R_·

011_{{τ <s}}(f(M_s^τ−(1−Z_s)) +M_s^τf⁰(M_s^τ −(1−Z_s)))dhY, Xi_s

is aQ-G local martingale.(For a later use, we denote by A^[Y;f](X)_t; 0≤t <∞ the sum of the three stochastic integrals so that the above formula can be written asX_t−X₀−A^[Y;f](X)_t) Proof : We could adopt the reasoning in Yor [8]. But we prefer to give another proof which reveals clearly the implication of the stochastic differential equation (\) in the semimartingale decomposition formula. We write

X_t−X₀ =X_τ∨t−X_τ+X_τ∧t−X₀

The decomposition forX_τ∧t−X₀ is given by Lemma 4.1 with B ≡0. We need only to prove that

X_τ∨t−X_τ −R_t

011_{{τ <s}}(−_1−Z^e−Λs_s)dhN, Xi_s

−R_t

0 11_{{τ <s}}(f(M_s^τ−(1−Z_s)) +M_s^τf⁰(M_s^τ −(1−Z_s)))dhY, Xi_s

is a Q-G local martingale. Without loss of generality we suppose that X is stopped so that everything in the calculus below is integrable. In particular we assume that

Q[

Z _∞

τ

e^−Λs

1−Z_s|dhN, Xi|_s] =Q[

Z _∞

0

e^−Λs|dhN, Xi|_s]