SOME EXISTENCE RESULTS FOR ADVANCED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH A JUMP TIME

S. Crépey, M. Jeanblanc and A. Nikeghbali Editors

SOME EXISTENCE RESULTS FOR ADVANCED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH A JUMP TIME

Monique JEANBLANC

, Thomas LIM

and Nacira AGRAM

Abstract. In this paper, we are interested by advanced backward stochastic differential equations (ABSDEs), in a probability space equipped with a Brownian motion and a single jump process, with a jump at timeτ. ABSDEs are BSDEs where the driver depends on the future paths of the solution. We show, that under immersion hypothesis between the Brownian filtration and its progressive enlargement withτ, assuming that the conditional law ofτis equivalent to the unconditional law ofτ, and a Lipschitz condition on the driver, the ABSDE has a solution.

Introduction

In the traditional approach, Backward Stochastic Differential Equations (BSDEs) are studied for a driver which depends in a Markovian way of the parameters; in particular, BSDEs with a single jump and driven by a Brownian motionB have the form

−dYt=f(t, Yt, Zt, Ut)dt−ZtdBt−UtdHt

where H is the process Ht =1{τ≤t} associated with a given random time τ. In this paper, we are interested by BSDEs in which the driver depends on the future of the solution, in such a way that the driver is adapted, for example if the driver depends on conditional expectation of the future of the solution, e.g., −dYt= (aYt+ bE^Gt(Yt+1))dt−ZtdWt−UtdHt, we use the notation E^Gt[X] :=E[X|Gt], whereG= (Gt)_t≥0 is the progressive enlargement filtration of the filtration of B by H. More precisely, we focus on BSDEs written in one of the following forms, called advanced backward stochastic differential equations (in short ABSDEs)















−dYt= f t, Y_t,E^Gt[Y_t+δ],(E^Gt[Y_t+s])_0≤s≤δ, Z_t,E^Gt[Z_t+δ],(E^Gt[Z_t+s])_0≤s≤δ, Ut,E^Gt[Ut+δ],(E^Gt[Ut+s])_0≤s≤δ

dt−ZtdBt−UtdHt, 0≤t≤T , Y_T+t= ξ_T+t, 0≤t≤δ ,

ZT+t= PT+t, UT+t = QT+t1{T+t≤τ}, 0< t≤δ ,

(0.1)

∗ This research was supported by Chaire Markets in Transition, (French Banking Federation) Institut Louis Bachelier and Labex ANR 11-LABX-0019.

∗∗ The research of N. Agram is also carried out with support of the Norwegian Research Council, within the research project Challenges in Stochastic Control, Information and Applications (STOCONINF), project number 250768/F20.

1 LaMME, University of Evry; [email protected]

2 LaMME, Ecole Nationale Supérieure d’Informatique pour l’Industrie et l’Entreprise; [email protected]

3 Department of Mathematics, University of Oslo; [email protected]

c

EDP Sciences, SMAI 2017

Article published online byEDP Sciencesand available athttp://www.esaim-proc.orgorhttps://doi.org/10.1051/proc/201756088

and











−dYt= E^Gt

f(t, Yt, Yt+δ,(Yt+s)_0≤s≤δ, Zt, Zt+δ,(Zt+s)_0≤s≤δ, U_t, U_t+δ,(U_t+s)_0≤s≤δ)]dt−Z_tdB_t−U_tdH_t, 0≤t≤T , YT+t= ξT+t, 0≤t≤δ ,

Z_T+t= P_T+t, U_T+t = Q_T+t1{T+t≤τ}, 0< t≤δ .

(0.2)

The solution is the triplet(Y, Z, U), living in some spaces we shall define later on. In this equation, for a process V, the notation (V_t+s)_0≤s≤δ means that we consider all the path of the process between t and t+δ. The terminal conditionsξ, P andQare given processes. We remark that the driversf of these ABSDEs depend on the values of the processes(Y, Z, U)for present timetas well as for future timet+δand also of the trajectory of the processes on the interval [t, t+δ]. The ABSDE (0.1) was introduced by Peng and Yang in [12] in a Brownian case setting (roughly speaking, forτ≡0). Øksendalet al.[11] have introduced ABSDEs of the form (0.2) - in a simpler case where not all the path of the solution is involved - when dealing with optimal control for delayed systems, taking into account a random Poisson measure, instead of a single jump process.

Using the methodology of BSDEs in an enlargement of filtration setting as in Kharroubi and Lim [8], we give conditions such that there exists a unique solution of (0.1) and of (0.2) under immersion hypothesis and in adequate spaces. This progressive enlargement is often considered as progressive adding of information given in form of a random timeτ in a way which transformsτ to a stopping time with respect to the filtrationG. The topic of enlargement of filtration was initiated by Jacod, Jeulin and Yor (see [6, 7]). Naturally, the enlargement of filtration appears in credit risk and it has also been related recently to stochastic optimal control by Pham [13]

and to mean-variance hedging by Kharroubiet al. [9] where the optimal strategy is described by non-standard BSDEs driven by a Brownian motion and a jump martingale in the enlarged filtration.

There are, in the literature, two approaches of BSDEs with a single jump. One of them considers BSDEs driven by martingales and is based on the predictable representation property (PRP), as in Dumitrescuet al. [3].

These authors consider a Brownian motionB and the martingaleM associated to the jump processH. Under some conditions, in particular that the Brownian motion is a Brownian motion in the filtration generated by the pair(B, M)(which is a consequence of the assumed immersion property), they prove that the pair(B, M)enjoys PRP and they solve BSDEs of the form −dYt=f(t, Yt, Zt, Ut)dt−ZtdWt−UtdMt using usual methodology.

Kharroubi and Lim’s method is different: they consider the case −dYt = g(t, Yt, Zt, Ut)dt−ZtdWt−UtdHt, assuming immersion property, and they show that this BSDE is equivalent to a system of BSDE in the Brownian filtration. In fact, assuming the more general hypothesis of existence of positive conditional density α(as in Hypothesis 1.2 below), they can solve the problem without immersion property as follows: The change of probability dP^∗ = (αt(τ))⁻¹dP is such that B is a P^∗-Brownian motion independent of H so that immersion holds underP^∗ and the solution of the BSDE is the same under PandP^∗. However, here, we can not use this methodology, since a change of probability will affect the conditional expectation, and the Lipschitz condition will be difficult to check. Nevertheless, working without immersion hypothesis is doable as we explain in the Appendix of the paper.

1. Framework 1.1. Classical results about progressive enlargement

Let(Ω,G,P)be a complete probability space. We assume that this space is equipped with a one-dimensional standard Brownian motion B and we denote by F := (F_t)_t≥0 the right-continuous and complete filtration generated by B. We consider on this space a random time τ and we introduce the right-continuous process H := 1{τ≤.}. Since τ is not supposed to be anF-stopping time, we use the standard approach of filtration enlargement by considering the smallest right-continuous extensionGofFthat turnsτ into aG-stopping time.

More precisely, the filtrationG:= (Gt)_t≥0 is defined by Gt := \

ε>0

G˜t+ε,

for anyt≥0, whereG˜_s:=F_s∨σ(H_u, u∈[0, s]), for anys≥0.

We denote by P(F) (resp. P(G)) the σ-algebra of F (resp. G)-predictable subsets of Ω×R+, i.e., the σ- algebra generated by the left-continuousF(resp. G)-adapted processes. We denote by O(F)(resp. O(G)) the σ-algebra of F (resp. G)-optional subsets ofΩ×R⁺, i.e., the σ-algebra generated by the right-continuous F (resp. G)-adapted processes.

We impose the following hypothesis introduced by Brémaud and Yor [2], which is classical in the filtration enlargement theory and is called(H)-hypothesis or immersion property.

Hypothesis 1.1. The processB remains aG-Brownian motion.

We observe that, since the filtrationFis generated by the Brownian motionB, Hypothesis 1.1 is equivalent to all F-martingales are alsoG-martingales. In particular, the stochastic integral Rt

0XsdBs is a well defined G-local martingale for allP(G)-measurable processesX such thatRt

0|Xs|²ds <∞a.s., for allt≥0.

We also introduce another hypothesis, often called the Jacod equivalence hypothesis (see, e.g., [1, chapter 4]), that the conditional law of τ is equivalent to the law ofτ and that τ admits a density w.r.t. Lebesgue’s measure, which will allow us to compute conditional expectations w.r.t. Gin terms of conditional expectations w.r.t. F.

Hypothesis 1.2. We assume that there exists a strictly positiveP(F)⊗ B(R)-measurable function(ω, t, u)→ αt(ω, u)continuous intsuch that

a) for anyθ≥0, the process(αt(θ))_t≥0 is anF-martingale,

b) for anyt≥0, the measure αt(ω, θ)dθis a version of P(τ∈dθ|Ft)(ω), that is for any Borel functionf such thatf(τ)is integrable, one has

E[f(τ)|Ft] = Z ∞

0

f(θ)α_t(θ)dθ , a.s.

In particular, the density ofτ isα0.

In all the paper, Hypotheses 1.1 and 1.2 are in force.

We now recall some standard results that will be important for our purpose and we refer to [4] for their proofs.

We introduce theF-supermartingaleG(called Azéma’s supermartingale) defined as G_t := P(τ > t| Ft) =

Z ∞ t

α_t(θ)dθ , t≥0.

The supermartingaleGis strictly positive, non-increasing and continuous. The processM defined by Mt := Ht−

Z t∧τ 0

αs(s) Gs

ds , t≥0,

is a G-martingale, with a single jump at timeτ. TheF-adapted processλdefined by λt := α_t(t)

Gt

, t≥0, (1.1)

is called theF-intensity ofτ. Under Hypotheses 1.1 and 1.2, we have, from [4, equality (11)],

α_t(θ) = α_θ(θ), ∀t≥θ , (1.2)

which implies

Gt = exp

− Z t

0

λsds

, (1.3)

since, by definition ofGandλand the fact that (1.2) holds, we have the following equalities, Gt=

Z ∞ t

αt(θ)dθ= 1− Z t

0

αt(θ)dθ= 1− Z t

0

αθ(θ)dθ= 1− Z t

0

Gθλθdθ andG₀= 1. Note that, from immersion

Gt = E^Fs(1{τ >t}), ∀s > t . (1.4)

Hypothesis 1.3. We assume that the processλis upper bounded by a constantk.

Lemma 1.4. For any t∈ [0, T], the random variableGt is lower bounded by e^−kt, and for anyθ ∈[0, T] we have 0< αt(θ)≤k.

Proof. The bound onGis obvious from (1.3). Letθ∈[0, T], then for anyt≥θ, (1.1) and (1.2) lead to αt(θ) = αθ(θ) = λθGθ ≤ k .

Moreover, since α(θ)is a martingale, we getαt(θ)≤E^Ft(αθ(θ))≤k for anyt≤θ.

Furthermore, ifY ∈ GT is integrable, then we have E^Gt[Y1{t<τ}] = 1

Gt1{t<τ}E^Ft(Y1{t<τ}).

We recall a decomposition result forP(G)-measurable processes, proved in [7, Lemma 4.4] for bounded processes.

It can be easily extended to the case of unbounded processes.

Proposition 1.5. Any P(G)-measurable processX = (X_t)_t≥0 can be represented as X_t = X_t^b1{t≤τ}+X_t^a(τ)1{t>τ},

for allt≥0, whereX^b isP(F)-measurable andX^a(·)isP(F)⊗ B(R+)-measurable.

Here, the superscriptbis forbeforeτ andaforafter τ. In particular, aG-predictable process is equal to an F-predictable process on the set{t≤τ}.

Song [14] has extended the previous result to the class of optional processes under some hypotheses, which are satisfied under equivalence Jacod’s hypothesis.

Proposition 1.6. Any O(G)-measurable processX= (Xt)t≥0 can be represented as Xt = X_t^b1{t<τ}+X_t^a(τ)1{t≥τ},

for allt≥0, whereX^b isO(F)-measurable and X^a(·)isO(F)⊗ B(R+)-measurable.

If the processX is bounded by a constantK, then the processX^b is bounded byKand one can also choose the processX^a(θ)bounded byKfor anyθ≥0. We remark that the uniqueness ofX_t^a(θ)is granted forθ≤t.

The processX^b is uniquely determined on[0, T]byX_t^b = _G¹

tE^Ft[Xt1{t<τ}], this quantity will be called the pre-default part.

Lemma 1.7. Let Y_T(τ)be a bounded F_T ⊗σ(τ)-measurable random variable. Then, for anyt≤T, we have E^Gt[YT(τ)] = Y_t^b1{t<τ}+Y_t^a(τ)1{τ≤t} a.s.

where

Y_t^b = E^Ft R∞

t YT(u)αT(u)du]

G_t a.s. (1.5)

Y_t^a(θ) = E^Ft YT(θ)

a.s. for any θ≤t which can be rewritten under the form

Y_t^a(τ) = E^Ft YT(θ)

|θ=τ a.s. on τ≤t . (1.6)

Proof. The proof of this lemma is an application of Proposition 1.6 and that Y_t^a(θ) = E^Ft

YT(θ)αT(θ)]

αt(θ) a.s.

andα_t(θ) =α_θ(θ)for anyt≥θ.

Therefore, ifYT(τ)is bounded by a constantKthen the processesY^b andY^a(θ)are bounded byK for any θ≥0.

We now give a decomposition result for some Stochastic Differential Equations (SDEs) inGin terms of SDEs in F.

Lemma 1.8. If the processX satisfies the following stochastic differential equation dX_t = µ(t, X_t, η_t)dt+σ(t, X_t, η_t)dB_t+ϕ(t, X_t−, η_t)dH_t,

whereµ,σ areO(G)⊗ B(R)⊗ B(R)-measurable maps,ϕis aP(G)⊗ B(R)⊗ B(R)measurable map andη is a G-predictable process, thenX^a andX^b satisfy







dX_t^a(τ) = µ^a(t, τ, X_t^a(τ), η_t^a(τ))dt+σ^a(t, τ, X_t^a(τ), η_t^a(τ))dBt, τ≤t≤T , dX_t^b= µ^b(t, X_t^b, η_t^b)dt+σ^b(t, X_t^b, η_t^b)dB_t, 0≤t≤T ,

X_t^a(t)−X_t^b= ϕ(t, X_t^b, η^b_t), 0≤t≤τ .

Proof. The proof of this lemma is an application of Proposition 1.6 and for the last equality, we have used that if an F-predictable processK satisfiesK_τ= 0, then K_t= 0on{t≤τ}(see [10, Lemma 3, Chapter 1]).

1.2. Notations

To define solutions to ABSDEs, we introduce the following spaces, wheres, t∈R+ with s≤t, andT <∞ is the terminal time andδis a strictly positive constant.

• S_G²[s, t] (resp. S_F²[s, t]) is the set of R-valuedO(G)(resp. O(F))-measurable processes(Y_u)_u∈[s,t] such that

kYk²_S2[s,t] := E[ sup

u∈[s,t]

|Yu|²] < ∞.

• L²

G[s, t] (resp. L²

F[s, t]) is the set of R-valuedP(G)(resp. P(F))-measurable processes (Z_u)_u∈[s,t] such that

kZk²_L2[s,t] := E hZ t

s

|Z_u|²dui

< ∞.

• L²(Ft)is the set ofR-valued square integrable Ft-measurable random variables.

• L²_τ is the set ofR-valuedP(F)-measurable processesU such that Ut= 0fort > τ and

||U||²_L2

τ := E

hZ T 0

|U_s|²dsi

< ∞.

• D[0, δ] is the set of càd-làg R-valued maps defined on [0, δ]. For Y ∈ D[0, δ], we denote |Y| :=

√1 δ

Rδ

0 |Y(s)|ds.

1.3. Existence results for ABSDE in a Brownian filtration

We extend the results of Peng and Yang [12] to more general drivers since we assume in our case that the driver depends on the trajectory of the processes on the interval [t, t+δ]. The proofs are based on standard methodologies, however they require some care to check the needed Lipschitz conditions. To simplify the writing we introduce some new notations for each Proposition, and the same notation~y∈Ais used in different meanings which are clear from the context.

Proposition 1.9. Let A:=R²×D[0, δ]×R²×D[0, δ]and, for any~y= (y,by,Y, z,z,b Z)∈Awe define |~y| by

|~y| = |y|+|y|b +|Y|+|z|+|z|b +|Z|,

where |Y| is defined in Section 1.2. Let f be a map from Ω×[0, T]×A valued in R. Let p and q be given boundedF-adapted processes.

The following ABSDE











−dYt= f t, Y_t,E^Ft[p_t+δY_t+δ],{E^Ft[p_t+sY_t+s]}0≤s≤δ, Z_t,E^Ft[q_t+δZ_t+δ],{E^Ft[q_t+sZ_t+s]}0≤s≤δ dt

−ZtdBt, 0≤t≤T , Y_T+t= ξ_T+t, 0≤t≤δ , ZT+t= PT+t, 0< t≤δ ,

(1.7)

has a unique solution in S_F²[0, T +δ]×L²_F[0, T+δ]if a) the mapf(·, ~y)is optional for any~y∈A,

b) there exists C >0 such that, for any t∈[0, T], any~y∈A, we have f(t, ~y)−f(t, ~y⁰)

≤ C|~y−~y⁰|, c) E[RT

0 |f(s, ~0)|²ds]<∞,

d) the terminal condition ξbelongs to S_F²[T, T+δ] andP belongs toL²_F[T, T+δ].

Proof. In the driver, the map Y = (Y_t(s) = E^Ft[p_t+sY_t+s],0 ≤s ≤ δ) is a family of F_t-measurable random variables. Let us first introduce a norm in the Banach spaceE :=S²

F[0, T +δ]×L²

F[0, T +δ] for β >0: for (Y, Z)∈E

||(Y, Z)||²_β := E hZ T+δ

0

e^βt(Y_t²+Z_t²)dti ,

and define the mapping Φ :E→E byΦ((y, z)) = (Y, Z)where(Y, Z)is defined by







−dYt= f(t, ~y_t)dt−Z_tdB_t, 0≤t≤T , YT+t= ξT+t, 0≤t≤δ ,

Z_T+t= P_T+t, 0< t≤δ ,

where ~y_t = (y_t,E^Ft[p_t+δy_t+δ],{E^Ft[p_t+sy_t+s]}_0≤s≤δ, z_t,E^Ft[q_t+δz_t+δ],{E^Ft[q_t+sz_t+s]}_0≤s≤δ). We now prove thatΦis a contraction inEunder the norm||.||_β. For two arbitrary elements(y, z)and(y⁰, z⁰), we denote their difference by

(ey,z)e = y−y⁰, z−z⁰ . We can prove by using classical estimates that we have

E hZ T

0

e^βtβ

2Ye_t²+Ze_t² dti

≤ 2 β E

hZ T 0

e^βt

f(t, ~yt)−f(t, ~y⁰_t)

2dti .

In the following inequalities, K is a constant which does not depend on β and may change from line to line.

By Lipschitz property of the mapf, the fact that the square of a sum (resp. integral) is bounded by the sum (resp. integral) of the square (up to a constant) and the boundedness ofpandq, it follows that

E hZ T

0

e^βtβ

2Ye_t²+Ze_t² dti

≤K β E

hZ T 0

e^βt

ye²_t+ze_t²+ey_t+δ² +ez²_t+δ+1 δ

Z δ 0

(ye_t+s² +ez²_t+s)ds dti

. (1.8) By the change of variableu=t+s, we get

E hZ T

0

e^βtβ

2Ye_t²+Ze_t² dti

≤ K β E

hZ T+δ 0

e^βt ye_t²+ez_t² dt+1

δ Z T

0

e^βt Z t+δ

t

(ye_u²+ze_u²)du dti

. (1.9) Fubini’s theorem leads to

1 δ

Z T 0

e^βt Z t+δ

t

(ye²_u+ez_u²)du dt ≤ 1 δ

Z T+δ 0

Z u u−δ

e^βtdt

(ey_u²+ze_u²)du

≤ 1−e^−βδ βδ

Z T+δ 0

e^βu(ye_u²+ze_u²)du

≤

Z T+δ 0

e^βu(ye²_u+ez²_u)du (1.10) where we have used that1−e^−βδ≤βδ.

Combining (1.10) with (1.9), we obtain forβ ≥2

E hZ T

0

e^βt

Ye_t²+Ze_t² dti

≤ K β E

hZ T+δ 0

e^βt ye_t²+ez_t² dti

.

Consequently, sinceYe =Ze= 0fort > T, we get

||(Y ,e Ze)||²_β ≤ K

β ||(y,ez)||e ²_β,

and Φ is a contraction on S²

F[0, T +δ]×L²

F[0, T +δ] for β large enough to ensure that K/β < 1, and β >2. AsΦis a contraction, using general results on BSDE (as in [5]) there exists a unique solution(Y, Z)in S²

F[0, T+δ]×L²

F[0, T +δ]to ABSDE (1.7).

We now give an estimation of the solution of the ABSDE.

Proposition 1.10. Suppose f satisfies the hypotheses of Proposition 1.9. Then there exists a strictly positive constant K that only depends on the Lipschitz constant C and on T such that for any ξ ∈ S_F²[T, T +δ] and P ∈L²

F[T, T+δ], the solution(Y, Z)of the ABSDE (1.7) satisfies E^Ft

h sup

t≤s≤T

Y_s²+ Z T

t

Z_s²dsi

≤ KE^Ft h

ξ_T² + Z T+δ

T

(ξ_s²+P_s²)ds+ Z T

t

f(s, ~0)²dsi ,

for any t∈[0, T].

Proof. The proof is obtained with standard computations. For the sake of completeness, we give details in the

Appendix.

Using the same methodology as in Proposition 1.9, one obtains the following result, where D(t,[0, δ])is the family of mapsY from[0, δ]toRsuch thatY(s)isFt+s-measurable, for anys∈[0, δ].

Proposition 1.11. For anyt∈[0, T], let A_t=R×L²(Ft+δ)×D(t,[0, δ])×R×L²(Ft+δ)×D(t,[0, δ])and for any ~y= (y, ζ,Y, z, η,Z)∈A_t, we introduce |~y|=|y|+|z|+E^Ft(|ζ|+|η|+|Y|+|Z|). For a mapf such that f(ω, t) :A_t→R, the following ABSDE







−dY_t= f(t, Y_t, Y_t+δ,(Y_t+s)_0≤s≤δ, Z_t, Z_t+δ,(Z_t+s)_0≤s≤δ)dt−Z_tdB_t, 0≤t≤T , YT+t= ξT+t, 0≤t≤δ ,

Z_T+t= P_T+t, 0< t≤δ ,

has a unique solution in S_F²[0, T +δ]×L²_F[0, T+δ]if the mapf satisfies:

a) for~yt∈At,f(t, ~yt)isFt-measurable,

b) there existsC such that for anyt∈[0, T], any~y, ~y⁰ in At, one has

|f(t, ~y)−f(t, ~y⁰)| ≤ C|~y−y~⁰|, c) E(RT

0 |f(t, ~0)|²dt)<∞,

d) the terminal conditionξ belongs toS_F²[T, T +δ]andP belongs toL²_F[T, T +δ].

Moreover, there exists a constantK such that we have

E^Ft h sup

t≤s≤T

Y_s²+ Z T

t

Z_s²dsi

≤ KE^Ft hξ_T² +

Z T+δ T

(ξ_s²+P_s²)ds+ Z T

t

f(s, ~0)²dsi ,

for any t∈[0, T].

Proof. We use similar arguments to the proofs of Proposition 1.9 and 1.10.

2. ABSDE with jump of type (0.1)

We assume that Hypotheses 1.1, 1.2 and 1.3 hold and thatf(., ~y)is optional. We consider in this section an ABSDE of the following form: find a triple(Y, Z, U)∈ S_G²[0, T +δ]×L²_G[0, T+δ]×L²_τ satisfying















−dYt= f t, Yt,E^Gt[Yt+δ],{E^Gt[Yt+s]}0≤s≤δ, Zt,E^Gt[Zt+δ],{E^Gt[Zt+s]}0≤s≤δ, Ut,E^Gt[Ut+δ],{E^Gt[Ut+s]}_0≤s≤δ

dt−ZtdBt−UtdHt, 0≤t≤T , YT+t= ξT+t, 0≤t≤δ ,

ZT+t= PT+t, UT+t = QT+t1{T+t≤τ}, 0< t≤δ .

From Propositions 1.5 and 1.6, all the involved processes can be decomposed in two parts, before and after τ.

In particular, since ξwill be given as a G-optional process and P as aG-predictable process, we have for any t∈[0, T]

f(t, ~y) = f^b(t, ~y)1{t<τ}+f^a(t, τ, ~y)1{t≥τ} (optional decomposition), and we have for anyt∈[T, T+δ]

(ξ_t= ξ^b_t1{t<τ}+ξ^a_t(τ)1{t≥τ} (optional decomposition) Pt= P_t^b1{t≤τ}+P_t^a(τ)1{t>τ} (predictable decomposition). We work under the following hypotheses:

Hypotheses 2.1. LetA:=R²×D[0, δ]×R²×D[0, δ]×R²×D[0, δ]and, for any~y∈A, we define |~y|by

|~y| = |y|+|ˆy|+|Y|+|z|+|ˆz|+|Z|+|u|+|ˆu|+|U |. a) The terminal conditions satisfyξ∈ S_G²[T, T +δ],P ∈L²

G[T, T+δ],Q∈L²

F[T, T +δ], there exists a constant K such thatE[|ξ^a_u(θ)|²]≤K andE[|P_u^a(θ)|²]≤K for any(θ, u)∈[0, T]×[T, T +δ].

b) The driverf : Ω×[0, T]×A→Rof the ABSDE is Lipschitz, i.e., there exists a constantC such that, for anyt∈[0, T], any~y and~y⁰ in A, we have

|f(t, ~y)−f(t, ~y⁰)| ≤ C|~y−~y⁰|. c) For any~y∈A, the processf(·, ~y)isG-optional.

d) There exists a constantC⁰ such that|f(s, ~0)|< C⁰. From Propositions 1.5 and 1.6, we can write

(Yt= Y_t^b1{t<τ}+Y_t^a(τ)1{t≥τ} (optional decomposition) Z_t= Z_t^b1{t≤τ}+Z_t^a(τ)1{t>τ} (predictable decomposition) It follows, from Lemma 1.8, that















−dY_t^a(τ) = f^a t, τ, Y_t^a(τ),E^Gt[Y_t+δ^a (τ)],{E^Gt[Y_t+s^a (τ)]}_0≤s≤δ, Z_t^a(τ),E^Gt[Z_t+δ^a (τ)], {E^Gt[Z_t+s^a (τ)]}0≤s≤δ,0,0,0

dt−Z_t^a(τ)dB_t, T∧τ ≤t≤T , Y_T+t^a (τ) = ξ_T^a_+t(τ), 0≤t≤δ ,

Z_T+t^a (τ) = P_T^a_+t(τ), 0< t≤δ ,

(2.1)

and















−dY_t^b= f^b t, Y_t^b,E^Ft[Y_t+δ],{E^Ft[Y_t+s]}_0≤s≤δ, Z_t^b,E^Ft[Z_t+δ],{E^Ft[Z_t+s]}_0≤s≤δ, Y_t^a(t)−Y_t^b,E^Ft[Ut+δ],{E^Ft[Ut+s]}_0≤s≤δ

dt−Z_t^bdBt, 0≤t≤T , Y_T^b_+t= ξ_T^b_+t, 0≤t≤δ ,

Z_T^b_+t= P_T^b_+t, U_T+t^b = QT+t, 0< t≤δ .

(2.2)

On the right-hand side of this equation, we still have to make precise ifYt+δ is part of the solution before τ or afterτ, that is to separate the case t+δ < τ and the caset+δ≥τ. Furthermore,Ut=

(Y_t^a(t)−Y_t^b)1{t≤T}+ Q_t1{T <t≤T+δ}

1{t≤τ}.

2.1. Study of the Equation (2.1)

Our aim is to write (2.1) as a family of ABSDEs in the filtrationF. For that purpose, we note that, on the set{t≥τ}, we have from (1.6)

E^Gt[Y_t+δ^a (τ)] = E^Ft[Y_t+δ^a (θ)]_|θ=τ .

The same equality holds for the part involvingf(t, ~y)andZ_t+δ^a (τ). Therefore, we study the family of ABSDE















−dY_t^a(θ) = f^a t, θ, Y_t^a(θ),E^Ft[Y_t+δ^a (θ)],{E^Ft[Y_t+s^a (θ)]}0≤s≤δ, Z_t^a(θ),E^Ft[Z_t+δ^a (θ)], {E^Ft[Z_t+s^a (θ)]}_0≤s≤δ,0,0,0

dt−Z_t^a(θ)dBt, 0≤t≤T , Y_T^a_+t(θ) = ξ^a_T+t(θ), 0≤t≤δ ,

Z_T^a_+t(θ) = P_T+t^a (θ), 0< t≤δ .

(2.3)

For any fixedθ∈[0, T], the mapF :=f^a(θ)defined asF(t, ~y) =f^a(t, θ, ~y)inherits the Lipschitz conditions of Proposition 1.9 from the one off. Due to the boundedness off(·, ~0), the mapF(·, ~0)is also bounded, and satisfies

sup

0≤θ≤TE hZ T

0

f^a(t, θ, ~0)

2dti

< ∞, and the existence of a solution follows from Proposition 1.9.

Using Proposition 1.10, there exists a constantK such that E^Ft

sup

t≤s≤T

(Y_s^a(θ))²+ Z T

t

(Z_s^a(θ))²ds

≤ KE^Ft

(ξ_T^a(θ))²+ Z T+δ

T

(ξ_s^a(θ))²+ (P_s^a(θ))² ds +

Z T t

(f^a(s, θ, ~0))²ds

. (2.4)

2.2. Study of the Equation (2.2)

Our aim is to write (2.2) as an ABSDE in the filtration F, that is to get rid of the quantities involving processes after time τ (as, e.g., Y_t+δ on{t+δ > τ}) and working only with conditional expectation w.r.t. F. Obviously, for anyt≤u≤t+δ, we have

E^Gt[Yu] = E^Gt[Yu1{u<τ}] +E^Gt[Yu1{u≥τ}]. Furthermore, from (1.5), we have

E^Gt[Yu1{u<τ}]1{t<τ} = E^Gt[Y_u^b1{u<τ}]1{t<τ}= 1

GtE^Ft[Y_u^bGu]1{t<τ}, (2.5)

and

E^Gt[Y_u1{u≥τ}]1{t<τ} = E^Gt[Y_u^a(τ)1{u≥τ}]1{t<τ}

= 1

GtE^Ft hZ u

t

Y_u^a(θ)αu(θ)dθi

1{t<τ} =: J_t^Ya(u). (2.6) The same equalities hold for the part involvingZ^a. Then, we introduce, relying on the uniqueness of pre-default parts, the following BSDE which is a transformation of (2.2)











−dY_t^b= g(t, Y_t^b, Y_t+δ^b ,{Y_t+s^b }_0≤s≤δ, Z_t^b, Z_t+δ^b ,{Z_t+s^b }_0≤s≤δ)dt−Z_t^bdB_t, 0≤t≤T , Y_T^b_+t= ξ_T^b_+t, 0≤t≤δ ,

Z_T^b_+t= P_T+t^b , 0< t≤δ .

(2.7)

Here, due to the equalities (2.5) and (2.6),gis the mapΩ×[0, T]×R×L²(F.+δ)×D(·,[0, δ])×R×L²(F.+δ)× D(·,[0, δ])→Rdefined, fory andz inR, ζandη in L²(F.+δ), andY andZ in D(·,[0, δ]), in terms of solution of the equation (2.3) by, for~y= (y, ζ,Y, z, η,Z)

g(t, ~y) = f^b t, ~I_t¹, ~I_t², ~I_t³ where, recalling that the quantitiesJ are defined in (2.6)

I~_t¹ = y, 1

G_tE^Ft[ζGt+δ] +J_t^Ya(t+δ),{ 1

G_tE^Ft(Yt(s)Gt+s) +J_t^Ya(t+s)}_0≤s≤δ , I~_t² = z, 1

G_tE^Ft[ηG_t+δ] +J_t^Za(t+δ),{ 1

G_tE^Ft(Z_t(s)G_t+s) +J_t^Za(t+s)}_0≤s≤δ , I~_t³ = Y_t^a(t)−y, 1

GtE^Ft[1{t+δ≤T}(Y_t+δ^a (t+δ)−ζ)G_t+δ+1{t+δ>T}Q_t+δG_t+δ], 1

Gt

n

E^Ft[(Y_t+s^a (t+s)− Yt(s))Gt+s1{t+s≤T}+Qt+sGt+s1{t+s>T}]o

0≤s≤δ

.

It is straightforward thatg isF-optional. We now show thatg satisfies Lipschitz conditions recalled in Propo- sition 1.9.

Since we have

f^b(t, ~y) = 1

G_tE^Ft[f(t, ~y)1{t<τ}],

we obtain that, using the Lipschitz condition forf and thatGis bounded, there exists a constantKsuch that

|g(t, ~y)−g(t, ~y⁰)| ≤ K G_t

(|y−y⁰|+|z−z⁰|)E^Ft(1{t<τ}) +E^Ft[(|ζ−ζ⁰|+|η−η⁰|)Gt+δ1{t<τ}] +E^Ft

(|YG− Y⁰G|+|ZG− Z⁰G|)1{t<τ}

. Since, forX ∈ Fsands > t, one has from (1.4)

E^Ft[X1{t<τ}] = E^Ft[XE^Fs(1{t<τ})] = E^Ft[XG_t] = G_tE^Ft[X],

we deduce

|g(t, ~y)−g(t, ~y⁰)| ≤ K

|y−y⁰|+|z−z⁰|+E^Ft[(|ζ−ζ⁰|+|η−η⁰|)G_t+δ] +E^Ft[|YG− Y⁰G|+|ZG− Z⁰G|]

.

Noting Gis upper bounded by 1, the Lipschitz property of Proposition 1.9 forg holds.

We now check the integrability condition on |g(t, ~0)|². We notice, using notation (2.6), we have g(t, ~0) = f^b

t, 0, J_t^Ya(t+δ),n

J_t^Ya(t+s)o

0≤s≤δ,0, J_t^Za(t+δ),n

J_t^Za(t+s)o

0≤s≤δ, Y_t^a(t), 1

G_t E^Ft[Y_t+δ^a (t+δ)Gt+δ1{t+δ≤T}+Qt+δGt+δ1{t+δ>T}], n 1

GtE^Ft[Y_t+s^a (t+s)G_t+s1{t+s≤T}+Q_t+sG_t+s1{t+s>T}]o

0≤s≤δ

.

From Lipschitz property off, sincef(t, ~0)is bounded andGt=E^Ft(1{t<τ}), we have f^b(t, ~y) ≤ 1

Gt

E^Ft[(f(t, ~0) +C|~y|)1{t<τ}]

≤ C₁+C|~y|.

Using again that the square of a sum is bounded (up to a constant) by the sum of the squares, and using again the fact that G is lower bounded, the integrability condition of |g(t, ~0)|² will follow from the boundedness of the quantities

E Z T

0

J_t^Y(t+δ)² dt

! , E

Z T 0

Z δ 0

(J_t^Y(t+s))²ds dt

!

(2.8) and similar expressions withJ^Z, as well as

















 E

RT

0 E^Ft(Y_t+δ^a (t+δ))² dt E

RT

0 (Y_t^a(t))²dt E

RT+δ

T E^Ft(Qt)² dt E

RT 0

Rδ

0 Y_t+s^a (t+s)²1t+s≤T +Q²_t+s1^t+s>T ds

dt . The quantities in (2.8) are bounded since αis bounded and

Z T+δ 0

dθ Z T

0

1{t<θ<t+δ}E((Y_t+δ^a (θ))²)dt

is bounded since sup

0≤θ≤TE(( sup

0≤s≤T

Y_s^a(θ))²) ≤ K sup

0≤θ≤TE

(ξ^a_T(θ))²+ Z T+δ

T

(ξ_s^a(θ))²+ (P_s^a(θ))² ds +

Z T 0

(f^a(s, θ, ~0))²ds

and the assumed boundedness of P andξ. The other quantities are studied using the same methodology and that Q∈L²[T, T+δ].

The existence of a unique solution(Y^b, Z^b)of the ABSDE (2.7) follows from Proposition 1.11. Moreover we have

E^Ft

sup

t≤s≤T

(Y_s^b)²+ Z T

t

(Z_s^b)²ds

≤ KE^Ft

|ξ^b_T|²+ Z T+δ

T

(ξ_s^b)²+ (P_s^b)² du +

Z T t

(f^b(s, ~0))²ds

. (2.9)

2.3. Integrability of the solutions

In this part we consider the integrability of the solutions(Y, Z, U)where Y_t = Y_t^b1{t<τ}+Y_t^a(τ)1{t≥τ}, Zt = Z_t^b1{t≤τ}+Z_t^a(τ)1{t>τ}, Ut = (Y_t^a(t)−Y_t^b)1{t≤τ}. From Subsections 2.1 and 2.2 we know(Y, Z, U)satisfy the ABSDE (0.1).

Proposition 2.2. The processU belongs toL²_τ. Proof. We have

E

hZ (T+δ)∧τ 0

U_s²dsi

= E

hZ T∧τ 0

(Y_s^a(s)−Y_s^b)²dsi

+hZ (T+δ)∧τ T∧τ

Q²_sdsi

≤ 2E hZ T

0

(Y_s^a(s))²dsi + 2E

hZ T 0

(Y_s^b)²dsi

+hZ T+δ T

Q²_sdsi

≤ 2 Z T

0

E h

(Y_s^a(s))²i

ds+ 2TE sup

0≤t≤T

(Y_t^b)² +E

hZ T+δ T

Q²_sdsi

and the quantities on the right-hand side are finite.

Proposition 2.3. There exists a strictly positive constant K such that the solution (Y, Z, U) of the ABSDE (0.1) satisfies

E^Gt h

sup

t≤s≤T

Y_s²+ Z T

t

Z_s²dsi

≤ KE^Ft h

(ξ_T^b)²+ Z T+δ

T

(ξ_s^b)²+ (P_s^b)² ds+

Z T t

(f^b(s, ~0))²dsi

+ K

αt(τ)E^Ft h

(ξ_T^a(θ))²+ Z T+δ

T

((ξ_s^a(θ))²+ (P_s^a(θ))²)ds+ Z T

t

(f^a(s, θ, ~0))²dsi

θ=τ1{τ <t}

+ K1{t≤τ}E^Ft hZ T

t

n

(ξ_T^a(θ))²+ Z T+δ

T

((ξ^a_s(θ))²+ (P_s^a(θ))²)ds+ Z T

t

(f^a(s, θ, ~0))²dso dθi

for any t∈[0, T].

Proof. In the proof, the constantKcan vary from line to line. We remark ¹ E^Gt

h sup

t≤s≤T

Y_s²+ Z T

t

Z_s²dsi

= E^Gt h

sup

t≤s≤T

Y_s²+ Z T∧τ

t

(Z_s^b)²ds+ Z T

T∧τ

(Z_s^a(τ))²dsi

≤ E^Gt h

sup

t≤s≤T

Y_s²+ Z T

t

(Z_s^b)²ds+ Z T

T∧τ

(Z_s^a(τ))²dsi . On the set{τ < t}, sinceλis bounded (Hypothesis 1.3), we use that

E^Gt h

sup

t≤s≤T

Y_s²i

= E^Gt h

sup

t≤s≤T

(Y_s^a(τ))²i

= 1 G_tE^Ft

h sup

t≤s≤T

(Y_s^a(θ))²α_T(θ)i

≤ ke^ktE^Ft h sup

t≤s≤T

(Y_s^a(θ))²i

≤KE^Ft h sup

t≤s≤T

(Y_s^a(θ))²i . On the set{t≤τ}, we remark

E^Gt h

sup

t≤s≤T

Y_s²i

≤ E^Ft h

sup

t≤s≤T

(Y_s^b)²i +E^Gt

h sup

T∧τ≤s≤T

(Y_s^a(τ))²i .

FromE^Gt h

sup_{T∧τ≤s≤T}(Y_s^a(τ))²i

= _α¹

t(τ)E^Ft h

sup_{T∧θ≤s≤T}(Y_s^a(θ))²αT(θ)i

θ=τ and that αis bounded, we have E^Gt

h sup

T∧τ≤s≤T

(Y_s^a(τ))²i

≤ K αt(τ)E^Ft

h sup

T∧θ≤s≤T

(Y_s^a(θ))²i

θ=τ. We proceed in the same way for the partRT

T∧τ(Z_s^a(τ))²ds.

Using (2.4)-(2.9) we can conclude.

2.4. Uniqueness of the solution

In this part we are concerned with the uniqueness of the solution of ABSDE (0.1). Suppose this AB- SDE has two solutions (Y, Z, U) and ( ¯Y ,Z,¯ U¯). Each process admits a unique decomposition under the form (Y^b, Z^b, U^b)-(Y^a(τ), Z^a(τ))and ( ¯Y^b,Z¯^b,U¯^b)-( ¯Y^a(τ),Z¯^a(τ)). Moreover we know (Y^b, Z^b)and( ¯Y^b,Z¯^b)are so- lution of ABSDE (2.2), thus by uniqueness of the solution of ABSDE (2.2) from Proposition 1.11 we get that Y^b = ¯Y^b and Z^b = ¯Z^b. We have with the same argumentsY^a(τ) = ¯Y^a(τ) andZ^a(τ) = ¯Z^a(τ). Moreover we have Ut= (Y_t^a(t)−Y_t^b)1{t≤τ}, thusU = ¯U. Finally we get the uniqueness of the solution of ABSDE (0.1).

3. ABSDE with jump of type (0.2)

We assume that Hypotheses 1.1, 1.2 and 1.3 hold. We define, for any t ∈ [0, T], A_t = R×L²(Ft+δ)× D(t,[0, δ])×R×L²(Ft+δ)×D(t,[0, δ])×R×L²(Ft+δ)×D(t,[0, δ]). We consider in this section an ABSDE of the following form: find a triple(Y, Z, U)∈ S²

G[0, T +δ]×L²

G[0, T +δ]×L²_τ satisfying











−dYt= E^Gt

f(t, Yt, Yt+δ,{Yt+s}_0≤s≤δ, Zt, Zt+δ,{Zt+s}_0≤s≤δ, Ut, Ut+δ, {Ut+s}_0≤s≤δ)]dt−ZtdBt−UtdHt, 0≤t≤T ,

YT+s= ξT+s, 0≤s≤δ

ZT+s= PT+s, UT+s = QT+s1{T+t≤τ}, 0≤s≤δ ,

(3.1)

1with the conventionRb

a.ds= 0ifb < a