S.Hamad`ene&M.HassaniLaboratoiredeStatistiqueetProcessus,Universit´eduMaine,72085LeMansCedex9,France. B SDEswithtworeflectingbarriersdrivenbyaBrownianandaPoissonnoiseandrelatedDynkingame

El e c t ro nic

Jo ur n a l o f

Pr

o ba b i l i t y

Vol. 11 (2006), Paper no. 5, pages 121–145.

Journal URL

http://www.math.washington.edu/∼ejpecp/

BSDEs with two reflecting barriers driven by a Brownian and a Poisson noise and related Dynkin game

S. Hamad`ene & M. Hassani Laboratoire de Statistique et Processus,

Universit´e du Maine, 72085 Le Mans Cedex 9, France.

e-mails: hamadene@univ-lemans.fr & mohammed.hassani@univ-lemans.fr

Abstract

In this paper we study BSDEs with two reflecting barriers driven by a Brownian motion and an independent Poisson process. We show the existence and uniqueness of local and global solutions. As an application we solve the related zero-sum Dynkin game.

Key words: Backward stochastic differential equation ; Poisson measure ; Dynkin game ; Mokobodzki’s condition.

.

AMS 2000 Subject Classification: Primary 91A15, 60G40, 91A60; Secondary 60G99.

Submitted to EJP on December 23, 2003. Final version accepted on January 25, 2006.

1 Introduction

Let (Ft)_t≤T be a filtration generated by a Brownian motion (Bt)_t≤T and an independent Pois- son measure µ(t, ω, de) defined on a probability space (Ω,F, P). A solution for the backward stochastic differential equation (BSDE for short) with two reflecting barriers associated with a coefficient f(t, ω, y, z), a terminal value ξ and a lower (resp. an upper) barrier (Lt)_t≤T (resp.

(Ut)_t≤T) is a quintuple of Ft-predictable processes (Yt, Zt, Vt, K_t⁺, K_t⁻)_t≤T which satisfies:











−dY_t=f(t, Y_t, Z_t, V_t)dt+dK_t⁺−dK_t⁻−Z_tdB_t− Z

E

V_t(e)˜µ(dt, de), t≤T ; Y_T =ξ Lt≤Yt≤Ut,∀t≤T

K^± are continuous non-decreasing and (Yt−Lt)dK_t⁺= (Yt−Ut)dK_t⁻= 0 (K₀^±= 0), (1)

where ˜µ is the compensated measure ofµ.

Nonlinear BSDEs have been first introduced by Pardoux and Peng [19], who proved the existence and uniqueness of a solution under suitable hypotheses on the coefficient and the terminal value of the BSDE. Since, these equations have gradually become an important mathematical tool which is encountered in many fields such as finance ([6], [4], [8], [9],...), stochastic games and optimal control ([10], [11], ...), partial differential equations ([1], [18], [20],...).

In the case when the filtration is generated only by a Brownian motion and when we consider just one lower barrier (setU ≡+∞andK⁻ ≡0 in (1)), the problem of existence and uniqueness of a solution for (1) is considered and solved by El-Karoui et al. in [6]. Their work has been generalized by Cvitanic & Karatzas in [4] where they deal with BSDEs with two reflecting barriers.

BSDEs without reflection (in (1) one should take L = −∞ and U = +∞, thereby K^± = 0) driven by a Brownian motion and an independent Poisson measure have been considered first by Tang & Li [17] then by Barles et al. in connection with partial-integral differential equations in [1]. In both papers the authors showed the existence and uniqueness of a solution.

The extension to the case of BSDEs with one reflecting barrier has been established by Hamad`ene

& Ouknine in [13]. The authors showed the existence and uniqueness of the solution when the coefficientf is Lipschitz. Two proofs have been given, the first one is based on the penalization scheme as for the second, it is obtained in using the Snell envelope notion. However both methods make use of a contraction argument since the usual comparison theorem fails to work in the general framework.

So in this work we study BSDEs with two reflecting barriers driven by a Brownian motion and an independent Poisson measure. This is the natural extension of Hamad`ene & Ouknine’s work.

However there are at least four motivations for considering this problem. The first one is related to Dynkin zero-sum game. The second is in connection with the real option area since the stopping and starting problem leads to a BSDE with two reflecting barriers (see e.g. [12]). The third one is that our problem can provide solutions for variational inequality problems with two obstacles when the generator is of partial-integral type. Finally we provide a condition easy to check in practice under which the well known Mokobodski’s hypothesis is satisfied.

In this paper we begin to show the existence of an adapted andrcll (right continuous and left limited) processY := (Yt)_t≤T which in a way is alocal solutionfor (1). Actually we prove that for any stopping timeτ there exist another stopping timeθ_τ ≥τ and a quadruple of processes (Z, V, K⁺, K⁻) which withY verify (1) on the interval [τ, θτ]. In addition the processY reaches the barriers U and Lbetween τ and θτ. In the proof of our theorem, the key point is that the predictable projection of a process π whose jumping times are inaccessible is equal to π₋, the process of left limits associated withπ.

This result is then applied to deal with the zero-sum Dynkin game associated withL,U,ξ and a process (gs)_s≤T which stands for the instantaneous payoff. We show that this game is closely related to the notion of localsolution for (1). Besides we obtain the existence of a saddle-point for the game under conditions out of the scope of the known results on this subject. Finally we give some feature of the value function of the game (see Remark 4.3). Our result can be applied in mathematical finance to deal with American game (or recallable) options whose underlying derivatives contain a Poisson part (see [15] for this type of option).

Further we consider the problem of existence and uniqueness of a global solution for (1). When the weak Mokobodski’s assumption [WM] is satisfied, which roughly speaking turns into the existence of a difference of non-negative supermartingales betweenL andU, we show existence and uniqueness of the solution. Then we address the issue of the verification of the condition [WM]. Actually we prove that under the fully separation of the barriers, i.e., Lt< Ut for any t≤T, the condition[WM] holds true.

This paper is organized as follows :

In Section 2, we deal withlocal solutions for (1) while Section 3 is devoted to zero-sum Dynkin games. At the end, in Section 4, we address the problem of existence of a global solution for (1). 2

2 Reflected BSDEs driven by a Brownian motion and an inde- pendent Poisson point process

Let (Ω,F,(Ft)_t≤T) be a stochastic basis such that F₀ contains all P-null sets of F, F_t+ = T

²>0F_t+² = F_t, ∀t < T, and suppose that the filtration is generated by the following two mutually independent processes :

- ad-dimensional Brownian motion (Bt)_t≤T

- a Poisson random measureµonIR⁺×E, whereE :=IR^l\ {0}is equipped with its Borel fields E, with compensator ν(dt, de) = dtλ(de), such that {˜µ([0, t]×A) = (µ−ν)([0, t]×A)}_t≤T is a martingale for every A ∈ E satisfying λ(A) <∞. The measure λis assumed to be σ-finite on (E,E) and verifies^R_E(1∧ |e|²)λ(de)<∞.

Now let:

- D be the set of F_t-adapted right continuous with left limits processes (Y_t)_t≤T with values in IR and D²:={Y ∈ D, IE[sup_t≤T|Yt|²]<∞}

- ˜P (resp. P) be theFt-progressive (resp. predictable) tribe on Ω×[0, T]

- H^2,k (resp. H^k) be the set of ˜P-measurable processes Z := (Zt)_t≤T with values in IR^k and dP ⊗dt-square integrable (resp. P-a.s. Z(ω) := (Zt(ω))_t≤T isdt-square integrable)

- L² (resp. L) be the set of mappings V : Ω×[0, T]×E →IRwhich areP ⊗ E-measurable and IE[^R₀^T kV_sk²ds]<∞(resp. ^R₀^T kV_sk²ds <∞, P-a.s.) wherekvk:= (^R_E|v(e)|²λ(de))¹² forv:E → IR

- C_ci² (resp. C_ci) the space of continuous F_t-adapted and non-decreasing processes (k_t)_t≤T such thatk₀= 0 and IE[k_T²]<∞(resp. kT <∞, P-a.s.)

- for a stopping timeτ,Tτ denotes the set of stopping times θ such thatθ≥τ

- for a given rcll process (w_t)_t≤T, w_t− = lim_s%tw_s, t ≤ T (w₀₋ = w₀) ; w₋ := (w_t−)_t≤T and 4w:=w−w₋ 2

We are now given four objects:

- a terminal valueξ ∈L²(Ω,F_T, P)

- a map f : Ω ×[0, T]× IR^1+d ×L²(E,E, λ;IR) → IR which with (ω, t, y, z, v) associates f(ω, t, y, z, v), ˜P ⊗ B(IR^1+d)⊗ B(L²(E,E, λ;IR))-measurable and satisfying :

(i) the process (f(t,0,0,0))_t≤T belongs to H^2,1

(ii) f is uniformly Lipschitz with respect to (y, z, v), i.e., there exists a constant C ≥ 0 such that for anyy,y⁰,z,z⁰ ∈IR andv, v⁰ ∈L²(E,E, λ;IR),

P −a.s., |f(ω, t, y, z, v)−f(ω, t, y⁰, z⁰, v⁰)| ≤C(|y−y⁰|+|z−z⁰|+kv−v⁰k)

- two obstaclesL:= (Lt)_t≤T and U := (Ut)_t≤T which areFt−progressively measurablercll, real valued processes satisfying Lt≤Ut, ∀t≤T and LT ≤ξ ≤UT, P-a.s.. In addition they belong toD²,i.e.,

IE[ sup

0≤t≤T

{|Lt|+|Ut|}²]<∞.

Besides we assume that their jumping times occur only at inaccessible stopping times which roughly speaking means that they are not predictable (see e.g. [2], pp.215 for the accurate definition). If this latter condition is not satisfied and especially if the upper (resp. lower) barrier U (resp. L) has positive (resp. negative) jumps then Y could have predictable jumps and the processes K^± would be no longer continuous. Therefore the setting of the problem is not the same as in (1).

Let us now introduce our two barrier reflected BSDE with jumps associated with (f, ξ, L, U).

A solution is a 5-uple (Y, Z, V, K⁺, K⁻) := (Y_t, Z_t, V_t, K_t⁺, K_t⁻)_t≤T of processes with values in IR^1+d×L²(E,E, λ;IR)×IR⁺×IR⁺ such that:























(i) Y ∈ D², Z∈ H^d, V ∈ Land K^±∈ Cci

(ii) Y_t=ξ+ Z T

t

f(s, Y_s, Z_s, V_s)ds+ (K_T⁺−K_t⁺)−(K_T⁻−K_t⁻)

− Z T

t

ZsdBs− Z T

t

Z

E

Vs(e)˜µ(ds, de),∀t≤T (iii) ∀t≤T, Lt≤Yt≤Ut and

Z T 0

(Yt−Lt)dK_t⁺= Z T

0

(Yt−Ut)dK_t⁻= 0.

(2) Note that equation (2) has not a solution in general. Actually one can take L=U with L not being a semimartingale, then obviously we cannot find a 5-uple which satisfies the relation (ii).

2

2.1 BSDEs with one reflecting barrier

To begin with we recall the following result by Hamad`ene & Ouknine [13] related to reflected BSDEs with one upper barrier (in (2),L≡ −∞andK⁺= 0) driven by a Brownian motion and an independent Poisson process.

Theorem 2.1 : There exits a quadruple (Y, Z, K, V) := (Yt, Zt, Kt, Vt)_t≤T of processes with values in IR^1+d×IR⁺× L² which satisfies :















Y ∈ D², Z ∈ H^2,d, K ∈ C_ci² and V ∈ L² Yt=ξ+

Z T t

f(s, Ys, Zs, Vs)ds−(K_T −Kt)− Z T

t

ZsdBs− Z T

t

Z

U

Vs(e)˜µ(ds, de), t≤T

∀t≤T, Yt≤Ut and Z T

0 (Ut−Yt)dKt= 0.2

(3) In general we do not have a comparison result for solutions of BSDEs driven by a Brownian motion and an independent Poisson process, reflected or not (see e.g. [1] for a counter-example).

However in some specific cases, when the coefficients have some features and especially when they do not depend on the variable v, we actually have comparison.

So let us give another pair (f⁰, ξ⁰) where f⁰ : (ω, t, y, z, v) 7−→ f⁰(ω, t, y, z, v) ∈ IR and ξ⁰ ∈ L²(Ω,FT, P;IR). On the other hand, assume there exists a quadruple of processes (Y⁰, Z⁰, V⁰, K⁰) which belongs toD²×H^2,d×L²×C_ci² and solution for the BSDE with one reflecting upper barrier associated with (f⁰(ω, t, y, z, v), ξ⁰, U). Then we have:

Lemma 2.1 : Assume that : (i) f is independent ofv

(ii) P-a.s. for anyt≤T, f(t, Y_t⁰, Z_t⁰)≤f⁰(s, Y_t⁰, Z_t⁰, V_t⁰) and ξ≤ξ⁰. Then P-a.s., ∀t≤T, Yt≤Y_t⁰.

Proof: Let X = (Xt)_t≤T be a rcll semi-martingale, then using Tanaka’s formula with the function (x⁺)²= (max{x,0})² reads:

(X_t⁺)² = (X_T⁺)²−2 Z T

t

X_s−⁺ dX_s− Z T

t

1_[Xs>0]d < X^c, X^c >_s− ^X

t<s≤T

{(X_s⁺)²−(X_s−⁺)²−2X_s−⁺∆X_s}.

But the function x∈IR7→(x⁺)² is convex then{(X_s⁺)²−(X_s−⁺)²−2X_s−⁺ ∆Xs} ≥0. It implies that

(X_t⁺)²+ Z T

t

1_[Xs>0]d < X^c, X^c >s≤(X_T⁺)²−2 Z T

t

X_s−⁺ dXs. Now using this formula withY −Y⁰ yields:

((Yt−Y_t⁰)⁺)²+ Z T

t

1_[Ys−Y0

s>0]|Zs−Z_s⁰|²ds≤ −2 Z T

t

(Y_s−−Y_s−⁰ )⁺d(Ys−Y_s⁰).

But^R_t^T(Y_s−−Y_s−⁰ )⁺d(Ks−K_s⁰) =^R_t^T(Ys−Y_s⁰)⁺d(Ks−K_s⁰) and as usual this last term is non- negative since (Ys−Y_s⁰)⁺dK_s⁰ = 0 (indeed when Yt > Y_t⁰ we cannot have Y_t⁰ = Ut). Besides

f(s, Y_s⁰, Z_s⁰) ≤ f⁰(s, Y_s⁰, Z_s⁰, V_s⁰) and f is Lipschitz continuous then there exist bounded and Ft- adapted processes (as)_s≤T and (bs)_s≤T such that:

f(s, Ys, Zs) =f(s, Y_s⁰, Z_s⁰) +as(Ys−Y_s⁰) +bs(Zs−Z_s⁰).

Therefore we have:

((Yt−Y_t⁰)⁺)²+^R_t^T1_[Ys−Y0

s>0]|Zs−Z_s⁰|²ds≤2^R_t^T(Y_s−−Y_s−⁰ )⁺{as(Ys−Y_s⁰) +bs(Zs−Z_s⁰)}ds

−2^R_t^T(Y_s−⁰ −Y_s−)⁺{(Zs−Z_s⁰)dBs+^R_EVs(e)˜µ(ds, de)}.

Taking now expectation, using in an appropriate way the inequality|a.b| ≤²|a|²+²⁻¹|b|²(² >0), and Gronwall’s one we obtainIE[((Y_t−Y_t⁰)⁺)²] = 0 for anyt≤T. The result follows thoroughly sinceY andY⁰ arercll. 2

2.2 Local solutions of BSDEs with two reflecting barriers

Throughout this part we assume that the function f does not depend on v. The main reason is, as pointed out in Lemma 2.1, in that case we can use comparison in order to deduce results for the two reflecting barrier BSDE associated with (f, ξ, L, U). Actually we have the following result related to the existence of local solutions for (2):

Theorem 2.2 : There exists a unique Ft-optional process Y := (Yt)_t≤T such that:

(i) YT =ξ and P-a.s. for any t≤T, Lt≤Yt≤Ut

(ii) for any Ft-stopping time τ there exists a quintuple (θτ, Z^τ, V^τ, K^τ,+, K^τ,−) which belongs to T_τ × H^2,d× L²× C_ci² × C²_ci such that: P-a.s.,

E(f, ξ, L, U) :



















∀t∈[τ, θτ], Yt=Yθτ + Z θτ

t

f(s, Ys, Z_s^τ)ds+ Z θτ

t

d(K_s^τ,+−K_s^τ,−)

− Z θτ

t

Z_s^τdB_s− Z θτ

t

Z

E

V_s^τ(e)˜µ(ds, de), Z θτ

τ

(Us−Ys)dK_s^τ,− = Z θτ

τ

(Ys−Ls)dK_s^τ,+ = 0

(4)

(iii) if we set ντ := inf{s≥τ, Ys=Us} ∧T and στ := inf{s≥τ, Ys=Ls} ∧T then ντ ∨στ ≤ θτ, Yντ =Uντon[ντ < T]andYστ =Lστon[στ < T].

Hereafter we call the process Y := (Yt)_t≤T a local solution for the BSDE with two reflecting barriers associated with (f, ξ, L, U) which we denote E(f, ξ, L, U).

P roof: Let us show uniqueness. Let Y and Y⁰ be two F_t-optional processes which satisfy (i)−(iii). Then for any stopping timeτ we have :

Yστ∧ν_τ⁰ =Yστ1_[στ≤ν0

τ]+Yν⁰_τ1_[ν0 τ<στ]

=Yστ1_[στ≤ν0

τ]∩[στ<T]+Yστ1_[στ=ν0

τ=T]+Y_ν0 τ1_[ν0

τ<στ]

≤Lστ1_[στ≤ν0

τ]∩[στ<T]+ξ1_[στ=ν0

τ=T]+Uν_τ⁰1_[ν0 τ<στ]

=L_στ1_[στ≤ν0

τ]∩[στ<T]+ξ1_[στ=ν0

τ=T]+Y_ν⁰0 τ1_[ν0

τ<στ]

≤Y_σ⁰

τ∧ντ⁰.

Besides P-a.s.for anyt∈[τ, στ∧ν_τ⁰] we have:

Yt=Yστ∧ντ⁰ +

Z στ∧ν_τ⁰ t

f(s, Ys, Z_s^τ)ds−

Z στ∧ν_τ⁰ t

dK_s^τ,−−

Z στ∧ν_τ⁰ t

{Z_s^τdBs+ Z

E

V_s^τ(e)˜µ(ds, de)}

and

Y_t⁰=Y_σ⁰_τ∧ν0 τ +

Z στ∧ν⁰_τ t

f(s, Y_s⁰, Z_s^0τ)ds+

Z στ∧ν_τ⁰ t

dK_s^0τ,+−

Z στ∧ν_τ⁰ t

{Z_s^0τdBs+ Z

E

V_s^0τ(e)˜µ(ds, de)}

since fort∈[τ, στ∧ν_τ⁰],dK_t^τ,+= 0 anddK_t^0τ,−= 0. Now arguing as in the proof of Lemma 2.1, for anyt≤T we obtain:

((Y_{(t∨τ)∧(στ∧ν}0

τ)−Y_{(t∨τ)∧(σ}⁰

τ∧ντ⁰))⁺)²+

Z (στ∧ντ⁰)

(t∨τ)∧(στ∧ν⁰τ)1_[Ys−Y0

s>0]|Z_s^τ−Z_s^0τ|²ds

≤ −2

Z _(στ∧ντ⁰₎

(t∨τ)∧(στ∧ν_τ⁰)

(Ys−Y_s⁰)⁺d(Ys−Y_s⁰)

≤2

Z (στ∧ν_τ⁰)

(t∨τ)∧(στ∧ντ⁰)(Y_s−−Y_s−⁰ )⁺{(f(s, Ys, Z_s^τ)−f(s, Y_s⁰, Z_s^0τ))ds−(dK_s^τ,−+dK_s^0τ,+)}

+M_(στ∧ν0

τ)−M_{(t∨τ)∧(στ∧ν}0

τ), whereM is a martingale.

But

Z _(στ∧ντ⁰₎

(t∨τ)∧(στ∧ν⁰_τ)

(Ys−Y_s⁰)⁺(dK_s^τ,−+dK_s^0τ,+) ≥ 0 then by the same lines as in the proof of Lemma 2.1, we get P-a.s. for any t≤ T, Y_{(t∨τ)∧(στ∧ν}0

τ) ≤ Y_{(t∨τ)∧(σ}⁰

τ∧ντ⁰). Taking now t = 0 to obtain Yτ ≤Y_τ⁰. However in a symmetric way we have alsoY_τ⁰ ≤Yτ and thenY_τ⁰ =Yτ. Finally the optional section theorem ([2], pp.220) implies that Y ≡Y⁰.

The proof of existence of Y will be obtained after Lemmas 2.2 & 2.4 below. So to begin with, forn≥0, let (Yⁿ, Zⁿ, K^n,−, Vⁿ) be the solution of the following reflected BSDE with just one upper barrier U (which exists according to Theorem 2.1):























(i) Yⁿ∈ D², Zⁿ∈ H^2,d, K^n,−∈ C_ci² and Vⁿ∈ L² (ii)Y_tⁿ=ξ+

Z T t

f(s, Y_sⁿ, Z_sⁿ)ds+ Z T

t

d(K_s^n,+−K_s^n,−)

− Z T

t

Z_sⁿdBs− Z T

t

Z

E

V_sⁿ(e)˜µ(ds, de),∀t≤T (iii)Yⁿ≤U and

Z T 0

(Us−Y_sⁿ)dK_s^n,−= 0 ; K_t^n,+:=n Z t

0

(Ls−Y_sⁿ)⁺ds.

(5)

Using comparison, we have for any n ≥ 0, P-a.s., and for all t ≤ T, Y_tⁿ ≤Y_tⁿ⁺¹ ≤ U_t. Then for any t ≤T let us set Yt := lim_n→∞Y_tⁿ. Therefore Y is an Ft-optional process since Yⁿ is so and obviously we have P-a.s., ∀ t ≤ T, Yt ≤ Ut. Besides, if we denote ^pX the predictable projection of a process X then fort ≤T, ^pY_t = lim_n→∞^pY_tⁿ = lim_n→∞Y_t−ⁿ. Indeed since the jumping times ofYⁿare the same as the ones of

Z t 0

Z

E

V_sⁿ(e)˜µ(de, ds), then they are inaccessible and ^pYⁿ=Y₋ⁿ ([3], pp.113).

Now for a stopping time τ, let ν_τⁿ := inf{s≥τ, Y_sⁿ=U_s} ∧T. Since Yⁿ ≤ Yⁿ⁺¹ then the sequence (ν_τⁿ)_n≥0 is non-increasing and converges to the stopping time δτ := lim_n→∞ν_τⁿ.

Lemma 2.2 : The following properties hold true : (i) for any stopping time τ, Yδτ =Uδτ1_[δτ<T]+ξ1_[δτ=T]. (ii) P-a.s. for any t≤T, Ut≥Yt≥Lt

(iii) for any stopping timeτ, there is a triple( ˜Z^τ,V˜^τ,K˜^τ,+)∈ H^2,d× L²× C_ci² such that: P-a.s.,



















∀t∈[τ, δ_τ], Y_t=Y_δτ + Z δτ

t

f(s, Y_s,Z˜_s^τ)ds+ Z δτ

t

dK˜_s^τ,+

− Z δτ

t

Z˜_s^τdBs− Z δτ

t

Z

E

V˜_s^τ(e)˜µ(ds, de), Z δτ

τ

(Y_s−L_s)dK˜_s^τ,+ = 0.

(6)

P roof: From equation (5) we have: P-a.s.,∀t∈[τ, ν_τⁿ]











Y_tⁿ=Y_νⁿⁿ

τ + Z ν_τⁿ

t

f(s, Y_sⁿ, Z_sⁿ)ds+ Z νⁿ_τ

t

n(Ls−Y_sⁿ)⁺ds

− Z νⁿ_τ

t

Z_sⁿdBs− Z ν_τⁿ

t

Z

E

V_sⁿ(e)˜µ(ds, de)

(7)

since the process K^n,− increases only when Yⁿ reaches the barrier U. Now for any n ≥ 0, Y⁰ ≤Yⁿ≤U then sup_n≥0IE[sup_t≤T|Y_tⁿ|²]<∞sinceY⁰ andU belong toD². Next using Itˆo’s formula with (Yⁿ)² we get : for anyt≤T,

(Y_{(t∨τ)∧ν}ⁿ n τ)² +

Z ν_τⁿ (t∨τ)∧ντⁿ

|Z_sⁿ|²ds+ ^X

(t∨τ)∧νⁿτ<s≤ντⁿ

(∆sYⁿ)² = (Y_νⁿn τ)²+ 2

Z νⁿ_τ (t∨τ)∧ντⁿ

Y_s−ⁿf(s, Y_sⁿ, Z_sⁿ)ds +2

Z ν_τⁿ

(t∨τ)∧τ_τⁿY_s−ⁿn(Ls−Y_sⁿ)⁺ds−2 Z νⁿ_τ

(t∨τ)∧ν_τⁿY_s−ⁿ{Z_sⁿdBs+ Z

E

V_sⁿ(e)˜µ(ds, de)}

≤(Y_νⁿn τ)²+ 2

Z νⁿ_τ (t∨τ)∧ντⁿ

Y_s−ⁿf(s, Y_sⁿ, Z_sⁿ)ds+²⁻¹sup

s≤T

|Y_sⁿ|²+

²{

Z ν_τⁿ

(t∨τ)∧νⁿ_τ n(Ls−Y_sⁿ)⁺ds}²−2 Z νⁿ_τ

(t∨τ)∧ν_τⁿY_s−ⁿ{Z_sⁿdBs+ Z

E

V_sⁿ(e)˜µ(ds, de)}

(8) for any² >0. But (7) implies the existence of a constatC≥0 such that for anyt≤T we have:

( Z ν_τⁿ

(t∨τ)∧ντⁿ

n(Ls−Y_sⁿ)⁺ds)²≤C{(Y_{(t∨τ)∧ν}ⁿ n

τ)²+ (Y_νⁿn τ)²+ (

Z ν_τⁿ (t∨τ)∧ντⁿ

f(s, Y_sⁿ, Z_sⁿ)ds)² +(

Z ν_τⁿ (t∨τ)∧ν_τⁿ

Z_sⁿdBs− Z νⁿ_τ

(t∨τ)∧ν_τⁿ

Z

E

V_sⁿ(e)˜µ(ds, de))²}.

(9)

Taking now expectation in both hand-sides, making use of the Burkholder-Davis-Gundy inequal- ity (see e.g. [3], pp.304) and the estimate sup_n≥0IE[sup_t≤T |Y_tⁿ|²]<∞, and finally taking into account the fact thatf is Lipschitz yield: ∀t≤T,

IE[(

Z ν_τⁿ

(t∨τ)∧ν_τⁿn(Ls−Y_sⁿ)⁺ds)²]≤C1(1 +IE[

Z ν_τⁿ τ

{|Z_sⁿ|²+kV_sⁿk²}ds]) (10) for some constantC₁. Next taking expectation in (8), plug (10) in (8), and using the inequality

∀δ >0,|Y_s−ⁿf(s, Y_sⁿ, Z_sⁿ)| ≤δC_f|Z_sⁿ|²+C_fδ⁻¹sup_s≤T(|Us|∨|Y_s⁰|)²+|f(s,0,0)|sup_s≤T(|Us|∨|Y_s⁰|)

(Cf is the Lipschitz constant off), we obtain by taking t= 0 and after an appropriate choice of ²andδ,

∀n≥0, IE[

Z ν_τⁿ τ

(|Z_sⁿ|²+kV_sⁿk²)ds]≤C₂, for some constantC₂ independent of n, since IE[^P_{τ <s≤ν}n

τ(∆sYⁿ)²] =IE[

Z ν_τⁿ τ

kV_sⁿk²ds]. Hence- forth there exists also a constantC such that for any n≥0,

IE[

Z ν_τⁿ τ

|f(s, Y_sⁿ, Z_sⁿ)|²ds+{ Z ν_τⁿ

τ

n(L_s−Y_sⁿ)⁺ds}²]≤C. (11) But equation (7) implies that

IE[Y_δⁿ_τ1_[δτ<T]]≥IE[Y_νⁿⁿ

τ1_[δτ<T]]−sup

n≥0

{IE[

Z νⁿ_τ τ

|f(s, Y_sⁿ, Z_sⁿ)|²ds]}^12qIE[ν_τⁿ−δτ]

sincen(Y_sⁿ−Ls)⁺ ≥0. Taking now the limit asn→ ∞we obtainIE[Yδτ1_[δτ<T]]≥IE[Uδτ1_[δτ<T]] which implies that Y_δτ =U_δτ1_[δτ<T]+ξ1_[δτ=T] sinceY ≤U.

Let us now show thatY ≥L.For this let us consider the following BSDE: P-a.s., ∀t∈[τ, δτ]











Y¯_tⁿ=Y_δⁿ_τ + Z δτ

t

f(s, Y_sⁿ, Z_sⁿ)ds+ Z δτ

t

n(Ls−Y¯_sⁿ)ds

− Z δτ

t

Z¯_sⁿdBs− Z δτ

t

Z

E

V¯_sⁿ(e)˜µ(ds, de).

(12)

Once again by comparison with (7) we have, ∀n≥0, ¯Y_τⁿ≤Y_τⁿ. But Y¯_τⁿ=IE[Y_δⁿ_τe^{−n(δτ−τ)}+

Z δτ

τ

e^−n(s−τ){f(s, Y_sⁿ, Z_sⁿ) +nLs}ds|Fτ]

and then ( ¯Y_τⁿ)n≥0converges toYδτ1_[δτ=τ]+Lτ1_[δτ>τ], P-a.s. Actually inL¹(dP),Y_δⁿ_τe^{−n(δτ−τ)}→ Y_τ1_[δτ=τ], ^R_τ^δτ e^−n(s−τ)f(s, Y_sⁿ, Z_sⁿ)ds → 0 through (11) and finally ^R_τ^δτe^−n(s−τ)nL_sds → Lτ1_[δτ>τ] sinceLis rcll. Therefore we have Yτ = limnY_τⁿ≥limnY¯_τⁿ=Yδτ1_[δτ=τ]+Lτ1_[δτ>τ]≥ Lτ. As τ is a whatever stopping time then the optional section theorem (see e.g. [2], pp.220) implies thatP-a.s.,∀t≤T, Y_t≥L_t.

Finally it remains to show (iii). Let ( ˜Y^τ,Z˜^τ,V˜^τ,K˜^τ,+)∈ D²× H^2,d× L²× C_ci² solution of the following reflected BSDE: P-a.s.,∀t∈[τ, δτ],



















Y˜_t^τ =Y_δτ + Z δτ

t

f(s,Y˜_s^τ,Z˜_s^τ)ds+ Z δτ

t

dK˜_s^τ,+

− Z δτ

t

Z˜_s^τdBs− Z δτ

t

Z

E

V˜_s^τ(e)˜µ(ds, de);

Z δτ

τ

( ˜Y_s^τ−L_s)dK˜_s^τ,+= 0 and ∀t∈[τ, δ_τ], Y˜_t^τ ≥L_t.

(13)

Write Itˆo’s formula for thercllprocess|Y˜_t^τ−Y_tⁿ|² witht∈[τ, δτ], taking expectation and finally let n → ∞ to obtain that P-a.s., ∀t ∈ [τ, δτ], ˜Y_t^τ = Yt. Actually let C = Cf, the Lipschitz constant of f, and let σ be a stopping time such thatσ ∈[τ, δτ], then :

IE[|Y˜_σ^τ−Y_σⁿ|²e^2(C+C2)σ+^P_σ<s≤δτe^2(C+C2)s(∆s( ˜Y_s^τ−Y_sⁿ))²] =IE[|Y_δτ −Y_δⁿ_τ|²e^2(C+C2)δτ] +2IE[

Z δτ

σ

[f(s,Y˜_s^τ,Z˜_s^τ)−f(s, Y_sⁿ, Z_sⁿ)−(C+C²)( ˜Y_s^τ−Y_sⁿ)]( ˜Y_s^τ−Y_sⁿ)e^2(C+C2)sds]

−IE[

Z δτ

σ

|Z˜_s^τ−Z_sⁿ|²e^2(C+C2)sds] + 2IE[

Z δτ

σ

e^2(C+C2)s( ˜Y_s^τ−Y_sⁿ)d( ˜K_s^τ,+−K_s^n,+)]

≤IE[|Yδτ −Y_δⁿ_τ|²e^2(C+C2)δτ] + 2IE[

Z δτ

σ

e^2(C+C2)s(Ls−Y_sⁿ)dK˜_s^τ,+].

Taking now the limit asn→ ∞ to obtain IE[|Y˜_σ^τ−Yσ|²e^2(C+C2)σ]≤2IE[

Z δτ

σ

e^2(C+C2)s(Ls−Ys)dK˜_s^τ,+]≤0 and the proof is complete. 2

We now give the following technical result.

Lemma 2.3 : Letκbe an inaccessible stopping time. Let(θⁿ)_n≥0 be a non-decreasing sequence of stopping times uniformly bounded by T and let us setθ:= sup_n≥0θⁿ. Then

P(∩_n≥0[θⁿ< θ]∩[θ=κ]) = 0.

P roof: LetK_t^p be the predictable dual projection ofKt:= 1_[t≥κ], which is continuous since for all predictable stopping time τ we have 4K_τ^p = IE[4Kτ|F_τ−] (see e.g. [3], pp.149-150), then 4K_τ^p =IE[1_[τ=κ]/F_τ−] = 0. But the process (1_]θⁿ_,θ](s))_s≤T is predictable, then we have

P([θⁿ< κ≤θ]) =IE[K_θ]−IE[K_θⁿ] =IE[K_θ^p]−IE[K_θ^pn]→0 as n→ ∞.

Finally to obtain the result it enough to remark that:

P(∩_n≥0[θⁿ< θ]∩[θ=κ]) = lim

n→∞P([θⁿ< θ]∩[θ=κ])≤ lim

n→∞P([θⁿ< κ≤θ]) = 0.2

Now letθⁿ_τ := inf{s≥δτ, Y_sⁿ≤Ls} ∧T. SinceYⁿ≤Yⁿ⁺¹ then the sequence of stopping times (θⁿ_τ)n≥0 is non-decreasing and converges to θτ := lim_n→∞θⁿ_τ.

Lemma 2.4 : We have the following properties:

(i) P-a.s., for any t∈[δτ, θτ],Y_t∧θⁿ n

τ →Yt as n→ ∞ (ii) P-a.s., Y_θτ =L_θτ1_[θτ<T]+ξ1_[θτ=T]

(iii) for all stopping time τ there is( ¯Z^τ,V¯^τ,K¯^τ,−)∈ H^2,d× L²× C_ci² such that :



















Y_t=Y_θτ + Z θτ

t

f(s, Y_s,Z¯_s^τ)ds− Z θτ

t

dK¯_s^τ,−

− Z θτ

t

Z¯_s^τdBs− Z θτ

t

Z

E

V¯_s^τ(e)˜µ(ds, de), ∀t∈[δτ, θτ] Z θτ

δτ

(Ys−Us)dK¯_s^τ,−= 0.

(14)