Generalized Dynkin games and doubly reflected BSDEs with jumps

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with jumps

Roxana Dumitrescu, Marie-Claire Quenez, Agnès Sulem

To cite this version:

Roxana Dumitrescu, Marie-Claire Quenez, Agnès Sulem. Generalized Dynkin games and doubly

reflected BSDEs with jumps. Electronic Journal of Probability, Institute of Mathematical Statistics

(IMS), 2016, �10.1214/16-EJP4568�. �hal-01388022�

El e c t ro nic J

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Electron. J. Probab.21(2016), no. 64, 1–32.

ISSN:1083-6489 DOI:10.1214/16-EJP4568

Generalized Dynkin games and doubly reflected BSDEs with jumps

Roxana Dumitrescu

Marie-Claire Quenez

Agnès Sulem

Abstract

We introduce a game problem which can be seen as a generalization of the classical Dynkin game problem to the case of a nonlinear expectationE^g, induced by a Backward Stochastic Differential Equation (BSDE) with jumps with nonlinear driverg. Letξ, ζ be two RCLL adapted processes withξ≤ζ. The criterium is given by

Jτ,σ=E_0,τ∧σ^g ξτ1{τ≤σ}+ζσ1{σ<τ}

,

whereτ andσare stopping times valued in[0, T]. Under Mokobodzki’s condition, we establish the existence of a value function for this game, i.e. infσsup_τJτ,σ = sup_τinfσJτ,σ. This value can be characterized via a doubly reflected BSDE. Using this characterization, we provide some new results on these equations, such as comparison theorems and a priori estimates. Whenξandζare left upper semicontinuous along stopping times, we prove the existence of a saddle point. We also study a generalized mixed game problem when the players have two actions: continuous control and stopping. We then study the generalized Dynkin game in a Markovian framework and its links with parabolic partial integro-differential variational inequalities with two obstacles.

Keywords: Dynkin game; mixed game problem; nonlinear expectation;g-evaluation; doubly reflected BSDEs; partial integro-differential variational inequalities; game option.

AMS MSC 2010:93E20; 60J60; 47N10.

Submitted to EJP on September 21, 2015, final version accepted on October 5, 2016.

1 Introduction

The classical Dynkin game has been widely studied: see e.g. Bismut [5], Alario- Nazaret et al. [1], Kobylanski et al. [31]. Letξ= (ξ_t), ζ = (ζ_t)be two right continuous left-limited (RCLL) adapted processes withξ≤ζandξT =ζT a.s. The criterium is given, for all pair(τ, σ)of stopping times valued in[0, T], by

Jτ,σ:=E ξτ1_{τ≤σ}+ζσ1_{σ<τ} .

*CEREMADE, Université Paris 9 Dauphine, CREST and INRIA Paris France. E-mail:roxana@ceremade.

dauphine.fr. The research leading to these results has received funding from the Région Ile-de-France.

†LPMA, Université Paris 7 Denis Diderot, France. E-mail:quenez@math.univ-paris-diderot.fr

‡INRIA Paris and Université Paris-Est, France. E-mail:agnes.sulem@inria.fr

Under Mokobodzki’s condition, which states that there exist two supermartingales such that their difference is betweenξandζ, the Dynkin game is fair, i.e. inf_σsup_τJ_τ,σ = sup_τinfσJτ,σ. When the barriersξ,ζare left upper semicontinuous, andξt< ζt,t < T, there exists a saddle point.

Using a change of variable, these results can be generalized to the case of a criterium with an instantaneous reward processg(t), of the form

J_τ,σ:=E

Z τ∧σ 0

g(t)dt+ξ_τ1_{τ≤σ}+ζ_σ1_{σ<τ}

. (1.1)

In the Brownian case and whenξandζare continuous processes, Cvitani´c and Karatzas have established links between these Dynkin games and doubly reflected Backward stochastic differential equations with driver processg(t)and barriersξandζ(see [9]).

In this paper, we introduce a new game problem, which generalizes the classical Dynkin game to the case ofE^g-expectations (org-evaluations in the terminology of Peng [33]). Given a Lipschitz driverg(t, y, z, k), a stopping timeτ≤T and a square integrable Fτ-measurable random variableη, the associated conditionalE^g-expectation process denoted by(E_t,τ^g (η),0 ≤ t ≤τ)is defined as the solution of the backward stochastic differential equation (BSDE) with driverg, terminal timeτand terminal conditionη. We thus consider here ageneralized Dynkin game, where the criterium is given, for each pair(τ, σ)of stopping times valued in[0, T], by

Jτ,σ:=E_0,τ∧σ^g ξτ1_{τ≤σ}+ζσ1_{σ<τ} , withξ, ζ two RCLL adapted processes satisfyingξ≤ζ.

When the driver gdoes not depend on the solution, that is, when it is given by a processg(t), the criteriumJ_τ,σcoincides withJ_τ,σgiven in (1.1). It is well-known that in this case, under Mokobodzki’s condition, the value function for the Dynkin game problem can be characterized as the solution of the Doubly Reflected BSDE (DRBSDE) associated with driver processg(t)and barriersξandζ(see e.g. [9], Hamadène-Lepeltier [25], Lepeltier-Xu [32], Hamadène-Ouknine [27]). We generalize here this result to the case of a nonlinear driverg(t, y, z, k)depending on the solution. More precisely, under Mokobodzki’s condition, we prove that

infσ sup

τ

Jτ,σ= sup

τ

infσ Jτ,σ,

and we characterize this common value function as the solution of the DRBSDE as- sociated with driver g and barriersξ and ζ. Moreover, when ξ and ζ are left-upper semicontinuous along stopping times, we show that there exist saddle points. Note that, contrary to the previous existence results given in the case of classical Dynkin games, we do not assume the strict separability of the barriers. We point out that the approach used in the classical case cannot be adapted to our case because of the nonlinearity of the driver. Using the characterization of the solution of a DRBSDE as the value function of a generalized Dynkin game, we prove some results on DRBSDEs, such as a comparison and a strict comparison theorem and a priori estimates which complete those given in the literature.

Moreover, we introduce a mixed game problem expressed in terms ofE^g-expectation/

g-evaluation, when the players have two possible actions: continuous control and stop- ping. The first (resp. second) player chooses a pair(u, τ)(resp. (v, σ)) of control and stopping time, and aims at maximizing (resp. minimizing) the criterium. This problem has been studied by [25] and [21] in the case of a classical expectation, that is when the criterium is given, for each quadruple(u, τ, v, σ)of controls and stopping times, by

EQ^u,v

Z τ∧σ 0

g(t, ut, vt)dt+ξτ1_{τ≤σ}+ζσ1_{σ<τ}

, (1.2)

whereQ^u,vare a priori probability measures, andg(t, u_t, v_t)represents the instantaneous reward process associated with controlsu, v. In [25], Hamadène and Lepeltier (see also Hamadène [21]) have established some links between this mixed game problem and DRBSDEs. Here, we consider ageneralized mixed gameproblem, where, for a given family of nonlinear driversg^u,v(t, y, z, k) :=g(t, ut, vt, y, z, k)depending on the controls u, v, the criterium is defined by

E_0,τ∧σ^u,v ξτ1_{τ≤σ}+ζσ1_{σ<τ}

, (1.3)

evaluated under the nonlinear expectationE^u,v =E^gu,v. Note that the criterium (1.3) corresponds to a criterium of the form (1.2) when the driversg^u,vare linear. We general- ize the results of [25] and [21] to the case of nonlinear expectations. We provide some sufficient conditions which ensure the existence of a value function of ourgeneralized mixed gameproblem, and show that the common value function can be characterized as the solution of a DRBSDE. Under additional regularity assumptions onξandζ, we prove the existence of saddle points.

The paper is organized as follows. In Section 2 we introduce notation and definitions and provide some preliminary results. In Section 3 we consider a classical Dynkin game problem and study its links with a DRBSDE associated with a driver which does not depend on the solution. We provide an existence result for this game problem under relatively weak assumptions onξandζ. Note that Section 3, although it contains new results, mainly situates our work and introduces the tools used in the sequel. In Section 4, we introduce thegeneralized Dynkin gamewithg-evaluation. We prove the existence of a value function for this game problem. We show that the common value function can be characterized as the solution of a nonlinear DRBSDE with jumps and RCLL barriersξand ζ. We then study in Section 5 ageneralized mixed gameproblem when the players have two actions: continuous control and stopping. In Section 6, using the characterization of the solution of a DRBSDE as the value function of ageneralized Dynkin game, we prove comparison theorems and a priori estimates for DRBSDEs. Finally in Section 7, we study thegeneralized Dynkin gamein a Markovian framework and its links with parabolic partial integro-differential variational inequalities (PIDVI) with two obstacles. We prove that the value function of the generalized Dynkin game is a viscosity solution of a PIDVI.

A uniqueness result is obtained under additional assumptions. Additional results and detailed proofs are given in the Appendix.

Motivating applications in mathematical finance As shown in El-Karoui-Quenez [18], in a market model with constraints such as taxes or large investor impact, the dynamics of the wealth process of an investor investing in this market can be written via a nonlinear driverg. In [18], anonlinear price system(later calledg-evaluation) is defined as follows: for eachS∈[0, T]and eachη∈L²(FS), the initial (hedging) price of an European option with maturitySand payoffηis given byE_0,S^g (η).

In the case of an American option associated with a payoff process ξ ∈ S², the buyer has the choice of the exercise (stopping) timeτ. Theg-value of the American option, defined assup_τE_0,τ^g (ξτ), is characterized as the initial value of the reflected BSDE associated with drivergand obstacleξ(see [18] for a proof in a Brownian framework and Quenez-Sulem [36] for the generalization to the case with jumps and irregular payoff).

In [14], we show that this price is thesuper-hedging priceof the American option.

Game options are an extension of American options introduced by Kifer in the case of a perfect market model (see [29]). Beside the holder’s ability to exercise at any (stopping) timeτ, the issuer of the option also cancel the option at any (stopping) time σand then pay the holder a cancellation feeζσ. The cancellation feeζis greater than or equal to the exercise payoffξat all time. The differenceζσ−ξσ ≥0is interpreted

as a penalty that the seller pays to the buyer for the cancellation of the contract.

Hence, the game option consists for the seller to select a cancellation timeσand for the buyer to choose an exercise timeτ, so that the seller pays to the buyer at time τ ∧σ the amount I(τ, σ) = ξτ1_τ≤σ +ζσ1σ<τ. The main result of the present paper yields that, under Mokobodzki’s condition, theg-valueof the game option, defined as infσsup_τE_0,τ∧σ^g (I(τ, σ)), is equal to thecommon value of ageneralizedDynkin game, and is characterized as the initial value of the solution of the doubly reflected BSDE associated with drivergand barriersξandζ(see (2.2)).¹ This result, which is new even in the Brownian case, can be seen as the analogous, in the case of a game option, of the characterization of theg-valueof an American option (i.e. sup_τE_0,τ^g (ξτ)), via a reflected BSDE. In [14], using this result, we show that theg-value of the game option is equal to thesuper-hedging price, that is the minimal initial wealth which allows the seller to be super-hedged.

2 Notation and definitions

Let(Ω,F,P)be a probability space. LetW be a one-dimensional Brownian motion.

LetE:=R^∗andB(E)be its Borelian filtration. Suppose that it is equipped with aσ-finite positive measureν and letN(dt, de)be a Poisson random measure with compensator ν(de)dt. LetN˜(dt, de)be its compensated process. LetIF ={Ft, t≥0}be the completed natural filtration associated withW andN.

Notation 2.1.LetP be the predictable σ-algebra onΩ×[0, T]. For eachT > 0, we introduce the following sets:

• L²(FT) : the set of random variables ξ which are FT-measurable and square integrable;

• IH²: the set of real-valued predictable processesφwithkφk²_IH2 :=Eh RT

0 φ²_tdti

<∞;

• S²: the set of real-valued RCLL adapted processesφwith kφk²_S2 :=E(sup_0≤t≤T|φ_t|²)<∞;

• A²(resp.A¹) : the set of real-valued non decreasing RCLL predictable processes AwithA0= 0andE(A²_T)<∞(resp. E(AT)<∞).

• L²_ν : the set of Borelian functions`:E→Rsuch thatR

E|`(e)|²ν(de)<+∞.

The setL²_νis a Hilbert space equipped with the scalar product h`, `⁰iν:=R

E`(e)`⁰(e)ν(de)for all`, `⁰∈L²_ν×L²_ν,and the normk`k²_ν :=R

E|`(e)|²ν(de).

• IH_ν² : the set of all mappings l : Ω×[0, T]×E → R that are P ⊗ B(E)/B(R) measurable and satisfyklk²_IH2

ν :=EhRT

0 kltk²_νdti

<∞,wherelt(ω, e) =l(ω, t, e)for all(ω, t, e)∈Ω×[0, T]×E.

Moreover,T0is the set of stopping timesτ such thatτ ∈[0, T]a.s. and for eachSin T0, we denote byTSthe set of stopping timesτsuch thatS≤τ≤T a.s.

Definition 2.2(Driver, Lipschitz driver).A functiongis said to be adriverif

• g: Ω×[0, T]×R²×L²_ν→R

(ω, t, y, z,˛(·))7→g(ω, t, y, z, k(·))isP ⊗ B(R²)⊗ B(L²_ν)−measurable,

• g(.,0,0,0)∈IH².

A drivergis called aLipschitz driverif moreover there exists a constantC≥0such that dP ⊗dt-a.s. , for each(y1, z1, k1),(y2, z2, k2),

|g(ω, t, y1, z1, k1)−g(ω, t, y2, z2, k2)| ≤C(|y1−y2|+|z1−z2|+kk1−k2kν).

1Note that in the literature, this characterization has only been proven whengis linear with respect to y, z, k. The proof is based on a change of probability measure and an actualization procedure (see [21]). This approach cannot be adapted to the nonlinear case.

Recall that for each Lipschitz driverg, and each terminal conditionξ∈L²(FT), there exists a unique solution(X^g, π^g, l^g)∈ S²×IH²×IH_ν²satisfying

−dX^g(t) =g(t, X_t^g−, π^g(t), l^g(t)(·))dt−π^g(t)dW_t− Z

E

l^g(t)(e) ˜N(dt, de); X_T =ξ. (2.1) The solution is denoted by (X_.^g(ξ, T), π^g_.(ξ, T), l^g_.(ξ, T)). The operator X^g : (ξ, T) 7→

X_·^g(ξ, T), callednonlinear pricing system(associated with driverg), was first introduced in [18]. In [33], this operator X^g is called the nonlinear evaluation (associated with driverg) and is denoted byE^g. In the sequel, we say thatE_·,T^g (ξ)is theE_.,T^g -conditional expectationprocess ofξ(or theg-evaluationof(ξ, T)). When there is no ambiguity on the driver,E^g will be simply denoted byE. Recall that this notion can be extended to the case whereT is replaced by a stopping time τ ∈ T0 and ξby a random variable η∈L²(F_τ).

We introduce a notion of mutually singular random measures associated with non decreasing RCLL predictable processes, which can be seen as a probabilistic version of a classical notion in analysis.

Definition 2.3.LetA= (At)_0≤t≤T andA⁰ = (A⁰_t)_0≤t≤T belonging toA¹. We say that the random measuresdAtanddA⁰_taremutually singular(in a probabilistic sense), and we writedAt⊥dA⁰_t, if there existsD∈ Psuch that:

E[

Z T 0

1_DcdA_t] =E[

Z T 0

1_DdA⁰_t] = 0,

which can also be written asRT

0 1D^c_tdAt=RT

0 1D_tdA⁰_t= 0 a.s.,where for eacht∈[0, T], D_tis thesection at timetofD, that is,D_t:={ω∈Ω,(ω, t)∈D}.²

We define now DRBSDEs with jumps, for which the solution is constrained to stay between two given RCLL processes called barriersξ≤ζ. Two nondecreasing processes AandA⁰are introduced in order to push the solutionY aboveξand belowζin a minimal way. This minimality property of A and A⁰ is ensured by the Skorohod conditions (condition(iii)below) together with the additional constraintdAt⊥dA⁰_t(condition(ii)).

Definition 2.4(Doubly Reflected BSDEs with Jumps).LetT >0be a fixed terminal time andg be a Lipschitz driver. Letξandζ be two adapted RCLL processes withζ_T =ξ_T a.s.,ξ∈ S²,ζ∈ S²,ξt≤ζt,0≤t≤T a.s.

A process(Y, Z, k(.), A, A⁰)inS²×IH²×IH_ν²× A²× A²is said to be a solution of the doubly reflected BSDE (DRBSDE) associated with drivergand barriersξ, ζ if

−dYt=g(t, Y_t, Z_t, k_t(·))dt+dA_t−dA⁰_t−Z_tdW_t− Z

E

k_t(e) ˜N(dt, de); Y_T =ξ_T, (2.2) with

(i) ξ_t≤Y_t≤ζ_t, 0≤t≤T a.s., (ii) dAt⊥dA⁰_t

(iii) Z T

0

(Y_t−ξ_t)dA^c_t= 0a.s. and Z T

0

(ζ_t−Y_t)dA^0c_t = 0a.s.

∆A^d_τ = ∆A^d_τ1_{Y

τ−=ξ_τ−}and ∆A^0d_τ = ∆A^0d_τ1_{Y

τ−=ζ_τ−} a.s.∀τ∈ T₀predictable.

HereA^c(respA^0c) denotes the continuous part ofA(respA⁰) andA^d(respA^0d) its discontinuous part.

Remark 2.5.WhenAandA⁰are not required to be mutually singular, they can simul- taneously increase on{ξ_t− = ζ_t−}. The constraintdA_t ⊥ dA⁰_tallows us to obtain the

2Note that if the random measuresdAtanddA⁰_tare mutually singular in the above probabilistic sense, then, for almost everyω, the deterministic measures on[0, T]dAt(ω)anddA⁰_t(ω)are mutually singular in the classical analysis sense. The converse is not straightforward. However, it holds by Proposition A.7.

uniqueness of the non decreasing RCLL processesAandA⁰, without the usual strict separability conditionξ < ζ (see Theorem 3.5).

We introduce the following definition.

Definition 2.6.A progressively measurable process(φt)(resp. integrable) is said to be left-upper semicontinuous (l.u.s.c.) along stopping times(resp. along stopping times in expectation ) if for allτ ∈ T0and for each non decreasing sequence of stopping times (τ_n)such thatτⁿ↑τa.s. ,

φ_τ ≥lim sup

n→∞

φ_τn a.s. (resp. E[φ_τ]≥lim sup

n→∞

E[φ_τn]). (2.3) Remark 2.7.When(φ_t)is left-limited, then(φ_t)isleft-upper semicontinuous (l.u.s.c.) along stopping timesif and only if for all predictable stopping timeτ ∈ T₀,φ_τ≥φ_τ− a.s.

3 Classical Dynkin games and links with doubly reflected BSDEs with a driver process

In this section, we suppose that the driver g does not depend on (y, z, k), that is g(ω, t, y, z, k) = g(ω, t), where g ∈ H^{2. Let} ξ and ζ be two adapted processes only supposed to be RCLL withζT =ξT a.s.,ξ∈ S²,ζ∈ S²,ξt≤ζt,0≤t≤T a.s.

We prove below that the doubly reflected BSDE associated with the driver process g(t)and the barriersξandζadmits a unique solution(Y, Z, k(·), A, A⁰), which is related to a classical Dynkin game problem. The results of this section complete previous works on classical Dynkin games and DRBSDEs (see e.g. [9, 22, 25, 7, 27, 26]). In particular, we provide an existence result of saddle points under weaker assumptions than those made in the previous literature.

For anyS∈ T0and any stopping timesτ, σ∈ TS, consider the gain (or payoff):

IS(τ, σ) = Z σ∧τ

S

g(u)du+ξτ1_{τ≤σ}+ζσ1_{σ<τ}. (3.1) For anyS∈ T₀, the upper and lower value functions at timeSare defined respectively by

V(S) := ess inf

σ∈T_S ess sup

τ∈TS

E[I_S(τ, σ)|F_S] (3.2)

V(S) := ess sup

τ∈TS

ess inf

σ∈TS

E[IS(τ, σ)|FS]. (3.3)

We clearly have the inequalityV(S)≤V(S)a.s. By definition, we say that the Dynkin game isfair(orthere exists a value function) at timeS ifV(S) =V(S)a.s.

Definition 3.1(S-saddle point).LetS∈ T0. A pair(τ^∗, σ^∗)∈ T_S²is called anS-saddle pointif for each(τ, σ)∈ T_S², we have

E[IS(τ, σ^∗)|FS]≤E[IS(τ^∗, σ^∗)|FS]≤E[IS(τ^∗, σ)|FS] a.s.

We introduce the following RCLL adapted processes (which depend on the process g):

ξ˜_t^g:=ξt−E[ξT + Z T

t

g(s)ds|Ft], ζ˜_t^g:=ζt−E[ζT + Z T

t

g(s)ds|Ft], 0≤t≤T. (3.4) Note that since g ∈ H^{2 and} ξ ∈ S², ξ˜^g and ζ˜_t^g belong to S². Moreover, we have ξ˜_T^g = ˜ζ_T^g = 0 a.s.

Definition 3.2.An optional processφ valued in[0,+∞]is said to be a strong super- martingaleif for anyθ, θ⁰∈ T0such thatθ≥θ⁰ a.s.,E[φθ| Fθ⁰]≤φθ⁰ a.s.

SetJ_·^g,0= 0andJ_·^0g,0= 0. We define recursively for alln∈N, the RCLL supermartin- galesJ_·^g,nandJ_·^0g,nsatisfying for allθ∈ T₀the equalities³

J_θ^g,n+1= ess sup

τ∈Tθ

Eh

J_τ^0g,n+ ˜ξ^g_τ|F_θi

a.s. and J_θ^0g,n+1= ess sup

τ∈Tθ

Eh

J_τ^g,n−ζ˜_τ^g|F_θi a.s.

(3.5) Lemma 3.3.The sequences of processes(J·^g,n)n∈N and(J^0g,n_· )n∈Nare non decreasing.

Moreover, the processesJ_·^g andJ_·^0gdefined for allt∈[0, T]byJ_t^g:= lim_n→+∞J_t^g,n andJ_t^0g := lim_n→+∞ J_t^0g,n are strong supermartingales valued in[0,+∞]. They satisfy J_T^g =J_T^0g= 0a.s. and for allθ∈ T0,

J_θ^g = ess sup

τ∈T_θ

Eh

J_τ^0g+ ˜ξ^g_τ|Fθ

i

a.s. and J_θ^0g = ess sup

σ∈T_θ

Eh

J_σ^g−ζ˜_σ^g|Fθ

i

a.s. (3.6) If J₀^g<+∞and J₀^0g<+∞, thenJ^g andJ^0gare RCLL supermartingales.

Proof. See Appendix.

Using this lemma, we derive that if J^g andJ^0g belong toS², then there exists a solution of the DRBSDE (2.2) associated with the driverg(t).

Theorem 3.4.Letξandζbe two adapted RCLL processes inS²withζT =ξT a.s. and ξt≤ζt,0≤t≤T a.s. Suppose that J^g, J^0g∈ S². LetY be the RCLL adapted process defined by

Yt:= J_t^g−J_t^0g+E[ξT + Z T

t

g(s)ds|Ft]; 0≤t≤T. (3.7) There exist(Z, k, A, A⁰)∈IH²×IH_ν²× A²× A²such that(Y , Z, k, A, A⁰)is a solution of DRBSDE(2.2)associated with the driver processg(t).

Proof. As usual in the literature on DRBSDEs (see e.g. [9, 22, 7]), the proof is based on some results of Optimal Stopping Theory. By assumption, the processesJ^gandJ^0g are finite. Hence, the differenceJ^g−J^0g and thus the processY are well defined. By Lemma 3.3, we haveJ_T^g =J_T^0ga.s. Hence,YT =ξT a.s. By (3.6), we haveJ^g ≥ J^0g+ ˜ξ^g andJ^0g≥ J^g−ζ˜^g. Using the definitions ofξ˜^g,ζ˜^gandY, we derive thatξ≤Y ≤ζ.

Moreover, by the last assertion of Lemma 3.3 and the assumption J^g, J^0g ∈ S², we obtain thatJ^gandJ^0gare indistinguishable from RCLL supermartingales. We thus can apply the Doob-Meyer decomposition, and derive the existence of two square integrable martingalesM andM⁰and two processesB andB⁰ ∈ A²such that:

dJ_t^g=dMt−dBt ; dJ_t^0g =dM_t⁰−dB_t⁰. (3.8) Set

M_t:=M_t−M_t⁰+E[ξ_T+ Z T

0

g(s)ds|Ft].

By (3.8), (3.7), we derivedYt=dMt−dαt−g(t)dt, withα:=B−B⁰.

Now, by the martingale representation theorem, there existZ∈H^2andk∈H²ν such thatdM_t=Z_tdW_t+R

Ek_t(e) ˜N(de, dt). Hence,

−dYt=g(t)dt+dαt−ZtdWt− Z

E

kt(e) ˜N(dt, de).

By the optimal stopping theory (see [28, Appendix Sect. D] in the case of a continuous reward process, and [30, Proposition B.7, B.11] in the right-continuous case), the process B^c increases only when the value function J^g is equal to the corresponding

3Note that these processes exist by aggregation results (cf. [17]).

rewardJ^0g+ ˜ξ^g. Now, {J_t^g = J_t^0g+ ˜ξ^g} ={Yt = ξt}. Hence, RT

0 (Yt−ξt)dB_t^c = 0 a.s.

Similarly the processB^0csatisfiesRT

0 (Y_t−ζ_t)dB_t^0c= 0a.s. Moreover, thanks to a result from optimal stopping theory (cf. [17, Proposition 2.34] or [30]), for each predictable stopping time τ ∈ T0 we have ∆B_τ^d = 1_Jg

τ−=J^0g

τ−+ ˜ξ^g

τ−∆B_τ^d = 1_Y

τ−=ξ_τ−∆B_τ^d a.s. and

∆B_τ^0d=1_Y

τ−=ζ_τ−∆B^0d_τ a.s.

By the canonical decomposition of an RCLL predictable process with integrable total variation (see Proposition A.7), there existA, A⁰ ∈ A²such thatα=A−A⁰withdAt⊥dA⁰_t. Also, dAt<< dBt. Hence, sinceRT

0 1_Y

t−>ξ_t−dBt = 0 a.s. , we getRT 0 1_Y

t−>ξ_t−dAt = 0 a.s. Similarly, we obtainRT

0 1_Y

t−<ζ_t−dA⁰_t= 0a.s. The processesAandA⁰ thus satisfy conditions (2.2)(iii) (withY replaced byY). The process(Y , Z, k, A, A⁰)is thus a solution of DRBSDE (2.2).

From this result, we derive the following uniqueness and existence result for the DRBSDE associated with the driver processg(t), as well as the characterization of the solution as the value function of the above Dynkin game problem.

Theorem 3.5.Letξandζbe two adapted RCLL processes inS²withζ_T =ξ_T a.s. and ξt ≤ ζt, 0 ≤ t ≤ T a.s. Suppose that J^g, J^0g ∈ S². The doubly reflected BSDE (2.2) associated with driver processg(t)admits a unique solution(Y, Z, k, A, A⁰)inS²×IH²× IH_ν²×(A²)².

For eachS∈ T0,YS is the common value function of the Dynkin game, that is

YS =V(S) =V(S) a.s. (3.9)

Moreover, if the processes A, A⁰ are continuous, then, for each S ∈ T0, the pair of stopping times(τ_s^∗, σ_s^∗)defined by

σ^∗_S := inf{t≥S, Yt=ζt}; τ_S^∗:= inf{t≥S, Yt=ξt} (3.10) is anS-saddle point for the Dynkin game problem associated with the gainIS.

A short proof is given in the Appendix.

Remark 3.6.The conditiondA_t⊥dA⁰_tensures that for each predictable stopping time τ∈ T0,∆A^d_τ∆A^0d_τ = 0a.s. Now, sinceY satisfies (2.2), we have∆Yτ= ∆A^0d_τ −∆A^d_τ a.s.

We thus have∆A^d_τ = (∆Yτ)⁻and∆A^0d_τ = (∆Yτ)⁺a.s. for each predictable stopping time τ∈ T0.

We now provide a sufficient condition onξandζfor the existence of saddle points.

By the last assertion of Theorem 3.5, it is sufficient to give a condition which ensures the continuity ofAandA⁰.

Theorem 3.7(Existence ofS-saddle points).Suppose that the assumptions of Theorem 3.5 are satisfied. Let(Y, Z, k(.), A, A⁰)be the solution of DRBSDE (2.2). We have

(i) Ifξ(resp. −ζ) is l.u.s.c. along stopping times, then the processA(resp. A⁰) is continuous.

(ii) When ξ and−ζ are l.u.s.c. along stopping time, for each S ∈ T₀, the pair of stopping times(τ_S^∗, σ_S^∗)defined by (3.10)is anS-saddle point.

Remark 3.8.For(ii), the assumptions made onξandζare weaker than the ones made in the literature where it is supposedξ_t< ζ_t, t < T a.s. (see e.g. [1, 9, 31]).

Proof. Suppose that ξ is l.u.s.c. along stopping times. Let τ ∈ T0 be a predictable stopping time. Let us show∆Aτ= 0a.s.

Since dAt ⊥ dA⁰_t, we have∆Aτ = (∆Yτ)⁻ a.s. (see Remark 3.6 above). SinceA satisfies the Skorohod condition, we have a.s.

∆Aτ =1_{Y

τ−=ξ_τ−}(Y_τ⁻−Yτ)⁺=1_{Y

τ−=ξ_τ−}(ξ_τ⁻−Yτ)⁺≤1_{Y

τ−=ξ_τ−}(ξτ−Yτ)⁺,

where the last inequality follows from the inequalityξ_τ− ≤ξ_τ a.s. (see Remark 2.7).

Sinceξ≤Y, we derive that∆A_τ ≤0a.s. Hence,∆A_τ = 0a.s. , and this holds for each predictable stopping timeτ. Consequently,Ais continuous. Similarly, one can show that if−ζis l.u.s.c. along stopping times, thenA⁰is continuous, which completes the proof of (i).

The assertion(ii)follows from(i)since, by the second assertion of Theorem 3.5, the continuity property ofAandA⁰ensures the existence of saddle points.

Definition 3.9(Mokobodzki’s condition).Letξandζbe adapted RCLL processes inS² withζT =ξT a.s. andξt≤ζt,0≤t≤T a.s. Mokobodzki’s condition is said to be satisfied when there exist two nonnegative RCLL supermartingalesH andH⁰ inS²such that:

ξt≤Ht−H_t⁰≤ζt 0≤t≤T a.s. (3.11) Note that Mokobodzki’s condition holds, for instance, whenξorζis a semimartingale satisfying some integrability conditions (see Remark A.8 for details).

Proposition 3.10.Letg∈IH². Letξandζbe two adapted RCLL processes inS²with ζT =ξT a.s. andξt≤ζt,0≤t≤T a.s. The following assertions are equivalent:

(i) J^g∈ S² (ii) J⁰∈ S²

(iii) Mokobodzki’s condition holds.

(iv) DRBSDE (2.2)with driver processg(t)has a solution.

A short proof is given in the Appendix. Note that the equivalence between(iii)and (iv)is well-known.

4 Generalized Dynkin games and links with nonlinear doubly re- flected BSDEs

In this section, we are given a Lipschitz driverg.

Theorem 4.1 (Existence and uniqueness for DRBSDEs).Suppose ξ and ζ are RCLL adapted process inS² such thatξT =ζT a.s. andξt≤ζt, 0 ≤t ≤T a.s. Suppose that J⁰∈ S²(or equivalently that Mokobodzki’s condition is satisfied).

Then, DRBSDE(2.2)admits a unique solution(Y, Z, k(.), A, A⁰)∈ S²×IH²×IH_ν²×(A²)². If ξ (resp. −ζ) is l.u.s.c. along stopping times, then the process A (resp. A⁰) is continuous.

Remark 4.2.Note that the solutionY of the DRBSDE (2.2) coincides with the value function of the classical Dynkin game (3.2) and (3.3) with the gain:

IS(τ, σ) = Z σ∧τ

S

g(u, Yu, Zu, ku)du+ξτ1_{τ≤σ}+ζσ1_{σ<τ}, (4.1) whereZ, kare the associated processes withY. However, this classical characterization ofY (see e.g. [9]) is not really exploitable because the instantaneous reward process g(u, Y_u, Z_u, k_u)depends on the value functionY (and Z, K) of the associated Dynkin game.

The proof of the existence and uniqueness of the solution is based on classical contraction arguments and is given in the Appendix.

We now introduce a game problem, which can be seen as ageneralized Dynkin game expressed in terms ofE^g-expectations.

In order to ensure that theE^g-expectation is non decreasing, we make the following assumption.

Assumption 4.3.Assume thatdP⊗dt-a.s for each(y, z, k₁, k₂)∈R²×(L²_ν)², g(t, y, z, k₁)−g(t, y, z, k₂)≥ hγ_t^y,z,k1,k2, k₁−k₂iν,

with γ: [0, T]×Ω×R²×(L²_ν)²→L²_ν; (ω, t, y, z, k₁, k₂)7→γ_t^y,z,k1,k2(ω, .) P · ⊗B(R²)⊗ B((L²_ν)²)-measurable and satisfying the inequalities

γ_t^y,z,k1,k2(e)≥ −1 and kγ_t^y,z,k1,k2kν≤K, (4.2) for each (y, z, k1, k2) ∈R²×(L²_ν)², respectivelydP ⊗dt⊗dν(e)-a.s. and dP ⊗dt-a.s.

(whereKis a positive constant).

Assumption 4.3 is satisfied for example whengisC¹with respect tokwith∇kg≥ −1 and|∇kg| ≤ ψ, where ψ∈L²_ν.Assumption 4.3 is also satisfied wheng is of the form g(ω, t, y, z, k) :=g(ω, t, y, z,R

Ek(e)ψ(e)ν(de))whereψis a nonnegative function inL²_νand g: Ω×[0, T]×R³→Ris Borelian and non-decreasing with respect tok, (see Proposition A.2 in the Appendix for details).

Assumption 4.3 ensures the non decreasing property ofE^gby the comparison theorem for BSDEs with jumps (see [35, Theorem 4.2]). When in (4.2), γt > −1, the strict comparison theorem (see in [35, Theorem 4.4]) implies thatE^gis strictly monotonous.

For eachτ, σ∈ T0, thereward(orpayoff) at timeτ∧σis given by the random variable I(τ, σ) :=ξτ1τ≤σ+ζσ1σ<τ. (4.3) Note thatI(τ, σ)is Fτ∧σ-measurable. Let S ∈ T0. For each τ ∈ TS and σ ∈ TS, the associatedcriteriumis given byE_S,τ^g _∧σ(I(τ, σ)), theg-evaluation of the payoffI(τ, σ).

Recall thatE_·,τ∧σ^g (I(τ, σ)) =X_·^τ,σ,where(X_·^τ,σ, π_·^τ,σ, l^τ,σ_· )is the solution of the BSDE associated with driverg, terminal timeτ∧σand terminal conditionI(τ, σ), that is

−dX_s^τ,σ=g(s, X_s^τ,σ, π_s^τ,σ, l^τ,σ_s )ds−π_s^τ,σdWs− Z

E

l^τ,σ_s (e) ˜N(ds, de); X_τ∧σ^τ,σ =I(τ, σ).

To simplify notation,E^gis denoted byEin the sequel.

For each stopping timeS ∈ T₀, theupperandlower value functionsat timeS are defined respectively by

V(S) := ess inf

σ∈TS

ess sup

τ∈TS

E_S,τ∧σ(I(τ, σ)); (4.4)

V(S) := ess sup

τ∈TS

ess inf

σ∈TS

ES,τ∧σ(I(τ, σ)). (4.5)

We clearly have the inequalityV(S)≤V(S)a.s. By definition, we say that the game is fair(orthere exists a value function) at timeS ifV(S) =V(S)a.s.

We now give the definition of anS-saddle point for this game problem.

Definition 4.4.LetS ∈ T₀. A pair(τ^∗, σ^∗)∈ T_S² is called an S-saddle point for the generalized Dynkin game if for each(τ, σ)∈ T_S²we have

E_S,τ∧σ^∗(I(τ, σ^∗))≤ ES,τ^∗∧σ^∗(I(τ^∗, σ^∗))≤ ES,τ^∗∧σ(I(τ^∗, σ)) a.s.

We provide a sufficient condition for the existence of an S-saddle point and the characterization of the common value function as the solution of the DRBSDE. We first introduce the following definition.

Definition 4.5.LetY ∈ S². The processY is said to be a strongE^g-supermartingale (respE^g-submartingale), if E_σ,τ^g (Yτ) ≤ Yσ (resp. E_σ,τ^g (Yτ) ≥ Yσ) a.s. onσ ≤ τ, for all σ, τ ∈ T0.

Lemma 4.6.Suppose that the drivergsatisfies Assumption(4.3). Letξandζbe RCLL adapted processes inS² such thatξ_T = ζ_T a.s. andξ_t ≤ζ_t, 0 ≤t ≤T a.s. Suppose that Mokobodzki’s condition is satisfied. Let (Y, Z, k(·), A, A⁰) be the solution of the DRBSDE (2.2). Let S ∈ T0. Let(ˆτ ,ˆσ)∈ T_S². Suppose that(Yt, S ≤t ≤τ)ˆ is a strong E-submartingale and that(Yt, S≤t≤σ)ˆ is a strongE-supermartingale withYˆτ=ξˆτand Yσˆ =ζσˆa.s.

The pair(ˆτ ,ˆσ)is then anS-saddle point for thegeneralized Dynkin game (4.4)-(4.5) and

YS =V(S) =V(S)a.s.

Remark 4.7.A well-known sufficient condition for a pair of stopping times(ˆτ ,σ)ˆ to be a saddle point for the classical Dynkin game (3.2)-(3.3) is thatτˆ(resp. ˆσ) is an optimal stopping time for the optimal stopping problemJ_S^g (resp. J_S^0g) (see e.g. [1, Theorem 2.4] or [31, Proposition 3.1]). In the nonlinear case, we cannot have an analogous sufficient condition since there is no coupled optimal stopping problem associated with our generalized Dynkin game. Note also that in the classical linear case, the sufficient condition given in Lemma 4.6 is weaker than the one given in the literature.

Proof. Since the process(Yt, S≤t≤τˆ∧σ)ˆ is a strongE-martingale (see Definition 4.5) and sinceYˆτ=ξˆτandYˆσ=ζσˆa.s. , we have

Y_S =E_{S,ˆτ∧ˆσ}(Y_ˆτ∧ˆσ) =E_{S,ˆτ∧ˆσ}(ξ_τˆ1_{τ≤ˆˆ σ}+ζ_σˆ1_{σ<ˆˆ τ}) =E_{S,ˆτ∧ˆσ}(I(ˆτ ,σ))ˆ a.s.

Letτ ∈ TS. We want to show that for eachτ ∈ TS

Y_S ≥ ES,τ∧ˆσ(I(τ,σ))ˆ a.s. (4.6) Since the process(Yt, S≤t≤τ∧σ)ˆ is a strongE-supermartingale, we get

YS ≥ ES,τ∧ˆσ(Yτ∧ˆσ) a.s. (4.7)

SinceY ≥ξandY_σˆ =ζ_σˆa.s. , we also have

Y_τ∧ˆσ =Yτ1_τ≤ˆσ+Yσˆ1σ<τˆ ≥ ξτ1_τ≤ˆσ+ζˆσ1σ<τˆ =I(τ,ˆσ) a.s.

By inequality (4.7) and the monotonicity property ofE, we derive inequality (4.6).

Similarly, one can show that for eachσ∈ T_S, we have:

YS ≤ ES,ˆτ∧σ(I(ˆτ , σ)) a.s.

The pair(ˆτ ,ˆσ)is thus anS-saddle point andYS =V(S) =V(S)a.s.

We now provide an existence result under an additional assumption.

Theorem 4.8(Existence ofS-saddle points).Suppose thatgsatisfies Assumption(4.3).

Letξandζbe RCLL adapted processes inS²such thatξT =ζT a.s. andξt≤ζt,0≤t≤T a.s. Suppose that Mokobodzki’s condition is satisfied.

Let (Y, Z, k, A, A⁰) be the solution of the DRBSDE (2.2). Suppose that A, A⁰ are continuous (which is the case ifξand−ζare l.u.s.c. along stopping times). For eachS∈ T0, let

τ_S^∗:= inf{t≥S, Y_t=ξ_t}; σ_S^∗ := inf{t≥S, Y_t=ζ_t}.

τS:= inf{t≥S, At> AS}; σS := inf{t≥S, A⁰_t> A⁰_S}.

Then, for eachS ∈ T₀, the pairs of stopping times(τ_S^∗, σ^∗_S)and (τ_S, σ_S)areS-saddle points for the generalized Dynkin game andYS=V(S) =V(S)a.s.

Moreover,Yσ_S^∗ =ζσ_S^∗,Yτ_S^∗ =ξτ_S^∗,Aτ_S^∗ =AS andA⁰_σ∗

S =A⁰_S a.s. The same properties hold forτS, σS.

Remark 4.9.Note thatσ^∗_S ≤σ_S andτ_S^∗≤τ_S a.s. Moreover, by Proposition A.6 in the Appendix,(Y_t, S≤t≤τ_S)is a strongE-submartingale and(Y_t, S≤t≤σ_S)is a strong E-supermartingale.

Proof. LetS ∈ T0. SinceY andξare right-continuous processes, we haveY_σ^∗

S =ζ_σ^∗

S and Y_τ^∗

S =ξ_τ^∗

S a.s. By definition ofτ_S^∗, for almost every ω, we haveY_t(ω)> ξ_t(ω)for each t ∈[S(ω), τ_S^∗(ω)[. Hence, sinceY is solution of the DRBSDE, the continuous process Ais constant on[S, τ_S^∗]a.s. becauseA is continuous. Hence,Aτ_S^∗ = AS a.s. Similarly, A⁰_σ∗

S =A⁰_S a.s. By Lemma4.6,(τ_S^∗, σ_S^∗)is anS-saddle point andY_S =V(S) =V(S)a.s.

It remains to show that(τ_S, σ_S)is anS-saddle point. By definition ofτ_S,σ_S, we have Aτ_S =AS a.s. andA⁰_σ

S =A⁰_S a.s. becauseAandA⁰ are continuous. Moreover, since the continuous processAincreases only on{Yt=ξt}, we haveYτ_S =ξτ_S a.s. Similarly, Yσ_S =ζσ_S a.s. The result then follows from Lemma 4.6.

In the case of irregular payoffsξandζ, there does not generally exist a saddle point.

However, we will now see that it is not necessary to have the existence of anS-saddle point to ensure the existence of a common value function and its characterization as the solution of a DRBSDE.

Theorem 4.10 (Existence and characterization of the value function).Suppose thatg satisfies Assumption (4.3). Let ξ andζ be RCLL adapted processes inS² such that ξT =ζT a.s. andξt≤ζt,0≤t≤T a.s. Suppose that Mokobodzki’s condition is satisfied.

Let(Y, Z, k, A, A⁰)be the solution of the DRBSDE (2.2). Then, the generalized Dynkin game isfair, and for each stopping timeS∈ T₀, we have

Y_S =V(S) =V(S) a.s. (4.8)

Proof. For eachS∈ T₀and for eachε >0, letτ_S^εandσ^ε_S be the stopping times defined by τ_S^ε:= inf{t≥S, Yt≤ξt+ε} and σ_S^ε := inf{t≥S, Yt≥ζt−ε}. (4.9) We first prove two lemmas.

Lemma 4.11. • We have

Y_τ^ε

S ≤ ξ_τ^ε

S+ε a.s. (4.10)

Yσ_S^ε ≥ ζσ^ε_S −ε a.s. (4.11)

• MoreoverA_τ^ε

S =A_S a.s. andA⁰_σε

S =A⁰_S a.s.

Remark 4.12.By the second point and PropositionA.6 in the Appendix, the process (Yt, S≤t≤τ_S^ε)is a strongE-submartingale and the process(Yt, S≤t≤σ^ε_S)is a strong

E-supermartingale.

Proof. The first point follows from the definitions ofτ_S^εandσ^ε_S and the right-continuity of ξ, ζ and Y. Let us show the second point. Note that τ_S^ε ∈ TS and σ_S^ε ∈ TS. Fix ε >0. For a.e. ω, ift∈[S(ω), τ_S^ε(ω)[, thenY_t(ω)> ξ_t(ω) +εand henceY_t(ω)> ξ_t(ω). It follows that almost surely,A^c is constant on[S, τ_S^ε]andA^d is constant on[S, τ_S^ε[. Also, Y_(τε

S)⁻ ≥ξ_(τε

S)⁻+ε a.s. Sinceε >0, it follows thatY_(τε

S)⁻ > ξ_(τε

S)⁻ a.s., which implies that∆A^d_τε

S = 0a.s. Hence, almost surely, Ais constant on[S, τ_S^ε]. Similarly,A⁰ is a.s.

constant on[S, σ_S^ε].

Lemma 4.13.Letε >0. For allS∈ T0and(τ, σ)∈ T_S², we have E_S,τ∧σ^ε

S(I(τ, σ_S^ε))−Kε ≤ YS ≤ ES,τ_S^ε∧σ(I(τ_S^ε, σ)) +Kε a.s., (4.12) whereKis a positive constant which only depends onT and the Lipschitz constantCof g.