E l e c t ro n ic J P r o b a b il i t y 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 expectationEg, 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= E0g,τ∧σ 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. infσsupτJτ,σ =
supτinfσJτ,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 = (ξt), ζ = (ζt)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. infσsupτJτ,σ =
supτinfσJτ,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
Jτ,σ:= E Z τ∧σ 0 g(t)dt + ξτ1{τ ≤σ}+ ζσ1{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 ofEg-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 conditionalEg-expectation process
denoted by(Et,τg (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 a generalized Dynkin game, where the criterium is given, for each pair(τ, σ)of stopping times valued in[0, T ], by
Jτ,σ:= E0,τ ∧σg 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 criteriumJτ,σcoincides withJτ,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
inf
σ supτ Jτ,σ= supτ 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 ofEg-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
EQu,v Z τ∧σ 0 g(t, ut, vt)dt + ξτ1{τ ≤σ}+ ζσ1{σ<τ } , (1.2)
whereQu,vare a priori probability measures, andg(t, u
t, vt)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 game problem, where, for a given
family of nonlinear driversgu,v(t, y, z, k) := g(t, u
t, vt, y, z, k)depending on the controls
u, v, the criterium is defined by
E0,τ ∧σu,v ξτ1{τ ≤σ}+ ζσ1{σ<τ } , (1.3)
evaluated under the nonlinear expectationEu,v = Egu,v
. Note that the criterium (1.3)
corresponds to a criterium of the form (1.2) when the driversgu,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 our generalized mixed game problem, 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 the generalized Dynkin game withg-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 a generalized mixed game problem 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 a generalized Dynkin game, we prove comparison theorems and a priori estimates for DRBSDEs. Finally in Section 7, we study the generalized Dynkin game in 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], a nonlinear price system (later calledg-evaluation) is
defined as follows: for eachS ∈ [0, T ]and eachη ∈ L2(F
S), the initial (hedging) price of
an European option with maturitySand payoffηis given byE0,Sg (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 assupτE0,τg (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 the super-hedging price of 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
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) = ξτ1τ≤σ + ζσ1σ<τ. The main result of the present paper
yields that, under Mokobodzki’s condition, theg-value of the game option, defined as
infσsupτE g
0,τ ∧σ(I(τ, σ)), is equal to the common value of a generalized Dynkin 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-value of an American option (i.e.supτE0,τg (ξτ)), via a reflected
BSDE. In [14], using this result, we show that theg-value of the game option is equal to
the super-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(F
T) : the set of random variables ξ which are FT-measurable and square
integrable;
• IH2: the set of real-valued predictable processesφwithkφk2
IH2 := E h RT 0 φ 2 tdt i < ∞;
• S2: the set of real-valued RCLL adapted processesφwith
kφk2
S2 := E(sup0≤t≤T|φt|2) < ∞;
• A2(resp.A1) : the set of real-valued non decreasing RCLL predictable processes
AwithA0= 0andE(A2T) < ∞(resp. E(AT) < ∞).
• L2
ν : the set of Borelian functionsℓ : E → Rsuch that
R E|ℓ(e)|
2ν(de) < +∞.
The setL2
νis a Hilbert space equipped with the scalar product
hℓ, ℓ′i ν:=
R Eℓ(e)ℓ
′(e)ν(de)for allℓ, ℓ′∈ L2
ν×L2ν,and the normkℓk2ν :=
R E|ℓ(e)|
2ν(de).
• IH2
ν : the set of all mappings l : Ω × [0, T ] × E → R that are P ⊗ B(E)/B(R)
measurable and satisfyklk2
IH2 ν := E hRT 0 kltk 2 νdt i < ∞,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 a driver if • 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 a Lipschitz driver if 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(Xg, πg, lg) ∈ S2× IH2× IH2
ν satisfying − dXg(t) = g(t, Xtg−, π g(t), lg(t)(·))dt − πg(t)dW t− Z E lg(t)(e) ˜N (dt, de); XT = ξ. (2.1)
The solution is denoted by (Xg
.(ξ, T ), πg.(ξ, T ), lg.(ξ, T )). The operator Xg : (ξ, T ) 7→ X·g(ξ, T ), called nonlinear pricing system (associated with driverg), was first introduced
in [18]. In [33], this operator Xg is called the nonlinear evaluation (associated with
driverg) and is denoted byEg. In the sequel, we say thatEg
·,T(ξ)is theE
g
.,T-conditional
expectation process ofξ(or theg-evaluation of(ξ, T )). When there is no ambiguity on
the driver,Eg 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 andA′ = (A′t)0≤t≤T belonging toA1. We say that the
random measuresdAtanddA′tare mutually singular (in a probabilistic sense), and we
writedAt⊥ dA′t, if there existsD ∈ P such that:
E[ Z T 0 1DcdAt] = E[ Z T 0 1DdA′t] = 0,
which can also be written asRT
0 1DctdAt=
RT 0 1DtdA
′
t= 0 a.s.,where for eacht ∈ [0, T ],
Dtis the section at timetofD, that is,Dt:= {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
AandA′are introduced in order to push the solutionY aboveξand belowζin a minimal
way. This minimality property of A and A′ is ensured by the Skorohod conditions
(condition(iii)below) together with the additional constraintdAt⊥ dA′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 withζT = ξT
a.s.,ξ ∈ S2,ζ ∈ S2,ξ
t≤ ζt,0 ≤ t ≤ T a.s.
A process(Y, Z, k(.), A, A′)inS2× IH2× IH2
ν× A2× A2is said to be a solution of the
doubly reflected BSDE (DRBSDE) associated with drivergand barriersξ, ζ if
−dYt= g(t, Yt, Zt, kt(·))dt + dAt− dA′t− ZtdWt− Z E kt(e) ˜N (dt, de); YT = ξT, (2.2) with (i) ξt≤ Yt≤ ζt, 0 ≤ t ≤ T a.s., (ii) dAt⊥ dA′t (iii) Z T 0
(Yt− ξt)dAct= 0a.s. and
Z T 0
(ζt− Yt)dA′ct = 0a.s. ∆Ad
τ = ∆Adτ1{Yτ −=ξτ −}and ∆Aτ′d= ∆A′dτ1{Yτ −=ζτ −} a.s.∀τ ∈ T0predictable.
HereAc(respA′c) denotes the continuous part ofA(respA′) andAd(respA′d) its
discontinuous part.
Remark 2.5.WhenAandA′are not required to be mutually singular, they can
simul-taneously increase on{ξt− = ζt−}. The constraintdAt ⊥ dA′tallows us to obtain the
2Note that if the random measuresdA
tanddA′tare mutually singular in the above probabilistic sense,
then, for almost everyω, the deterministic measures on[0, T ] dAt(ω)anddA′t(ω)are mutually singular in the
uniqueness of the non decreasing RCLL processesAandA′, 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
(τn)such thatτn↑ τ a.s. ,
φτ ≥ lim sup n→∞
φτn a.s. (resp. E[φτ] ≥ lim sup
n→∞
E[φτn]). (2.3)
Remark 2.7.When(φt)is left-limited, then(φt)is left-upper semicontinuous (l.u.s.c.)
along stopping times if and only if for all predictable stopping timeτ ∈ T0,φτ≥ φτ− 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 ∈ H2. 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, A′), 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∈TS ess sup τ∈TS E[IS(τ, σ)|FS] (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 is fair (or there exists a value function) at timeS ifV (S) = V (S)a.s.
Definition 3.1 (S-saddle point).LetS ∈ T0. A pair(τ∗, σ∗) ∈ TS2is called anS-saddle
point if for each(τ, σ) ∈ T2
S, 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): ˜ ξtg:= ξt− E[ξT + Z T t g(s)ds|Ft], ζ˜tg:= ζt− E[ζT + Z T t g(s)ds|Ft], 0 ≤ t ≤ T. (3.4)
Note that since g ∈ H2 and ξ ∈ S2, ξ˜g and ζ˜g
t belong to S2. Moreover, we have
˜
ξTg = ˜ζTg = 0 a.s.
Definition 3.2.An optional processφ valued in[0, +∞]is said to be a strong
super-martingale if for anyθ, θ′∈ T
SetJ·g,0= 0andJ·′ g,0
= 0. We define recursively for alln ∈ N, the RCLL
supermartin-galesJ·g,nandJ·′g,nsatisfying for allθ ∈ T0the equalities3
Jθg,n+1= ess sup τ∈Tθ EhJ′ τ g,n + ˜ξg τ|Fθ i a.s. and J′ θ g,n+1 = ess sup τ∈Tθ EhJg,n τ − ˜ζτg|Fθ i a.s. (3.5)
Lemma 3.3.The sequences of processes(J·g,n)n∈N and(J′g,n· )n∈Nare non decreasing.
Moreover, the processesJ·g andJ·′
gdefined for all
t ∈ [0, T ]byJtg:= limn→+∞Jtg,n
andJ′
t
g := lim
n→+∞ Jt′
g,n are strong supermartingales valued in[0, +∞]. They satisfy
JTg = J′ T
g
= 0a.s. and for allθ ∈ T0,
Jθg = ess sup τ∈Tθ EhJτ′ g + ˜ξgτ|Fθ i a.s. and Jθ′ g = ess sup σ∈Tθ EhJσg− ˜ζσg|Fθ i a.s. (3.6) If J0g< +∞and J0′ g
< +∞, thenJg andJ′gare RCLL supermartingales.
Proof. See Appendix.
Using this lemma, we derive that if Jg andJ′g 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 inS2withζ
T = ξT a.s. and
ξt≤ ζt,0 ≤ t ≤ T a.s. Suppose that Jg, J′g ∈ S2. LetY be the RCLL adapted process
defined by Yt:= Jtg− Jt′ g + E[ξT + Z T t g(s)ds|Ft]; 0 ≤ t ≤ T. (3.7) There exist(Z, k, A, A′) ∈ IH2× IH2
ν× A2× A2such that(Y , Z, k, A, A′)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 processesJgandJ′g
are finite. Hence, the differenceJg− J′g and thus the processY are well defined. By
Lemma 3.3, we haveJTg = J′
T g
a.s. Hence,YT = ξT a.s. By (3.6), we haveJg ≥ J′g+ ˜ξg
andJ′g≥ Jg− ˜ζg. Using the definitions ofξ˜g,ζ˜gandY, we derive thatξ ≤ Y ≤ ζ.
Moreover, by the last assertion of Lemma 3.3 and the assumption Jg, J′g ∈ S2, we
obtain thatJgandJ′g are indistinguishable from RCLL supermartingales. We thus can
apply the Doob-Meyer decomposition, and derive the existence of two square integrable
martingalesM andM′and two processesB andB′ ∈ A2such that:
dJtg= dMt− dBt ; dJt′ g = dMt′− dBt′. (3.8) Set Mt:= Mt− Mt′+ E[ξT+ Z T 0 g(s)ds|Ft]. By (3.8), (3.7), we derivedYt= dMt− dαt− g(t)dt, withα := B − B′.
Now, by the martingale representation theorem, there existZ ∈ H2andk ∈ H2
ν such
thatdMt= ZtdWt+REkt(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 Bc increases only when the value function Jg is equal to the corresponding
rewardJ′g+ ˜ξg. Now, { Jg t = Jt′ g + ˜ξg} = {Y t = ξt}. Hence, R T 0 (Yt− ξt)dB c t = 0 a.s.
Similarly the processB′csatisfiesRT
0 (Yt− ζt)dB ′ t
c= 0
a.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 ∆Bτd = 1Jg
τ −=J ′g τ −+ ˜ξ g τ − ∆Bd τ = 1Yτ −=ξτ −∆Bτd a.s. and ∆Bτ′d=1Yτ −=ζτ −∆B′dτ a.s.
By the canonical decomposition of an RCLL predictable process with integrable total
variation (see Proposition A.7), there existA, A′ ∈ A2such thatα = A−A′withdA
t⊥ dA′t.
Also, dAt<< dBt. Hence, since
RT
0 1Yt−>ξt−dBt = 0 a.s. , we get RT
0 1Yt−>ξt−dAt = 0
a.s. Similarly, we obtainRT
0 1Yt−<ζt−dA′t= 0a.s. The processesAandA′ thus satisfy
conditions (2.2)(iii) (withY replaced byY). The process(Y , Z, k, A, A′)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 = ξT a.s. and
ξt ≤ ζt, 0 ≤ t ≤ T a.s. Suppose that Jg, J′g ∈ S2. The doubly reflected BSDE (2.2)
associated with driver processg(t)admits a unique solution(Y, Z, k, A, A′)inS2× IH2×
IH2
ν× (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, A′ are continuous, then, for each S ∈ T
0, the pair of
stopping times(τ∗
s, σs∗)defined by
σ∗S := inf{t ≥ S, Yt= ζt}; τS∗:= 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 conditiondAt⊥ dA′tensures that for each predictable stopping time τ ∈ T0,∆Adτ∆A′dτ = 0a.s. Now, sinceY satisfies (2.2), we have∆Yτ= ∆A′dτ − ∆Adτ a.s.
We thus have∆Ad
τ = (∆Yτ)−and∆A′dτ = (∆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 ofAandA′.
Theorem 3.7 (Existence ofS-saddle points).Suppose that the assumptions of Theorem
3.5 are satisfied. Let(Y, Z, k(.), A, A′)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. A′) 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, σS∗)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 supposedξt< ζt, t < T a.s. (see e.g. [1, 9, 31]).
Proof. Suppose thatξ is l.u.s.c. along stopping times. Let τ ∈ T0 be a predictable
stopping time. Let us show∆Aτ= 0a.s.
Since dAt ⊥ dA′t, we have∆Aτ = (∆Yτ)− a.s. (see Remark 3.6 above). SinceA
satisfies the Skorohod condition, we have a.s.
where the last inequality follows from the inequalityξτ− ≤ ξτ a.s. (see Remark 2.7).
Sinceξ ≤ Y, we derive that∆Aτ ≤ 0a.s. Hence,∆Aτ = 0a.s. , and this holds for each
predictable stopping timeτ. Consequently,Ais continuous. Similarly, one can show that
if−ζ is l.u.s.c. along stopping times, thenA′is 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 ofAandA′ensures 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 andH′ inS2such that:
ξt≤ Ht− Ht′≤ 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) Jg∈ 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 thatξ
T = 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, A′) ∈ S2×IH2×IH2
ν×(A2)2.
If ξ (resp. −ζ) is l.u.s.c. along stopping times, then the process A (resp. A′) 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, Yu, Zu, ku)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 a generalized Dynkin game
expressed in terms ofEg-expectations.
In order to ensure that theEg-expectation is non decreasing, we make the following
Assumption 4.3.Assume thatdP ⊗ dt-a.s for each(y, z, k1, k2) ∈ R2× (L2ν)2, g(t, y, z, k1) − g(t, y, z, k2) ≥ hγty,z,k1,k2, k1− k2iν,
with γ : [0, T ] × Ω × R2× (L2ν)2→ L2ν; (ω, t, y, z, k1, k2) 7→ γty,z,k1,k2(ω, .) P · ⊗B(R2) ⊗ B((L2
ν)2)-measurable and satisfying the inequalities
γy,z,k1,k2
t (e) ≥ −1 and kγ
y,z,k1,k2
t kν≤ K, (4.2)
for each (y, z, k1, k2) ∈ R2× (L2ν)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 ∈ L2ν.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 inL 2 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 ofEgby 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 thatEgis strictly monotonous.
For eachτ, σ ∈ T0, the reward (or payoff) 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
associated criterium is given byES,τg ∧σ(I(τ, σ)), theg-evaluation of the payoffI(τ, σ).
Recall thatE·,τ ∧σg (I(τ, σ)) = X·τ,σ,where(X·τ,σ, π·τ,σ, lτ,σ· )is the solution of the BSDE
associated with driverg, terminal timeτ ∧ σand terminal conditionI(τ, σ), that is
−dXτ,σ
s = g(s, Xsτ,σ, πsτ,σ, lτ,σs )ds − πsτ,σdWs− Z
E lτ,σ
s (e) ˜N (ds, de); Xτ∧στ,σ = I(τ, σ).
To simplify notation,Egis denoted byEin the sequel.
For each stopping timeS ∈ T0, the upper and lower value functions at timeS are
defined respectively by V (S) := ess inf σ∈TS ess sup τ∈TS ES,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 (or there 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∗) ∈ TS2 is called an S-saddle point for the
generalized Dynkin game if for each(τ, σ) ∈ T2
S we have
ES,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 strongEg-supermartingale
(respEg-submartingale), if Eg
σ,τ(Yτ) ≤ Yσ (resp. Eσ,τg (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 = ζT a.s. andξt ≤ ζt, 0 ≤ t ≤ T a.s. Suppose
that Mokobodzki’s condition is satisfied. Let (Y, Z, k(·), A, A′) be the solution of the
DRBSDE (2.2). LetS ∈ T0. Let(ˆτ , ˆσ) ∈ TS2. Suppose that(Yt, S ≤ t ≤ ˆτ )is a strong
E-submartingale and that(Yt, S ≤ t ≤ ˆσ)is a strongE-supermartingale withYˆτ= ξˆτand
Yσˆ = ζσˆa.s.
The pair(ˆτ , ˆσ)is then anS-saddle point for the generalized 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 , ˆσ)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 problemJSg (resp. J′
S g
) (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 ≤ ˆτ ∧ ˆσ)is a strongE-martingale (see Definition 4.5)
and sinceYˆτ= ξˆτandYˆσ= ζσˆa.s. , we have
YS = ES,ˆτ∧ˆσ(Yˆτ∧ˆσ) = ES,ˆτ∧ˆσ(ξτˆ1τ≤ˆˆ σ+ ζσˆ1σ<ˆˆ τ) = ES,ˆτ∧ˆσ(I(ˆτ , ˆσ)) a.s.
Letτ ∈ TS. We want to show that for eachτ ∈ TS
YS ≥ ES,τ∧ˆσ(I(τ, ˆσ)) a.s. (4.6)
Since the process(Yt, S ≤ t ≤ τ ∧ ˆσ)is a strongE-supermartingale, we get
YS ≥ ES,τ∧ˆσ(Yτ∧ˆσ) a.s. (4.7)
SinceY ≥ ξandYσˆ = ζσˆa.s. , we also have
Yτ∧ˆσ = Yτ1τ≤ˆσ+ Yσˆ1σ<τˆ ≥ ξτ1τ≤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 ∈ TS, 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 thatξ
T = ζT a.s. andξt≤ ζt,0 ≤ t ≤ T
a.s. Suppose that Mokobodzki’s condition is satisfied.
Let (Y, Z, k, A, A′) be the solution of the DRBSDE (2.2). Suppose that A, A′ are
continuous (which is the case ifξand−ζ are l.u.s.c. along stopping times). For eachS ∈
T0, let
τ∗
S := inf{t ≥ S, Yt= ξt}; σS∗ := inf{t ≥ S, Yt= ζt}. τS:= inf{t ≥ S, At> AS}; σS := inf{t ≥ S, A′t> A′S}.
Then, for eachS ∈ T0, the pairs of stopping times(τS∗, σ∗S)and (τS, σS)areS-saddle
points for the generalized Dynkin game andYS= V (S) = V (S)a.s.
Moreover,Yσ∗ S = ζσS∗,YτS∗ = ξτS∗,AτS∗ = AS andA ′ σ∗ S = A ′
S a.s. The same properties
Remark 4.9.Note thatσ∗S ≤ σS andτS∗≤ τS a.s. Moreover, by Proposition A.6 in the
Appendix,(Yt, S ≤ t ≤ τS)is a strongE-submartingale and(Yt, S ≤ t ≤ σS)is a strong
E-supermartingale.
Proof. LetS ∈ T0. SinceY andξare right-continuous processes, we haveYσ∗
S = ζσ∗S and
Yτ∗
S = ξτS∗ a.s. By definition ofτ
∗
S, for almost every ω, we haveYt(ω) > ξt(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,
A′ σ∗
S = A
′
S a.s. By Lemma4.6,(τS∗, σS∗)is anS-saddle point andYS = V (S) = V (S)a.s.
It remains to show that(τS, σS)is anS-saddle point. By definition ofτS,σS, we have
AτS = AS a.s. andA
′ σS = A
′
S a.s. becauseAandA′ are continuous. Moreover, since
the continuous processAincreases only on{Yt= ξt}, we haveYτS = ξτS a.s. Similarly,
YσS = ζσS 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). Letξ 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, A′)be the solution of the DRBSDE (2.2). Then, the generalized Dynkin
game is fair, and for each stopping timeS ∈ T0, we have
YS = V (S) = V (S) a.s. (4.8)
Proof. For eachS ∈ T0and for eachε > 0, letτSεandσεS be the stopping times defined by
τε
S := inf{t ≥ S, Yt≤ ξt+ ε} and σSε := inf{t ≥ S, Yt≥ ζt− ε}. (4.9)
We first prove two lemmas.
Lemma 4.11. • We have Yτε S ≤ ξτSε+ ε a.s. (4.10) Yσε S ≥ ζσεS − ε a.s. (4.11) • MoreoverAτε S = AS a.s. andA ′ σε S = A ′ S a.s.
Remark 4.12.By the second point and PropositionA.6in the Appendix, the process
(Yt, S ≤ t ≤ τSε)is a strongE-submartingale and the process(Yt, S ≤ t ≤ σεS)is a strong
E-supermartingale.
Proof. The first point follows from the definitions ofτε
S andσεS and the right-continuity
of ξ, ζ and Y. Let us show the second point. Note that τε
S ∈ TS and σSε ∈ TS. Fix
ε > 0. For a.e. ω, ift ∈ [S(ω), τε
S(ω)[, thenYt(ω) > ξt(ω) + εand henceYt(ω) > ξt(ω). It
follows that almost surely,Ac is constant on[S, τε
S]andAd is constant on[S, τSε[. Also,
Y(τε
S)− ≥ ξ(τ ε
S)−+ ε a.s. Sinceε > 0, it follows thatY(τ
ε
S)− > ξ(τ ε
S)− a.s., which implies
that∆Ad
τε
S = 0a.s. Hence, almost surely, Ais constant on[S, τ
ε
S]. Similarly,A′ is a.s.
constant on[S, σε
S].
Lemma 4.13.Letε > 0. For allS ∈ T0and(τ, σ) ∈ TS2, we have ES,τ∧σε S(I(τ, σ ε S)) − Kε ≤ YS ≤ ES,τε S∧σ(I(τ ε S, σ)) + Kε a.s., (4.12)
whereKis a positive constant which only depends onT and the Lipschitz constantCof
Proof. Letτ ∈ TS.By Remark4.9, the process(Yt, S ≤ t ≤ σSε)is a strongE -supermartin-gale. Hence, YS ≥ ES,τ∧σε S(Yτ∧σSε) a.s. (4.13) SinceY ≥ ξandYσε S ≥ ζσ ε
S − εa.s. (see Lemma 4.11), we have:
Yτ∧σε S ≥ ξτ 1τ≤σε S+ (ζσSε − ε)1σSε<τ ≥ I(τ, σ ε S) − ε a.s.
where the last inequality follows from the definition ofI(τ, σ). Hence, using (4.13) and
the monotonicity property ofEg, we get
YS ≥ ES,τ∧σε S(I(τ, σ
ε
S) − ε) a.s. (4.14)
Now, by a priori estimates on BSDEs (see [35, Proposition A.4]), we have
|ES,τ∧σε S(I(τ, σ ε S) − ε) − ES,τ∧σε S(I(τ, σ ε S))| ≤ Kε a.s. It follows that YS ≥ ES,τ∧σε S(I(τ, σ ε S)) − Kε a.s.
Similarly, one can show thatYS ≤ ES,τε
S∧σ(I(τ
ε
S, σ)) + Kε.
End of proof of Theorem 4.10. Using Lemma 4.13, we derive that for eachε > 0,
ess sup τ∈TS ES,τ∧σε S(I(τ, σ ε S)) − Kε ≤ YS ≤ ess inf σ∈TS ES,τε S∧σ(I(τ, σ ε S)) + Kε a.s., which implies V (S) − Kε ≤ YS ≤ V (S) + Kε a.s.
SinceV (S) ≤ V (S)a.s. , we getV (S) = YS = V (S)a.s. The proof of Theorem 4.10 is
thus complete.
Remark 4.14.Inequality (4.12) shows that(τε
S, σεS)defined by (4.9) is anε′-saddle point
at timeS withε′= Kε.
Remark 4.15.Note that contrary to the classical Dynkin game with payoff (4.1) (see Remark 4.2), the generalized Dynkin game is well-posed in the sense that the criterium
does not depend on the value function. This new characterization of the solutionY of
the nonlinear DRBSDE (2.2) in terms of the value function of the generalized Dynkin game is thus more interesting and exploitable than the one given in the literature. We will see in Section 6 that this result allows us to show a comparison theorem and a strict comparison theorem for DRBSDEs, as well as some new estimates with universal constants.
5
Generalized mixed game problems
We now introduce a game problem, which can be seen as a generalization of a mixed
game problem studied in [4, 25, 21] to the case ofEg-expectation/g-evaluation. The
players have two actions: continuous control and stopping. Let(gu,v; (u, v) ∈ U × V)be a
family of Lipschitz drivers satisfying Assumption (4.3). LetS ∈ T0.For each quadruple
(u, τ, v, σ) ∈ U × TS× V × TS, the criterium at timeSis given byES,τu,v∧σ(I(τ, σ)),whereEu,v
corresponds to thegu,v-evaluation. 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.
For each stopping timeS ∈ T0, the upper and lower value functions at timeS are
defined respectively by V (S) := ess inf v∈V,σ∈TS ess sup u∈U ,τ ∈TS ES,τ∧σu,v (I(τ, σ)); (5.1)
V (S) := ess sup u∈U ,τ ∈TS
ess inf v∈V,σ∈TS
ES,τ∧σu,v (I(τ, σ)). (5.2)
We say that the game is fair (or there exists a value function) at timeSifV (S) = V (S)
a.s. We now introduce the definition of anS-saddle point for this game problem.
Definition 5.1.LetS ∈ T0. A quadruple(u, τ , v, σ) ∈ U ×TS×V ×TSis called an S-saddle
point for the generalized mixed game problem if for each(u, τ, v, σ) ∈ U × TS× V × TS
we have
ES,τ∧σu,v (I(τ, σ)) ≤ ES,τu,v∧σ(I(τ ∧ σ)) ≤ ES,τ∧σu,v (I(τ , σ)) a.s.
We prove below that when the obstacles are l.u.s.c. along stopping times, there exist saddle points for the above generalized mixed game problem.
Theorem 5.2.Let(gu,v; (u, v) ∈ U × V)be a family of Lipschitz drivers satisfying
As-sumptions (4.3). Letξandζbe RCLL adapted processes inS2and l.u.s.c. along stopping
times, such thatξT = ζT a.s. andξt≤ ζt, 0 ≤ t ≤ T a.s. Suppose that Mokobodzki’s
condition is satisfied and that there exist controlsu ∈ U andv ∈ V such that for each
(u, v) ∈ U × V,
gu,v(t, Yt, Zt, kt) ≤ gu,v(t, Yt, Zt, kt) ≤ gu,v(t, Yt, Zt, kt) dt ⊗ dP a.s. , (5.3)
where (Y, Z, k, A, A′)is the solution of the DRBSDE (2.2) associated with drivergu,v.
Consider the stopping times
τS∗:= inf{t ≥ S : Yt= ξt} ; σ∗S:= inf{t ≥ S : Yt= ζt}.
The quadruple(u, τ∗
S, v, σ∗S)is then an S-saddle point for the generalized mixed game
problem (5.1)-(5.2), and we haveYS = V (S) = V (S)a.s.
Proof. By the last assertion of Theoreom4.8, the process(Yt, S ≤ t ≤ τS∗∧ σ∗S)is a strong
Eu,v-martingale andY
τ∗
S = ξτS∗,YσS∗ = ζσ∗S a.s. , which implies
YS = ES,τu,v∗ S∧σ∗S(Yτ ∗ S∧σ∗S) = E u,v S,τ∗ S∧σS∗(ξτ ∗ S1τS∗≤σS∗ + ζσS∗1σ∗S<τS∗) = E u,v S,τ∗ S∧σ∗S(I(τ ∗ S, σ∗S)) a.s.
Letτ ∈ TS. SinceY ≥ ξ andYσ∗
S = ζσS∗ a.s. , we have Yτ∧σ∗ S = Yτ1τ≤σ∗S+ Yσ∗S1σS∗<τ ≥ ξτ1τ≤σ∗S + ζσ∗S1σ∗S<τ = I(τ, σ ∗ S) a.s. Moreover, by Theorem 4.8,A′ σ∗ S = A ′
S a.s., which implies that:
−dYt= gu,v(t, Yt, Zt, kt)dt + dAt− ZtdWt− Z
E
kt(e) ˜N (dt, de); S ≤ t ≤ σS∗, dt ⊗ dP a.s.
Hence, (Yt)S≤t≤τ ∧σ∗
S is the solution of the BSDE associated with generalized driver
gu,v(·)dt + dA
tand terminal conditionYτ∧σ∗
S . By using Assumption (5.3), the inequality
Yτ∧σ∗
S ≥ I(τ, σ
∗
S)and the comparison theorem for BSDEs with jumps, we obtain that for
eachu ∈ U:
YS ≥ ES,τ∧σu,v ∗ S(I(τ, σ
∗ S)) a.s.
Similarly, one can prove that for eachv ∈ V, σ ∈ TS, we have:
YS ≤ ES,τu,v∗ S∧σ(I(τ
∗
S, σ)) a.s.
The quadruple(u, τ∗
S, v, σS∗)is thus anS-saddle point andYS = V (S) = V (S)a.s.
Under less restricted assumptions on the obstacles, we prove below that the above game problem is fair, and the common value function can be characterized as the solution of a DRBSDE.
Theorem 5.3 (Existence and characterization of the value function).Let(gu,v; (u, v) ∈ U × V)be a family of drivers satisfying Assumptions (4.3) and uniformly Lipschitz with
common Lipchitz constantC. Letξandζbe RCLL adapted processes inS2such that
ξT = ζT a.s. andξt≤ ζt,0 ≤ t ≤ T a.s. Suppose that Mokobodzki’s condition is satisfied
and that there exist controlsu ∈ U andv ∈ V such that for eachu ∈ U, v ∈ V:
gu,v(t, Y
t, Zt, kt) ≤ gu,v(t, Yt, Zt, kt) ≤ gu,v(t, Yt, Zt, kt), dt ⊗ dP a.s. (5.4)
where(Y, Z, k, A, A′)is the solution of the DRBSDE (2.2) associated with drivergu,v.
Then, the generalized mixed game problem (5.1)-(5.2) is fair, and for each stopping timeS ∈ T0, we have
YS = V (S) = V (S) a.s.
Proof. For eachS ∈ T0and for eachε > 0, letτSεandσεS be the stopping times defined by
τε
S := inf{t ≥ S, Yt≤ ξt+ ε}; σεS := inf{t ≥ S, Yt≥ ζt− ε}.
Letτ ∈ TS. SinceY ≥ ξ andYσε
S ≥ ζσSε − εa.s. (see Lemma 4.11), we have:
Yτ∧σε S ≥ ξτ 1τ≤σε S+ (ζσSε − ε)1σSε<τ ≥ I(τ, σ ε S) − ε a.s. By Lemma4.11,A′ σε S = A ′
S a.s. which implies that:
−dYt= gu,v(t, Yt, Zt, kt)dt + dAt− ZtdWt− Z
E
kt(e) ˜N (dt, de), S ≤ t ≤ σSε, dt ⊗ dP a.s.
Hence, (Yt)S≤t≤τ ∧σε is the solution of the BSDE associated with generalized driver
f (·)dt+ dAtand terminal conditionYτ∧σε. Using Assumption (5.4), the inequalityYτ∧σε≥
I(τ, σε) − εand the comparison theorem for BSDEs with jumps, we obtain YS ≥ ESu,v(I(τ, σ
ε) − ε) ≥ Eu,v S (I(τ, σ
ε)) − Kε a.s. ,
where the second inequality follows from a priori estimates for BSDEs with jumps.
Here, the constant K only depends on T andC, the common Lipschitz constant.
Consequently, we get
YS ≥ ess inf v∈V,σ∈TS
ess sup u∈U ,τ ∈TS
ES,τu,v∧σ(I(τ, σ)) − Kε a.s.
Similarly, one can prove that for eachε > 0,
YS ≤ ess sup u∈U ,τ ∈TS
ess inf v∈V,σ∈TS
ES,τ∧σu,v (I(τ, σ)) + Kε a.s.
Hence,V (S) ≤ V (S)a.s. SinceV (S) ≤ V (S)a.s., the equality follows.
Remark 5.4.Theorem 5.3 still holds ifgu,vis replaced by any Lipschitz drivergwhich
satisfies (5.4).
Application: the case of control processes LetU, V be compact Polish spaces. In
the following,Ωis the canonical space defined in [13] (in Section 2). We are given a map
F : Ω × [0, T ] × U × V × R2× L2
ν → R, (ω, t, u, v, y, z, k) 7→ F (ω, t, u, v, y, z, k), supposed to
be measurable with respect toP ⊗B(U )⊗B(V )⊗B(R2)⊗B(L2
ν),continuous, concave with
respect touand convex with respect tov, and uniformly Lipchitz with respect to(y, z, k).
Suppose thatF isC1 with respect tokwith ∇
kF ≥ −1, and thatF (ω, t, u, v, 0, 0, 0) is
uniformly bounded. LetU (resp.V) be the set of predictable processes valued inU (resp.
V). For each(u, v) ∈ U × V, letgu,vbe the driver defined by
Letξandζbe RCLL adapted processes inS2such thatξT = ζT a.s. andξt≤ ζt,0 ≤ t ≤ T
a.s. Suppose that Mokobodzki’s condition is satisfied. Let us consider the associated
generalized mixed game problem. Define for each(t, ω, y, z, k)the map
g(t, ω, y, z, k) = sup u∈U
inf
v∈VF (t, ω, u, v, y, z, k). (5.6)
SinceU andV are Polish spaces, there exist some dense countable subsetsU (resp. V)
ofU (resp. V). SinceF is continuous with respect tou, v, the sup (resp. inf) can be
taken overU (resp.V). Hence,gis a Lipschitz driver.
Let(Y, Z, k, A, A′) ∈ S2× H2× H2
ν× (A2)2be the solution of the DRBSDE associated
with drivergand obstaclesξandζ. By classical convex analysis, for each(t, ω)there
exist(u∗, v∗) ∈ (U, V )such that
F (ω, t, u, v∗, Yt−(ω), Zt(ω), kt(ω)) ≤ F (ω, t, u∗, v∗, Yt−(ω), Zt(ω), kt(ω)) (5.7) ≤ F (ω, t, u∗, v, Y
t−(ω), Zt(ω), kt(ω)), ∀(u, v) ∈ U × V ; g(ω, t, Yt−(ω), Zt(ω), kt(ω))) = F (ω, t, u∗, v∗, Yt−(ω), Zt(ω), kt(ω))
Let (u, v) ∈ U × V. Since the processes Yt−, Zt and kt are predictable, the map (ω, t, u∗, v∗) 7→ (ω, t, u, v∗, Y
t−(ω), Zt(ω), kt(ω))is measurable with respect to theσ
-alge-brasP ⊗B(U )⊗B(V )andP ⊗B(U )⊗B(V )⊗B(R2)⊗B(L2
ν). Using the measurability
prop-erty ofF, it follows by composition that the map(ω, t, u∗, v∗) 7→ F (ω, t, u, v∗, Y
t−(ω), Zt(ω), kt(ω))isP ⊗ B(U ) ⊗ B(V )-measurable, and similarly for the other maps which appear
in (5.7). Hence, the set of all(ω, t, u∗, v∗) ∈ Ω × [0, T ] × U × V satisfying conditions (5.7)
belongs toP ⊗ B(U ) ⊗ B(V ). SinceΩis a Polish space for the Skorohod metric (see
[13] sect. 2), by applying a measurable selection theorem (see e.g. Section 81 in the Appendix of Ch. III in [10]) and Lemma 1.2 in [8], we derive the existence of a pair of
predictable process(u∗, v∗) ∈ U × V such thatdt ⊗ dP a.s., for all(u, v) ∈ U × Vwe have:
F (t, ut, vt∗, Yt, Zt, kt) ≤ F (t, u∗t, vt∗, Yt, Zt, kt) ≤ F (t, u∗t, vt, Yt, Zt, kt)
andg(t, Yt, Zt, kt) = F (t, u∗t, vt∗, Yt, Zt, kt). Hence, Assumption (5.3) is satisfied. By
apply-ing Theorems 5.3 and 5.2, we derive the followapply-ing result:
Proposition 5.5.The generalized mixed game problem, associated with the controlled
drivers gu,v given by (5.5), is fair. LetY be the solution of the DRBSDE associated
with obstaclesξ,ζand the drivergdefined by (5.6). For each stopping timeS ∈ T0, we
haveYS = V (S) = V (S)a.s. Suppose thatξandζare l.u.s.c. along stopping times, and
consider the stopping times
τ∗
S := inf{t ≥ S : Yt= ξt} ; σ∗S:= inf{t ≥ S : Yt= ζt}.
The quadruple(u∗, τ∗
S, v∗, σS∗)is then anS-saddle point for this generalized mixed game
problem.
We give now an example of application of the above proposition.
Example 5.6 (the classical linear case).Consider the particular case when F takes
the following form:F (t, ω, u, v, y, z, k) = β(t, ω, u, v)z+ < γ(t, ω, u, v, ·), k >ν +c(t, ω, u, v),
withβ, γ, cbounded. By classical results on linear BSDEs (see [35]), the criterium can be written
ES,τu,v∧σ(I(τ, σ)) = EQu,v
Z τ∧σ S
c(t, ut, vt)dt + I(τ, σ)|FS
,
withQu,vthe probability measure which admitsZu,v
T as density with respect toP, where
(Ztu,v)is the solution of the following SDE:
dZtu,v= Z u,v t [β(t, ut, vt)dWt+ Z E γ(t, ut, vt, e) ˜N (dt, de)]; Z0u,v= 1.
The processc(t, ut, vt)can be interpreted as an instantaneous reward associated with
controls u, v. This linear model takes into account ambiguity on the model via the
probability measuresQu,vas well as ambiguity on the instantaneous reward. This case
corresponds to the “classical” mixed game problems studied in [4, 25, 21].
6
Comparison theorems for DRBSDEs with jumps and a priori
estimates
Thanks to the characterization of the solution of the nonlinear DRBSDE as the value function of a generalized Dynkin game (Theorem 4.10), we now establish a comparison theorem and a strict comparison theorem for DRBSDEs, as well as some new estimates with universal constants.
6.1 Comparison theorems
Theorem 6.1 (Comparison theorem for DRBSDEs).Letξ1,ξ2,ζ1,ζ2be processes inS2
such thatξi
T = ζTi a.s. andξti ≤ ζti,0 ≤ t ≤ T a.s. fori = 1, 2. Suppose that fori = 1, 2,
ξi, ζi satisfies Mokobodzki’s condition. Let g1and g2 be Lipschitz drivers satisfying
Assumption (4.3). Let(Yi, Zi, ki, Ai, A′i
)be the solution of the DRBSDE associated with
(ξi, ζi, gi),i = 1, 2. Suppose that (i) ξ2 t ≤ ξ1t andζt2≤ ζt1,0 ≤ t ≤ T a.s. (ii) g2(t, Y2 t, Zt2, kt2) ≤ g1(t, Yt2, Zt2, k2t), 0 ≤ t ≤ T dP ⊗ dt − a.s. We then have: Yt2≤ Yt1, 0 ≤ t ≤ T a.s.
Remark 6.2.A comparison theorem has been provided in [7] in the case of jumps under stronger assumptions, with a different proof based on Itô’s calculus.
Proof. We give a short proof based on the characterization of solutions of DRBSDEs (Theorem 4.10) via generalized Dynkin games.
Step 1: Let us first suppose that condition (i) holds and that g1 and g2 satisfy: g2(t, y, z, k) ≤ g1(t, y, z, k)for all (y, z, k) ∈ R2× L2
ν dP ⊗ dt-a.s. (which is a stronger
assumption than(ii)). Lett ∈ [0, T ]. For eachτ, σ ∈ Tt, let us denote byE.,τ∧σi (Ii(τ, σ))the
unique solution of the BSDE associated with drivergi, terminal timeτ ∧ σand terminal
conditionIi(τ, σ) := ξi
τ1τ≤σ+ ζσi1σ<τ fori = 1, 2. Sinceg2≤ g1, andI2(τ, σ) ≤ I1(τ, σ),
by the comparison theorem for BSDEs, the following inequality
Et,τ∧σ2 (I2(τ, σ)) ≤ Et,τ∧σ1 (I1(τ, σ)) a.s.
holds for eachτ,σinTt. Hence, by taking the essential supremum overτinTtand the
essential infimum overσinTt, and by using Theorem 4.10, we get
Y2 t = ess infσ∈T t ess sup τ∈Tt E2 t,τ∧σ(I2(τ, σ)) ≤ ess inf σ∈Tt ess sup τ∈Tt E1 t,τ∧σ(I1(τ, σ)) = Yt1a.s.
Step 2: Suppose that conditions(i)and(ii)hold. Letδgbe the process defined byδgt:= g2(t, Y2
t, Zt2, k2t) − g1(t, Yt2, Zt2, kt2). Note that (Y2, Z2, k2)is the solution the DRBSDE
associated with barriers ξ2, ζ2 and driver g1(t, y, z, k) + δg
t. Now, by (ii), we have
g1(t, y, z, k) + δg
t≤g1(t, y, z, k)for all(y, z, k). By Step 1 applied to the driverg1and the
driverg1(t, y, z, k) + δg
t(instead ofg2), we getY2≤ Y1.
We now provide a strict comparison theorem for DRBSDEs. Note that no strict comparison theorem exists in the literature even in the Brownian case. The first assertion addresses the particular case when the non decreasing processes are continuous and the second one deals with the general case.
Theorem 6.3 (Strict comparison for DRBSDEs.).Suppose that the assumptions of
Theo-rem 6.1 hold and that the driverg1satisfies Assumption 4.3 withγ
t> −1in (4.2). LetS
inT0and suppose thatYS1= YS2a.s.
1. Suppose thatAi, A′i,i = 1, 2are continuous. Fori = 1, 2, let
τi = τi,S:= inf{s ≥ S; Ais> AiS}andσi= σi,S:= inf{s ≥ S; A′is > A′iS}. Then Y1
t = Yt2, S ≤ t ≤ τ1∧ τ2∧ σ1∧ σ2 a.s.
and
g2(t, Yt2, Zt2, k2t) = g1(t, Yt2, Zt2, kt2) S ≤ t ≤ τ1∧ τ2∧ σ1∧ σ2, dP ⊗ dt − a.s. (6.1)
2. Consider the case whenAi, A′i,i = 1, 2are not necessarily continuous. Fori = 1, 2,
define for eachε > 0,
τiε:= inf{t ≥ S, Yti≤ ξti+ ε} ; σiε:= inf{t ≥ S, Yti≥ ζti− ε}.
Settingτ˜i:= limε↓0↑ τiεandσ˜i:= limε↓0↑ σiε,we have
Yt1= Yt2, S ≤ t < ˜τ1∧ ˜τ2∧ ˜σ1∧ ˜σ2. a.s. (6.2)
Moreover, equality (6.1) holds on[S, ˜τ1∧ ˜τ2∧ ˜σ1∧ ˜σ2].
Proof. We adopt the same notation as in the proof of the comparison theorem.
1. Suppose first thatAi, A′i, i = 1, 2are continuous. By Theorem 4.8, for i = 1, 2,
(τi, σi)is a saddle point for the game problem associated withg = gi,ξ = ξi andζ = ζi.
By Remark 4.9,(Yi
t, S ≤ t ≤ τi∧ σi)is anEimartingale. Hence we have Yti= Et,τi i∧σi(I(τi, σi)), S ≤ t ≤ τi∧ σi a.s.
Settingθ = τ1∧ τ2∧ σ1∧ σ2, we thus have
Yti= Et,θi (Y i
θ), S ≤ t ≤ θ a.s.fori = 1, 2.
By hypothesis, Y1
S = YS2 a.s. Now, we apply the strict comparison theorem for non
reflected BSDEs with jumps (see [35, Th 4.4]) for terminal time θ. Hence, we get
Y1
t = Yt2, S ≤ t ≤ θ a.s. , as well as equality (6.1), which provides the desired result.
2. Consider now the general case. Letε > 0. By Remark 4.12,(Yi
t, S ≤ t ≤ τiε∧ σiε)is anEimartingale. Hence we have Yi
t = Et,τi ε i∧σεi(I(τ
ε
i, σεi)), S ≤ t ≤ τiε∧ σiε a.s.
By the same arguments as above with τ1∗,τ2∗ and σ∗1,σ2∗ replaced by τ1ε,τ2ε and σ1ε,σ2ε
respectively, we deriveY1
t = Yt2, S ≤ t ≤ τ1ε∧ τ2ε∧ σ1ε∧ σ2εa.s. , and equality (6.1) holds
on[S, τε
1∧ τ2ε∧ σ1ε∧ σ2ε],dt ⊗ dP-a.s. By lettingεtend to0, we obtain the desired result.
We now give an application of the comparison theorem to a control game problem for DRBSDEs.
Proposition 6.4 (Control game problem for DRBSDEs).Suppose that the assumptions of
Theorem5.3hold. For each(u, v) ∈ U × V, letYu,vbe the solution of the DRBSDE (2.2)
associated with drivergu,v. Then, for eachS ∈ T
0,YSu,v≤ YSu,v≤ YSu,va.s.
Proof. By using Assumption (5.3) and by applying the comparison theorem for DRBSDEs
(Theorem 6.1), we get that for eachu ∈ U, YSu,v≤ YSu,v a.s. Similarly, for allv ∈ V, we
haveYSu,v≤ YSu,va.s.
Remark 6.5.We point out that the above control game problem for DRBSDEs is different from the generalized mixed game problem studied in Section 5. However, from the above proposition, it follows that, under Assumption (5.3), the value functions of these two game problems coincide.
6.2 A priori estimates with universal constants
Using Theorem 4.10, we now prove the following estimates on the spread of the solutions of two DRBSDEs, where the constants are universal (i.e. they only depend on
the terminal timeT and the common Lipschitz constantC).
Proposition 6.6 (A priori estimates for DBBSDEs).Letξ1, ξ2, ζ1, ζ2∈ S2such thatξi T = ζi
T a.s. and ξit ≤ ζti, 0 ≤ t ≤ T a.s. Suppose that for i = 1, 2, ξi and ζi satisfy
Mokobodzki’s condition. Letg1, g2be Lipschitz drivers satisfying Assumption 4.3 with
common Lipschitz constantC > 0. Fori = 1, 2, letYi be the solution of the DRBSDE
associated with drivergi and barriersξi,ζi.
LetY := Y1− Y2,ξ := ξ1− ξ2,ζ = ζ1− ζ2. Letη, β > 0withβ ≥ 3
η + 2Candη ≤ 1 C2.
Letδgs= g2(t, Ys2, Zs2, k2s) − g1(t, Ys2, Zs2, ks2). For eacht, we have Y2t ≤ eβ(T −t)E[sup s≥t ξs 2 + sup s≥t ζs 2 |Ft] + ηE[ Z T t eβ(s−t)(δgs)2ds|Ft] a.s. (6.3)
Proof. The proof is divided into two steps.
Step 1: Fori = 1, 2and for eachτ, σ ∈ T0, let(Xi,τ,σ,πi,τ,σ, li,τ,σ)be the solution of the
BSDE associated with drivergi, terminal timeτ ∧ σand terminal conditionIi(τ, σ), where
Ii(τ, σ) = ξi
τ1τ≤σ+ ζσi1σ<τ. SetX τ,σ
:= X1,τ,σ− X2,τ,σ andIτ,σ := I1(τ, σ) − I2(τ, σ) = ξτ1τ≤σ+ ζσ1σ<τ. By an estimate on BSDEs (see PropositionA.4in [36]), we have a.s.: (Xτ,σt )2≤ eβ(T −t)E[I(τ, σ)2| Ft] + ηE[
Z T t
eβ(s−t)[(g1− g2)(s, Xs2,τ,σ, π2,τ,σs , l2,τ,σs )]2ds | Ft]
(6.4) from which we derive that
(Xτ,σt )2≤ eβ(T −t)E[sup s≥t ξ2s+ sup s≥t ζ2s|Ft] + ηE[ Z T t eβ(s−t)g2sds|Ft] a.s. , (6.5)
where gs := supy,z,k|g1(s, y, z, k) − g2(s, y, z, k)|. Now, by using inequality (4.12), we
obtain that for eachε > 0and for all stopping timesτ, σ,
Yt1− Yt2≤ X 1,τǫ 1,σ t − X 2,τ,σǫ 2 t + 2Kǫ.
Applying this inequality toτ = τǫ
1, σ = σǫ2we get Y1 t − Yt2≤ X 1,τǫ 1,σ ǫ 2 t − X 2,τǫ 1,σ ǫ 2 t + 2Kǫ ≤ |X 1,τǫ 1,σ ǫ 2 t − X 2,τǫ 1,σ ǫ 2 t | + 2Kǫ. (6.6) By (6.5) and (6.6), we have: Yt1− Yt2≤ s eβ(T −t)E[sup s≥t ξs 2 + sup s≥t ζs 2 |Ft] + ηE[ Z T t eβ(s−t)g2 sds|Ft] + 2Kǫ.
By symmetry, the last inequality is also verified byY2
t − Yt1. We thus derive that
Y2t ≤ eβ(T −t)E[sup s≥t ξs 2 + sup s≥t ζs 2 |Ft] + ηE[ Z T t eβ(s−t)g2 sds|Ft] a.s. (6.7)
This result holds for all Lipschitz driversg1andg2satisfying Assumption 4.3.
Step 2: Note that(Y2, Z2, k2)is the solution the DRBSDE associated with barriersξ2, ζ2
and driverg1(t, y, z, k) + δg
t. By applying the result of Step 1 to the driverg1(t, y, z, k)
and the driverg1(t, y, z, k) + δg
Remark 6.7.The arguments of the above proof are different from those used in the literature. Based on Theorem 4.10, they allow us to obtain universal constants, which is not the case for the a priori estimates on DRBSDEs given in the literature (for details see Remark A.5 in the Appendix). This new estimate with universal constants is useful to study the Markovian case (see the next section), in particular to obtain the continuity of the value function. Moreover, this estimate is a powerful tool which allows us to study a mixed generalized Dynkin game problem (see [15]), in particular to obtain a weak dynamic programming principle under mild assumptions and a classical one in the regular case.
We also state the following estimate on the common value functionY of our
gen-eralized Dynkin game problem (4.4)-(4.5) (or equivalently the solution of the DRBSDE
associated with driverg).
Proposition 6.8.Letg be a driver satisfying Assumption (4.3). Letξ andζ be RCLL
adapted processes inS2such thatξ
T = ζT a.s. andξt≤ ζt,0 ≤ t ≤ T a.s. Suppose that
Mokobodzki’s condition is satisfied.
Let(Y, Z, k, A, A′)be the solution of the DRBSDE (2.2). For eacht ∈ [0, T ], we have: Yt2≤ eβ(T −t)E[sup s≥t ξs2+ sup s≥t ζs2|Ft] + ηE[ Z T t eβ(s−t)g(s, 0, 0, 0)2ds|Ft] a.s. (6.8)
Proof. LetXτ,σbe the solution of the BSDE associated with driverg, terminal timeτ ∧ σ
and terminal conditionI(τ, σ). By applying inequality (6.4) withg1 = g,ξ
1 = ξ,ζ1= ζ, g2= 0,ξ2= 0andζ2= 0, we get: (Xtτ,σ)2≤ eβ(T −t)E[I(τ, σ)2|Ft] + ηE[ Z T t eβ(s−t)(g(s, 0, 0, 0))2|Ft]. (6.9)
By using the same procedure as in the proof of Proposition6.6, the result follows.
7
Relation with partial integro-differential variational
inequali-ties (PIDVI)
We consider now the Markovian case, and we study the links between Markovian generalized Dynkin games (or equivalently DRBSDEs) and obstacle problems.
Let b : R → R , σ : R → Rbe continuous mappings, globally Lipschitz and β : R× E → Ra measurable function such that for some nonnegative realC, and for all e ∈ E
|β(x, e)| ≤ Cϕ(e), |β(x, e) − β(x′, e)| ≤ C|x − x′|ϕ(e), x, x′ ∈ R,
whereϕis a bounded map belonging toL2
ν. For each(t, x) ∈ [0, T ]×R, let(Xst,x, t ≤ s ≤ T )
be the uniqueR-valued solution of the SDE with jumps:
Xst,x= x + Z s t b(Xrt,x)dr + Z s t σ(Xrt,x)dWr+ Z s t Z E β(Xrt,x−, e) ˜N (dr, de), and setXt,x
s = xfors ≤ t. We consider the DRBSDE associated with obstaclesξt,x,ζt,x
of the following form:
ξt,xs := h1(s, Xst,x), ζst,x:= h2(s, Xst,x), s < T ; ξ t,x T = ζ t,x T := g(X t,x T ).
We suppose thatg ∈ C(R),h1, h2: [0, T ] × R → Rare continuous with respect totand
Lipschitz continuous with respect tox, uniformly int and thatg, h1, h2 have at most
polynomial growth with respect tox. Moreover, the obstaclesξt,xandζt,xare supposed
to satisfy Mokobodzki’s condition, which holds for example whenh1orh2 isC1,2(see
