arXiv:2110.02075v1 [math.PR] 2 Oct 2021 Robust optimal problem for dynamic risk measures governed by BSDEs with jumps and delayed generator Navegué Tuo * and Auguste Aman † UFR de Mathématiques et Informatique Université Félix H. Boigny, Cocody 22 BP 582 Abidjan, Côte d’Ivoire Abstract The aim of this paper is to study an optimal stopping problem for dynamic risk measures induced by backward stochastic differential equations with jumps and de- layed generator. Firstly, we connect the value function of this problem to reflected BS- DEs with jump and delayed generator. Furthermore, after establishing existence and uniqueness result for this reflected BSDE, we use its to address through a mixed/optimal stopping game problem for the previous dynamic risk measure in ambiguity case. MSC:Primary: 60F05, 60H15, 47N10, 93E20; Secondary: 60J60 Keywords: Backward stochastic differential equations; Delayed generators Reflected back- ward stochastic equations; Jump processes; Optimal stopping; Dynamic risk measures; Game problems. 1 Introduction The risk measures start with the work of Artzner et al. [1]. Later, there has been a lot of studies on risk measures. See e.g Follmer and Shied [12], Frittelli and Gianin [13], Bion-Nadal [4], Barrieu and El Karoui [2], Bayraktar E, I. Karatzas and Yao [3]. After these, around the year 2005, various authors established the links between continuous time dynamic risk measures and backward differential equations. They have introduced dynamic risk measures in the Brownian case, defined as the solutions of BSDEs (see [13, 14, 2]). Clearly, let consider f and ξ respectively a function and random variable. The risk measure * [email protected]† [email protected], corresponding author
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arX
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0207
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Robust optimal problem for dynamic risk measures
governed by BSDEs with jumps and delayed
generator
Navegué Tuo* and Auguste Aman†
UFR de Mathématiques et Informatique
Université Félix H. Boigny, Cocody
22 BP 582 Abidjan, Côte d’Ivoire
Abstract
The aim of this paper is to study an optimal stopping problem for dynamic risk
measures induced by backward stochastic differential equations with jumps and de-
layed generator. Firstly, we connect the value function of this problem to reflected BS-
DEs with jump and delayed generator. Furthermore, after establishing existence and
uniqueness result for this reflected BSDE, we use its to address through a mixed/optimal
stopping game problem for the previous dynamic risk measure in ambiguity case.
of the position ξ denoted by ρt(ξ) is described by the process −Xt where X(t), t ≥ 0 is the
first component solution of BSDEs associated to generator f and terminal value ξ. Many
studies have been done on such risk measures, dealing with optimal stopping problem and
robust optimization problems (see for example [18, 3, 2]).
Recently, in [7], Delong and Imkeller introduced the theory of nonlinear backward
stochastic differential equations (BSDEs, in short) with time delayed generators. Precisely,
given a progressively measurable process f , so-called generator and a square integrable
random variable ξ, BSDEs with time delayed generator are BSDEs of the form:
X(t) = ξ+∫ T
tf (s,Xs,Zs)ds−
∫ T
tZ(s)dW(s),0 ≤ t ≤ T,
where the process (Xt ,Zt) = (X(t+u),Z(t+u))−T≤u≤0 represents all the past values of the
solution until t. Under some assumptions, they proved existence and uniqueness result of
such a BSDEs. In this dynamic, the same authors study, in an accompanying paper (see
[8]), BSDE with time delayed generator driven both by a Brownian motion and a Poisson
random measure. Existence and uniqueness of a solution and its Malliavin’s differentia-
bility has been established. A few year later, in [6], Delong proved that BSDEs with time
delayed generator is a important tool to formulate many problems in mathematical finance
and insurance. For example, he proved that the dynamic of option based portfolio assurance
is the following time delayed BSDE:
X(t) = X(0)+(X(T)−X(0))+−
∫ T
tZ(s)dW(s).
From these works, and given the importance of applications related to BSDEs with time de-
layed generator, in your opinion, it is very judicious to expect to study an optimal stopping
problem for dynamic risk measures governed by backward stochastic differential equa-
tions with delayed generator. Better, this paper is dedicated to resolve an optimal stopping
problem for dynamic risk measure governed by backward stochastic differential equations
driven with both Brownian motion and Poison random measure. For more detail, let con-
sider (ψ(t))t≥0 a given right continuous left limited adapted process and τ be a stopping
time in [0,T ]. Our objective is to solve an optimal stopping problem related to risk mea-
sure of the position ψ(τ) denoted by ρψ,τ with dynamic follows as the process −Xψ,τ where
(Xψ,τ,Zψ,τ,Uψ,τ) satisfied the following BSDE
Xψ,τ(t) = ψ(τ)+∫ T
tf (s,Z
ψ,τs ,U
ψ,τs (.))ds−
∫ T
tZψ,τ(s)dW(s)
−
∫ T
t
∫R∗
Uψ,τ(s,z)N(ds,dz), 0 ≤ t ≤ τ,
where R∗ = R\0.
Roughly speaking, for all stopping time σ with values on [0,T ], our aim is to minimize
the risk measures at time σ i.e we want to find a unique stopping time τ∗ such that setting
v(σ) = ess infσ≤τ≤T
ρψ,τ(σ),
2
we have
v(σ) = ρψ,τ∗(σ). (1.1)
Our method is essentially based on the link establish between the value function v and the
first component of the solution of a reflected BSDEs with jump and delayed generator.
Notion of reflected BSDEs has been introduced for the first time by N. EL Karoui et al. in
[9] with a Brownian filtration. The solutions of such equations are constrained to be greater
than given continuous processes called obstacles. Later, different extensions have been
performed when we add the jumps process and/or suppose the obstacle not continuous.
One can cite works of Tang and Li [20], Hamadène and Ouknine [15, 16], Essaky [11] and
Quenez and Sulem [18]. More recently, reflected BSDEs without jump and with delayed
generator have been introduced respectively by Zhou and Ren [22], and Tuo et al. [21].
Our study takes place in two stages. First, we provide an optimality criterium, that is a
characterization of optimal stopping times and when the obstacle is right continuous and
left limited (rcll, in short), we show the existence of an optimal stopping time. Thereafter,
we address the optimal stopping problem when there is ambiguity on the risk measure.
It means that there exists a given control δ that can influence the dynamic risk measures.
More precisely, given the dynamic position ψ this situation consists to focus on the robust
optimal stopping problem for the family of risk measures ρδ, δ ∈ A of this position ψinduced by the BSDEs associated with generators f δ, δ ∈ A. To this purpose and in
view of the first part, we study the following optimal control problem related to Y δ the
first component solution of reflected BSDEs with jumps and delayed generator f δ, δ ∈ A)with a RCLL obstacle ψ. In other words, we want to determine a stopping time τ∗, which
minimizes over all stopping times τδ, the risk of the position ψ. This is equivalent to derive
a saddle points to a mixed control/optimal stopping game problem.
The paper is organized as follows. We give the notation and formulation of the optimal
problem for risk measures problem in Section 2. Existence and uniqueness results for
RBSDEs with jumps and delayed generator with right continuous left limit (rcll) obstacle
is provided in Section 3. In both section 4 and 5, we deal with the robust optimal stopping
problem.
2 Formulation of the problem
Let consider a probability space (Ω,F ,P). For E = Rd\0 equipped with its Borel field
E , let N be a Poisson random measure on R+×E with compensator ν(dt,dx) = λ(dx)dt
where λ is σ-finite measure on (E,E) satisfying
∫E(1∧|x|2)dλ(x)<+∞.
such that ((N −ν)([0, t]×A))t≥0 is a martingale. Let also consider acd-dimensional stan-
dard Brownian motion (Wt)t≥0 independent of N. Let finally consider the filtration F =
3
Ft t≥t defined by
Ft = F W ∧F N ∧N ,
where N is the set of all P-null element of F .
2.1 BSDEs with time delayed generators driven by Brownian motions
and Poisson random measures
This subsection is devoted to recall existence and uniqueness result for BSDEs with jump
and time-delayed generator
X(t) = ξ+∫ T
tf (s,Xs,Zs,Us(.))ds−
∫ T
tZ(s)dW (s)
−∫ T
t
∫E
U(s,z)N(ds,dz), 0 ≤ t ≤ T, (2.1)
studied by Delong and Imkeller in [8] and derive a comparison principle associated to this
BSDE. In this instance, let us describe following spaces of processes:
• L2−T (R) denotes the space of measurable functions z : [−T,0]→ R satisfying
∫ 0
−T| z(v) |2 dv <+∞,
• L∞−T (R) denotes the space of bounded, measurable functions y : [−T,0]→R
satisfying
supv∈[−T,0]
| y(v) |2<+∞,
L2−T,m(R) denotes the space of product measurable functions u : [−T,0]×R/0 → R
such that ∫ 0
−T
∫E|u(t,z)|2m(dz)dt <+∞.
• L2(Ω,FT ,R) is the Banach space of FT -measurable random variables ξ : Ω→R normed
by ‖ξ‖L2 =[E(|ξ|2)
]1/2
• H 2(R) denotes the Banach space of all predictable processes ϕ with values in R such
that E[∫ T
0 |ϕ(s)|2ds]<+∞.
• Let H 2m(R) denote the space of P ⊗E-mesurable processes φ satisfyingE
(∫ T0
∫E |φ(t,z)|
2m(dz)dt)<
+∞, where P is the sigma algebra of (Ft)t≥0-predictable set on Ω× [0,T ].
4
• S 2(R) denotes the Banach space of all (Ft)0≤t≤T -adapted right continuous left limit
(rcll) processes η with values in R such that E(sup0≤s≤T |η(s)|2
)<+∞
• K 2(R) denotes the Banach space of all (Ft)0≤t≤T -predictable right continuous left limit
(rcll) increasing processes η with values in R such that η(0) = 0 and E(|η(T )|2
)<
+∞
The spaces H 2(R), H 2m(R) and S 2(R) are respectively endowed with the norms
‖ϕ‖2H 2,β = E
[∫ T
0eβs|ϕ(s)|2ds
]
‖φ(t,z)‖2β,m = E
(∫ T
0
∫E|φ(t,z)|2m(dz)dt
)
‖η(s)‖2S 2,β = E
(sup
0≤s≤T
eβs|η(s)|2
).
Our two results has been done under the following hypotheses: For a fix T > 0,
(A1) τ is a finite (Ft)0≤t≤T -stopping time.
(A2) ξ ∈ L2(Fτ,R)
(A3) f : Ω × [0,T ]× L∞−T (R)× L2
−T (R)× L2−T,m(R) → R is a product measurable, F-
adapted function satisfying
(i) There exists a probability measure α on ([−T,0],B([−T,0])) and a positive
constant K, such that
| f (t,yt,zt ,ut(.)− f (t, yt, zt , ut(.)|2
≤ K
∫ 0
−T
[|y(t + v)− y(t + v)|2 + |z(t + v)− z(t + v)|2
+
∫E|u(t + v,ζ)− u(t + v,ζ)|2m(dζ)
]α(dv)
for P⊗λ a.e, (ω, t) ∈ Ω× [0,T ], for any (xt ,zt ,ut(.)), (xt , zt , ut(.)) ∈ L∞−T (R)×
L2−T (R)×L2
−T,m(R)
(ii) E
(∫ T
0| f (t,0,0,0)|2dt
)<+∞
(iii) f (t, ., ., .) = 0 a.s, for t < 0
For the sake of good understanding, we give in the following the notion of solution of
BSDE (2.1).
5
Definition 2.1. The triple processes (X ,Z,U) is called solution of BSDE (2.1) if (X ,Z,U)belongs in S 2(R)×H 2(R)×H 2
m(R) and satisfies (2.1).
We recall the existence and uniqueness result established in [8].
Theorem 2.2. Assume that (A1)-(A3) hold. If T a terminal time or K a Lipschitz constant
are sufficiently small i.e
9T Kemax(1,T )< 1,
(2.1) has a unique solution.
The concept of comparison principle is a very important in the theory of BSDE without
delay. Unfortunately, as point out by Example 5.1 in [7], this principle cannot be extended
in general form to BSDEs with delayed generators. Nevertheless, according to Theorem
3.5 appear in [21], the comparison principle for BSDEs without jump and with delayed
generator, still hold on stochastic intervals in where the strategy process Z stays away from
0. The following theorem is an extension to BSDEs with jump and delayed generator. To
do it, we need this additional assumption
(A4 ) f : Ω× [0,T ]× L∞−T (R)× L2
−T (R)× L2−T,m(R) → R is a product measurable, F-
adapted function satisfying:
f (t,xt,zt,ut(.))− f (t,xt,zt ,u′t(.))≥
∫ 0
−T〈θxt ,zt ,ut(.),u
′t(.),u(t + v, .)−u′(t + v, .)〉mα(dv),
for P⊗λ a.e, (ω, t) ∈ Ω× [0,T ] and each (xt ,zt ,ut(.),u′t(.)) ∈ L∞
−T (R)×L2−T (R)×
L2−T,m(R)×L2
−T,m(R), where θ : Ω×[0,T ]×L∞−T (R)×L2
−T (R)×L2−T,m(R)×L2
−T,m(R)→
L2−T,m(R) is a measurable an bounded function such that there exists ϕ belongs to
L2−T,m(R), verifying
θxt ,zt ,ut(.),u′t(.)(ζ)≥−1 and |θxt ,zt ,ut(.),u
′t(.)(ζ)| ≤ ϕ(ζ).
Theorem 2.3. Consider BSDE (2.1) associated to delayed generators f1, f2 and corre-
sponding terminal values ξ1, ξ2 at terminal time τ satisfying the assumptions (A1)-(A3).Let (X τ,1,Zτ,1,U τ,1) and (X τ,2,Zτ,2,U τ,2) denote respectively the associated unique solu-
tions. Let consider the sequence of stopping time (σn)n≥1 define by
σn = inf
t ≥ 0 , |X τ,1(t)−X τ,2(t)| ∨ |Zτ,1(t)−Zτ,2(t)| ∨
∫E|U τ,1(t,z)−U τ,2(t,z)|m(dz)≤
1
nor
|X τ,1(t)−X τ,2(t)| ∨ |Zτ,1(t)−Zτ,2(t)| ∨∫
E|U τ,1(t,z)−U τ,2(t,z)|m(dz)≥ n.
∧ T (2.2)
6
and set
σ = supn≥1
σn. (2.3)
Moreover we suppose that
• X τ,1(σ)≥ X τ,2(σ)
• f1(t,Xτ,1t ,Z
τ,1t ,U
τ,1t (.))≥ f2(t,X
τ,1t ,Z
τ,1t ,U
τ,1t (.)) or
• f1(t,Xτ,2t ,Zτ,2
t ,U τ,2t (.))≥ f2(t,X
τ,2t ,Zτ,2
t ,U τ,2t (.)).
Then X τ,1(t)≥ X τ,2(t), P-a.s. for all t ∈ [0,σ].
Proof. We follow the ideas from Theorem 5.1 for BSDEs without jumps and with delayed
generator established in [7]. For each t ∈ [0,T ] let
Let consider the real processes δ,β and γ defined respectively by
δ(t) =
f 1(t,Xτ,1
t ,Zτ,1t ,Uτ,1
t (.))− f 1(t,Xτ,2t ,Zτ,1
t ,Uτ,1t (.))
∆Xτ(t) if ∆X τ(t) 6= 0
0 otherwise,
β(t) =
f 1(t,Xτ,2
t ,Zτ,1t ,Uτ,1
t (.))− f 1(t,Xτ,2t ,Zτ,2
t ,Uτ,1t (.))
∆Zτ(t) if ∆Zτ(t) 6= 0
0 otherwise.
and
γ(t) =
f 1(t,X
τ,2t ,Z
τ,2t ,U
τ,1t (.))− f 1(t,X
τ,2t ,Z
τ,2t ,U
τ,2t (.))∫
E ∆Uτ(t,z)m(dz)if
∫E ∆U τ(t,z)m(dz) 6= 0
0 otherwise.
Hence, since f 1 and f 2 are Lipschitz with respect x, z and in u, we have
|δ(t)|2 ≤ K
∫ 0
−T
(|∆X τ(t +u)|2
|∆X τ(t)|2
)α(du),
|β(t)|2 ≤ K
∫ 0
−T
(|∆Zτ(t +u)|2
|∆Zτ(t)|2
)α(du)
7
and
|γ(t)|2 ≤ K
∫ 0
−T
∫E|∆U τ(t +u,z)|2m(dz)∫
E|∆U τ(t,z)|2m(dz)
α(du).
Next, in view of (2.10) and (2.3), for t ∈ [0,σ], there exist a constant C such that φ = δ,β,γ,
|φ(t)| ≤C, a.s.
On other hand, we have
∆X τ(t) = ∆X τ(σ)+
∫ σ
tδ(s)∆X τ(s)ds+
∫ σ
tβ(s)∆Z(s)ds
+
∫ σ
t
∫E
γ(s)∆U τ(t,z)m(dz)ds+
∫ σ
t∆ f (s,X τ,2
s ,Zτ,2s ,U τ,2
s (.))ds
−∫
∆Z(s)dW(s)∫ σ
t
∫E
∆U(s,z)N(ds,dz)
and setting R(t) =
∫ t
0δ(s)ds, it follows from Itô’s formula applied to R(s)∆X τ(s) between
t to σ that
R(t)∆X τ(t) = R(σ)∆X τ(σ)+∫ σ
tR(s)β(s)∆Zτ(s)ds
+∫ σ
t
∫E
R(s)γ(s)∆U τ(t,z)m(dz)ds+∫ σ
tR(s)∆ f (s,X τ,2
s ,Zτ,2s ,U τ,2
s (.))ds
−∫ σ
tR(s)∆Zτ(s)dW(s)−
∫ σ
t
∫E
R(s)∆U τ(s,z)N(ds,dz).
Taking into consideration the assumptions on generators and terminal values, we obtain
R(t)∆X τ(t) ≤∫ σ
tR(s)β(s)∆Zτ(s)ds
+∫ σ
t
∫E
R(s)γ(s)∆U τ(t,z)m(dz)ds
−∫ σ
tR(s)∆Zτ(s)dW (s)−
∫ σ
t
∫E
R(s)∆U τ(s,z)N(ds,dz). (2.4)
Let denote by D(t) the right hand side of (2.4) and set M(t)=∫ t
0 β(s)dW(s)+∫ t
0
∫E γ(s)N(ds,dz).
In view of Girsanov theorem, the process (D(t))0≤t≤T is a martingale under the probability
measure Q defined by Q = Eσ(M).P, where Eσ(M) is called a Doléan-Dade exponential.
Taking conditional expectation with respect to Ft under Q both sides of (2.4), we obtain
R(t)∆X τ(t) ≤ 0 Q-a.s., and hence P-a.s. Finally, since the process (R(t), t ≥ 0) is non-
negative, we have t ∈ [0,σ], X τ,1(t)≥ X τ,2(t) P-a.s.
8
2.2 Properties of dynamic risk measures
2.3 Optimal stopping problem for dynamic risk measures
Let T > 0 be a time horizon and f be delayed generator satisfied (A2). For each stopping
time τ with values in [0,T ] and (ψ(t))t≥0 a (Ft)t≥0-adapted square integrable stochastic
process, we consider the risk of ψ(τ) at time t defined by
ρψ,τ(t) =−Xψ,τ(t), 0 ≤ t ≤ τ,
where Xψ,τ satisfy BSDE (2.1) with driver f 1[0,τ], terminal condition ψ(τ) and terminal
time τ. The functional ρ : (ψ,τ) 7→ ρψ,τ(.) defines then a dynamic risk measure induced by
the BSDE (2.1) with driver f 1[0,τ]. Let us now deal with some optimal stopping problem
related to the above risk measure. Contrary to the case without delay, there is a real dif-
ficulty in setting up the problem for the BSDE with delayed generator. Indeed, since the
comparison principle of delayed BSDEs failed at the neighborhood of 0, we are no longer
able to construct the supremum of this risk on [0,T ]. To work around this difficulty, we
need to construct a stochastic interval in which, we can derive a comparison theorem. For
a stopping time δ, let also consider (Xψ,δ,Zψ,δ) the solution of BSDE (2.1) with driver
f 1[0,δ], terminal condition ψ(δ) and terminal time δ. We consider following stopping times
σn = inf(An)∧T,
where
An =
t ≥ 0, infτ,δ
(|Xψ,τ(t)−Xψ,δ(t)| ∨ |Zψ,τ(t)−Zψ,δ(t)| ∨∫
E|Uψ,τ(t,z)−Uψ,δ(t,z)|m(dz)≤
1
n
or
infτ,δ
(|Xψ,τ(t)−Xψ,δ(t)| ∨ |Zψ,τ(t)−Zψ,δ(t)| ∨
∫E|Uψ,τ(t,z)−Uψ,δ(t,z)|m(dz))≥ n
and set
σ = supn≥1σn. (2.5)
For a stopping time σ ≤ σ, let consider Fσ-measurable random variable v(σ) (unique for
the equality in the almost sure sense) defined by
v(σ) = ess infσ≤τ≤T
ρψ,τ(σ). (2.6)
Since ρψ,τ =−Xψ,τ, we get
v(σ) = ess infσ≤τ≤T
(−Xψ,τ(σ)) =−ess supσ≤τ≤T
Xψ,τ(σ), (2.7)
for each stopping time σ ∈ [0,σ], which characterize the minimal risk-measure. We then
provide an existence result of an σ-optimal stopping time τ∗ ∈ [σ,T ], satisfies v(σ) =ρψ,τ∗(σ) a.s.
In order to characterize minimal risk measure by reflected BSDEs with jump and de-
layed generators, let’s derive first the notion of solution of this type of equations.
9
Definition 2.4. The triple of processes (Y (t),Z(t),U(t,z),K(t))0≤t≤T,z∈E is said to be a
solution of the reflected delayed BSDEs with jumps associated to delayed generator f ,
stochastic terminal times τ, terminal value ξ and obstacle process (S(t))t≥0, if it satisfies
the following.
(i) (Y,Z,U,K) ∈ S 2(R)×H 2(R)×H 2m(R)×K 2(R).
(ii)
Y (t) = ξ+
∫ τ
tf (s,Ys,Zs,Us(.))ds+K(τ)−K(t)−
∫ τ
tZ(s)dW (s)
−
∫ τ
t
∫E
U(s,z)N(ds,dz), 0 ≤ t ≤ τ (2.8)
(iii) Y dominates S, i.e. Y (t)≥ S(t), 0 ≤ t ≤ τ
(iv) the Skorohod condition holds:
∫ τ
0(Y (t−)−S(t−))dK(t) = 0 a.s.
In our definition, the jumping times of process Y is not come only from Poisson process
jumps (inaccessible jumps) but also from the jump of the obstacle process S (predictable
jumps).
Remark 2.5. Let us point out that condition (iv) is equivalent to : If K = Kc +Kd , where
Kc and Kd denote respectively continuous and discontinuous part of K, then
∫ τ
0(Y (t)−
S(t))dKc(t) = 0 a.s. and for every predictable stopping time σ ∈ [0,T ], ∆Y (σ) = Y (σ)−Y (σ−) =−(S(σ−)−Y (σ))+1[Y (σ−)=S(σ−)]. On the other hand, since the jumping times of
the Poisson process are inaccessible, for every predictable stopping time σ ∈ [0,T ],∆Y (σ) =−∆K(σ) =−(S(σ−)−Y (σ))+1[Y (σ−)=S(σ−)]
The following theorem will be state in special context that ξ = ψ(τ) and S = ψ in order
to establish a link between the risk measure associated with the EDSR (τ,ψ(τ), f ) and the
solution of the reflected EDSR associated with (τ,ψ(τ), f ,ψ).
Theorem 2.6. Let τ be a stopping time belonging on [0,T ], ψ(t), 0 ≤ t ≤ T and f be
respectively a terminal time, an rcll process in S 2(R) and a delayed generator satisfying
Assumption (A3)− (A4). Suppose (Y,Z,U,K) be the solution of the reflected BSDE asso-
ciated to (τ,ψ(τ), f ,ψ).
(i) For each stopping time σ ≤ σ, we have
v(σ) =−Y (σ) =−ess supτ∈[σ,T ]
Xψ,τ(σ), (2.9)
where v(σ) is defined by (2.6).
10
(ii) For each stopping time σ with values on [0,σ] and each ε > 0, let Dεσ be the stopping
time defined by
Dεσ = inft ∈ [σ,T ], Y (t)≤ ψ(t)+ ε . (2.10)
We have
Y (σ)≤ Xψ,Dεσ(σ)+Cε a.s.,
where C is a constant which only depends on T and the Lipschitz constant K. In other
words, Dεσ is a (Cε)-optimal stopping time for (4.2).
Remark 2.7. Note that Property (ii) implies that for all stopping times σ and τ with values
on [0,σ] and [0,T ] respectively such that σ ≤ τ ≤ Dεσ, we have Y (σ) = E
fσ,τ(Y (τ)) a.s. In
other words, the process (Y (t), σ ≤ t ≤ Dεσ) is an E f -martingale.
Proof of Theorem 2.6. Let consider σ and τ two stopping time with values in [0,T ] such
that σ ≤ τ. Let consider (Y,Z,U,K) be solution of the reflected BSDE associated to
(ψ(τ), f ,ψ). We have
Y (σ) = ψ(τ)+
∫ τ
σf (s,Ys,Zs,Us(.))ds+K(τ)−K(σ)−
∫ τ
σZ(s)dW (s)
−
∫ τ
σ
∫E
U(s,z)N(ds,dz)
According to reflected BSDEs framework, we know that the process K is non-decreasing,
hence K(τ)−K(σ)≥ 0. Therefore,
Y (σ) ≥ ψ(τ)+∫ τ
σf (s,Ys,Zs,Us(.))ds−
∫ τ
σZ(s)dW (s)−
∫ τ
σ
∫E
U(s,z)N(ds,dz).(2.11)
Let (Y , Z,U) satisfy equation
Y (σ) = ψ(τ)+∫ τ
σf (s,Ys, Zs,Us(.))ds−
∫ τ
σZ(s)dW(s)−
∫ τ
σ
∫E
U(s,z)N(ds,dz).(2.12)
It follows from (2.11) that Y (σ) ≥ Y (σ). On other hand, thanks to uniqueness of solution
for BSDE (2.1), we obtain Y =Xψ,τ which implies Y (σ)≥Xψ,τ(σ) for all τ∈ [σ,T ]. Finally
we get
Y (σ) ≥ ess supτ∈[σ,T ]
Xψ,τ(σ). (2.13)
Let us show now the reversed inequality. In view of it definition, Dεσ belongs in [σ,T ]
and for each t ∈ [σ(ω),Dσ(ω)[ for almost all ω ∈ Ω, we have Y (t)> ψ(t) a.s. Therefore,
11
recalling reflected BSDEs framework, the function t 7→ K(t) is almost surely constant on
[σ(ω),Dσ(ω)] so that K(Dσ)−K(σ) = 0. This implies that
Y (σ) = ψ(Dσ)+
∫ Dσ
σf (s,Ys,Zs,Us(.))ds−
∫ Dσ
σZ(s)dW (s)−
∫ Dσ
σ
∫E
U(s,z)N(ds,dz).
Using again comparison principle, we derive that Y (σ) = Xψ,Dσ(σ) which leads
Y (σ) ≤ ess supτ∈[σ,T ]
Xψ,τ(σ) (2.14)
According to (2.13) and (2.14), we prove (i). We will prove now (ii). According to (2.10)
and comparison theorem of BSDE with delayed generator, we get that for all stopping times
σ ≤ σ,
Y (σ) = XY,Dεσ(σ)≤ Xψ+ε,Dε
σ(σ) as. (2.15)
On the other hand, using some appropriate estimate on BSDE with delayed generator, we
derive
|XY,Dεσ(σ)−Xψ+ε,Dε
σ(σ)|2 ≤ eβ(T−S)ε2, as,
where β is a constant depending only on the time horizon T and a Lipschitz constant K.
Finally, in view of (2.15) we get the result.
To end this subsection let now derive an optimality criterium for the optimal stopping
time problem based on the strict comparison theorem. Before let us give what we mean by
an optimal stopping time.
Definition 2.8. A stopping time τ ∈ [σ,T ] is an σ-optimal stopping time if
Y (σ) = ess supτ∈[σ,T ]
Xψ,τ(σ) = Xψ,τ(σ).
On the other word, the process (Y (t))σ≤t≤τ is the solution of the non reflected BSDE asso-
ciated with terminal time τ and terminal value ψ(τ).
Theorem 2.9. Let a rcll process (ψ(t))t≥0 be l.u.s.c along stopping times and belong to
S 2(R). We assume (A1)-(A4) holds and suppose (Y,Z,U(.),K) is a solution of the reflected
BSDE with jump and delayed (2.8). Setting for all stopping time σ ≤ σ (σ is the same
defined by (2.5)), the following stopping times:
τσ = limε↓0
↑ τεσ, (2.16)
where τεσ = infσ ≤ t ≤ T, Y (t)≤ ψ(t)+ ε,
τ∗σ = infσ ≤ t ≤ T, Y (t) = ψ(t), (2.17)
and
τσ = infσ ≤ t ≤ T, K(t)−K(σ)> 0 . (2.18)
Then τσ, τ∗σ and τσ are σ-stopping times of the optimal problem (2.6) such that
12
(i) τσ ≤ τ∗σ and we have Y (s) = Xψ,τ∗σ(s) for all σ ≤ s ≤ τ∗σ a.s.
(ii) τσ is the minimal σ-stopping time
(iii) τσ is the maximal σ-stopping time.
(iv) Moreover if in (A3)(iv), we have |θxt ,zt ,ut(.),u′t(.)|>−1, then τ∗σ = τσ.
Since the proof follows the same argument used in its proof and to avoid unnecessarily
lengthening the writing, we will refer the reader to the proof of Theorem 3.7 appear to [18].
3 Reflected BSDEs with jumps and time-delayed genera-
tor
This section is devoted to study in general framework of the reflected BSDEs with jumps,
right continuous and left limit (rcll) obstacle process and delayed generator. More precisely,
for a fixed T > 0 and a stopping time τ in value on [0,T ], we consider
Y (t) = ξ+∫ τ
tf (s,Ys,Zs,Us(.))ds+K(τ)−K(t)−
∫ τ
tZ(s)dW (s)
−∫ τ
t
∫E
U(s,z)N(ds,dz), 0 ≤ t ≤ τ. (3.1)
We derive an existence and uniqueness result under the following additional hypothesis
related to the obstacle process.
(A5) The obstacle process S(t), 0 ≤ t ≤ T is a rcll progressively measurable R-valued
process satisfies
(i) E(sup0≤t≤T (S
+(t))2)<+∞,
(ii) ξ ≥ S(τ) a.s.
To begin with, let us first assume f to be independent of (yt ,zt ,ut) ∈, that is, it is a given
(Ft)0≤t≤τ-progressively measurable process satisfying that E(∫ τ
0 f (t)dt)<+∞. A solution
to the backward reflection problem (BRP, in short) is a triple (Y,Z,U,K) which satisfies
(i),(iii),(iv) of the Definition 2.4. and
(ii’)
Y (t) = ξ+∫ τ
tf (s)ds+K(τ)−K(t)−
∫ τ
tZ(s)dW (s)−
∫ τ
t
∫E
U(s,z)N(ds,dz), 0 ≤ t ≤ τ.
The following proposition is from Hamadène and Ouknine [15] (Theorem 1.2.a and 1.4.a)
or Essaky [11].
13
Proposition 3.1. The reflected BSDE with jump associated with (ξ,g,S) has a unique so-
lution (Y,Z,K,U).
Theorem 3.2. Assume (A1)-(A3) and (A5) hold. For a sufficiently small time horizon T
or for a sufficiently small Lipschitz constant K of the generator f i.e
KTemax1,T< 1, (3.2)
the reflected BSDE with jumps and delayed generator (2.8) admits a unique solution (Y,Z,U,K)∈S 2(R)×H 2(R)×H 2
m(R)×K 2(R).
Proof. Let us begin with the uniqueness result. In this fact, assume (Y,Z,U,K) and (Y ′,Z′,U ′,K′)be two solutions of RBSDE associated to data (ξ, f ,S) and set θ = θ−θ′ for θ =Y,Z,U,K.
Applying Itô’s formula to the discontinuous semi-martingale |Y |2, we have
|Y (t)|2+
∫ T
t|Z(s)|2ds+
∫ T
t
∫E|U(s,z)|2m(dz)ds
= 2
∫ T
tY (s)( f (s,Ys,Zs,Us(.))− f (s,Y ′
s ,Z′s,U
′s(.)))ds+2
∫ T
tY (s)dK(s)
−2
∫ T
tY (s)Z(s)dW(s)−2
∫ T
t
∫E
Y (s−)U(s,z)N(ds,dz). (3.3)
In view of Skorohod condition (iv), we get∫ T
tY (s)dK(s) =
∫ T
t(Y (s−)−S(t−))dK(t)+
∫ T
t(S(s−)−Y ′(t−))dK(t)
+
∫ T
t(Y ′(s−)−S′(t−))dK′(t)+
∫ T
t(S′(s−)−Y (t−))dK′(t)
≤ 0. (3.4)
Next, since the third and fourth term of (3.3) are (Ft)t≥0-martingales together with (3.4),
we have
E
(|Y (t)|2+
∫ T
t|Z(s)|2ds+
∫ T
t
∫E|U(s,z)|2m(dz)ds
)
= 2E
(∫ T
tY (s)( f (s,Ys,Zs,Us(.))− f (s,Y ′
s ,Z′s,U
′s(.)))ds
)
≤ βE
(∫ T
t|Y (s)|2ds
)+
1
βE
(∫ T
t| f (s,Ys,Zs,Us(.))− f (s,Y ′
s ,Z′s,U
′s(.)))|
2ds
)(3.5)
According to assumptions (A3)(i), change of variable and fubini’s theorem, we obtain∫ T
t| f (s,Ys,Zs,Us(.))− f (s,Y ′
s ,Z′s,U
′s(.)))|
2ds
≤ K
∫ T
t
(∫ 0
−T
[|Y (s+u)|2+ |Z(s+u)|2+
∫E|U(s+u,z)|2m(dz)
]α(du)
)ds
≤ K
∫ T
−T
[|Y (s)|2+ |Z(s)|2+
∫E|U(s,z)|2m(dz)
]ds. (3.6)
14
Putting the last inequality into (3.5) yields
E
(|Y (t)|2+
∫ T
t|Z(s)|2ds+
∫ T
t
∫E|U(s,z)|2m(dz)ds
)
≤
(β+
K
β
)E
∫ T
−T|Y (s)|2ds+
K
βE
∫ T
0
(|Z(s)|2+
∫E|U(s,z)|2m(dz)
)ds. (3.7)
If we choose β such that Kβ≤ 1, inequality (3.7) becomes
E
(|Y (t)|2+
∫ T
0|Z(s)|2ds+
∫ T
0
∫E|U(s,z)|2m(dz)ds
)
≤ CE
∫ T
−T|Y (s)|2ds. (3.8)
According the above estimate, using Gronwall’s lemma and in view of the right continuity
of the process Y , we have Y =Y ′. Therefore (Y,Z,U,K) = (Y ′,Z′,U ′,K′), whence reflected
BSDE with jump and delayed generator (3.1) admit a uniqueness solution.
It remains to show the existence which will be obtained via a fixed point method. For
this let consider D = S 2(R)×H 2(R)×Hm(R) endowed with the norm ‖(Y,Z,U)‖β de-
fined by
‖(Y,Z,U)‖β = E
(sup
0≤t≤τeβt |Y (t)|2+
∫ τ
0eβt
(|Z(t)|2+
∫E
U(s,z)m(dz)
)ds
).
We now consider a mapping Φ : D into itself defined by Φ((Y,Z,U)) = (Y , Z,U) which
means that there is a process K such as (Y , Z,U , K) solve the reflected BSDE with jump as-
sociated to the data ξ, f (t,Y,Z,U) and S. More precisely, (Y , Z,U , K) satisfies (i), (iii), (iv)of Definition 2.4 such that
Y (t) = ξ+∫ τ
tf (s,Ys,Zs,Us(.))ds+ K(τ)− K(t)−
∫ τ
tZ(s)dW(s)−
∫ τ
t
∫E
U(s,z)N(ds,dz).
For another process (Y ′,Z′,U ′) belonging in D let set Φ(Y ′,Z′,U ′) = (Y ′, Z′,U ′). In the
sequel and for a generic process θ, we denote δθ = θ−θ′. Next, applying Ito’s formula to
eβt |∆Y (t)|2 yields
eβt |δY (t)|2+β
∫ T
teβs|δY (s)|2ds+
∫ T
teβs|δZ(s)|2ds
+∫ T
teβs
∫E|δU(s,z)|2m(dz)ds+ ∑
t≤s≤T
eβs(∆s(δY )−∆s(δY ′))2
= 2
∫ T
teβsδY (s)( f (s,Ys,Zs,Us(.))− f (s,Y ′
s ,Z′s,Us(.)))ds+2
∫ T
teβsδY (s)dδK(s)
+M(T )−M(t),
15
where (M(t))0≤t≤T is a martingale. On the other hand, in view of uniqueness proof and
young inequality, we have respectively
∫ T
teβsδY (s)dδK(s)≤ 0 and
eβsδY (s)( f (s,Ys,Zs,Us(.))− f (s,Y ′s ,Z
′s,Us(.)))ds
≤ βeβt |δY (s)|2+1
β| f (s,Ys,Zs,Us(.))− f (s,Y ′
s ,Z′s,Us(.))|
2,
which allow us to get
eβt |δY (t)|2+
∫ T
teβs|δZ(s)|2ds+
∫ T
teβs
∫E|δU(s,z)|2m(dz)ds+ ∑
t≤s≤T
eβs(∆s(δY )−∆s(δY ′))2
≤1
β
∫ T
0eβs| f (s,Ys,Zs,Us(.))− f (s,Y ′
s ,Z′s,Us(.))|
2ds+M(T )−M(t). (3.9)
Then taking the conditional expectation with respect (Ft)t≥0 in both side of the previous
inequality, we obtain
eβt |δY (t)|2 ≤1
βE
(∫ T
0eβs| f (s,Ys,Zs,Us(.))− f (s,Y ′
s ,Z′s,Us(.))|
2ds|Ft
),
which together with Doob inequality yields
E
(sup
0≤t≤T
eβt |δY (t)|2
)≤
1
βE
(∫ T
0eβs| f (s,Ys,Zs,Us(.))− f (s,Y ′
s ,Z′s,Us(.))|
2ds
).(3.10)
Taking expectation in both side of (3.9) for t = 0, it follows from (3.10) that
E
(sup
0≤t≤T
eβt |δY (t)|2+∫ T
teβs|δZ(s)|2ds+
∫ T
teβs
∫E|δU(s,z)|2m(dz)ds
)
≤1
βE
(∫ T
0eβs| f (s,Ys,Zs,Us(.))− f (s,Y ′
s ,Z′s,Us(.))|
2ds
). (3.11)
Let us now derive the estimation of right side of inequality (3.11). In view of assumption
(A1), we have∫ T
0eβs| f (s,Ys,Zs,Us(.))− f (s,Y ′
s ,Z′s,Us(.))|
2ds
≤ K
∫ T
0
∫ 0
−Teβs
(|δY (s+u)|2+ |δZ(s+u)|2+
∫E|δU(s+u,z)|2m(dz)
)α(du)ds.
Next, since Z(t) = 0,U(t, .)≡ 0 and Y (t) = Y (0), for t < 0, we get respectively with Fu-
bini’s theorem, changing the variables that∫ T
0eβs| f (s,Ys,Zs,Us(.))− f (s,Y ′
s ,Z′s,Us(.))|
2ds
≤ K max(1,T )eβT
(sup
0≤t≤T
eβt |δY (t)|2+∫ T
0eβs
(|δZ(s)|2+
∫E|δU(s,z)|2m(dz)
)ds
).
(3.12)
16
Thereafter, it follows from (3.11), (3.12) and β = 1T