Reflected backward stochastic differential equations ... - Project Euclid

Electron. Commun. Probab. 20 (2015), no. 26, 1–11. DOI: 10.1214/ECP.v20-3771 ISSN: 1083-589X

ELECTRONIC COMMUNICATIONS in PROBABILITY

Reflected backward stochastic differential equations driven by countable Brownian motions with continuous coefficients Jean-Marc Owo*

Abstract In this note, we study one-dimensional reflected backward stochastic differential equations (RBSDEs) driven by Countable Brownian Motions with one continuous barrier and continuous generators. Via a comparison theorem, we provide the existence of minimal and maximal solutions to this kind of equations. Keywords: Backward doubly stochastic differential equations; Countable Brownian Motions; comparison theorem; continuous and linear growth conditions. AMS MSC 2010: 60J70. Submitted to ECP on September 4, 2014, final version accepted on March 13, 2015.

1

Motivation

Recently, Pengju Duan et al. [10] studied a new class of reflected backward stochastic differential equations driven by countable Brownian motions. That is :

Z Yt = ξ +

T

f (s, Ys , Zs )ds+ t

∞ Z X j=1 t

T

←− gj (s, Ys , Zs )dB js +KT −Kt −

Z

T

Zs dWs .

(1.1)

t

Under the global Lipschitz continuity condition, they proved in [10] via Snell envelope and fixed point theorem, the existence and uniqueness of the solution for RBSDEs (1.1). Unfortunately, the global Lipschitz continuity condition can not be satisfied in certain models that limits the scope of the result of Pengju Duan et al. [10] for several applications (finance, stochastic control, stochastic games, SPDEs, etc,...). In finance, this kind of RBSDEs is useful to describe the wealth/strategy of a particular investor who, trading continuously, has countable extra information which can not be detected in the market. In this case, ξ can be regarded as the contingent claim; Yt is the wealth of the investor at time t, Zt is his strategy at time t to make the wealth Yt be measurable to the information inside or outside the market at time t and to allow the realization of the option YT = ξ , and Kt is the subsidy injected by the government in the market at time t to allow the wealths of investors to remain above a threshold price St at time t; f (t, Yt , Zt ) is the appreciation of the wealth/strategy at time t and gj (t, Yt , Zt ) is the impact of the j th additional information on the wealth/strategy at time t. For example, consider a particular investor who trades continuously in a financial market. Suppose that this investor has countable additional information which is not * UFR

de Mathématiques et Informatique; Université Félix Houphouët-Boigny, Abidjan, Côte d’Ivoire. E-mail: [email protected]

RBDSDEs driven by countable Brownian motions

available in the market. If the appreciation of his wealth/strategy is

q Yt 1{Yt ≥0} , then

this investor has to solve the following RBSDEs :

Yt = ξ +

Z Tq Z ∞ Z T X ←− Ys 1{Ys ≥0} ds+ gj (s, Ys , Zs )dB js +KT −Kt − t

T

Zs dWs .

t

j=1 t

p

Since y 7→ y1{y≥0} is not Lipschitz in y , then we can not apply the existence result in Pengju Duan et al. [10] to get the existence of solution of the above RBSDE. Consequently, the problem of such an investor can not be solved at present. To correct this shortcoming, we relax in this paper the global Lipschitz continuity condition on the coefficients f to a continuity with sub linear condition and derive the existence of minimal and maximal solutions to RBSDEs (1.1). The paper is organized as follows. In section 2, we give some notations and preliminaries Section 3 is devoted to the main result.

2

Preliminaries

The scalar product of the space Rd (d ≥ 2) will be denoted by < ., . > and the associated Euclidian norm by k.k. Throughout this paper, T is a positive constant and (Ω, F, P) is a probability space on j which, {Bt , 0 ≤ t ≤ T }∞ j=1 are mutual independent one-dimensional standard Brownian motions and {Wt , 0 ≤ t ≤ T } is a Rd − valued standard Brownian motion which is j independent of {Bt , 0 ≤ t ≤ T }. Let N denote the class of P-null sets of F and define

 Ft = 

∞ _

 j

B  Ft,T

_

FtW

_

N, 0 ≤ t ≤ T

j=1 η

η

η

where Fs,t = σ{ηr − ηs , s ≤ r ≤ t} for any ηt , and Ft = F0,t . Since {FtW , t ∈ [0, T ]} j

B is an increasing filtration and {Ft,T , t ∈ [0, T ]} is a decreasing filtration, the collection {Ft , t ∈ [0, T ]} is neither increasing nor decreasing so that it does not constitute a filtration.

For any n ∈ N, let M2 (0, T ; Rn ) denote the set of (class of dP ⊗ dt a.e. equal) ndimensional jointly measurable random processes {ϕt ; 0 ≤ t ≤ T } which satisfy: (i) (ii)

kϕk2M2 = E(

RT 0

|ϕt |2 dt) < ∞

ϕt is Ft -measurable, for a.e. t ∈ [0, T ].

We denote by S 2 ([0, T ]; R) the set of continuous one-dimensional random processes which satisfy: (i)

kϕk2S 2 = E( sup | ϕt |2 ) < ∞ 0≤t≤T

(ii)

ϕt is Ft -measurable, for any t ∈ [0, T ].

We set by A2 (0, T, R+ ) the space of a real positive continuous and increasing process, such that ϕ0 = 0 and (i) (ii)

kϕk2A2 = E(|ϕT |2 ) < ∞ ϕt is Ft -measurable, for a.e. t ∈ [0, T ].

Finally, we set E 2 (0, T ) = S 2 ([0, T ], R) × M2 (0, T, Rd ) × A2 (0, T, R+ ). ECP 20 (2015), paper 26.

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Definition 2.1. A solution of a (1.1) is a triplet of (R × Rd × R+ )-valued process (Yt , Zt , Kt )0≤t≤T , which satisfies (1.1), and

(i) (Y, Z, K) ∈ E 2 (0, T ); (ii) Yt ≥ St ; Z

T

(Yt − St )dKt = 0.

(iii) K is a continuous and increasing process with K0 = 0 and 0

Definition 2.2. A triplet of processes (Y , Z, K) (resp. (Y , Z, K)) of E 2 (0, T ) is said to be a minimal (resp. a maximal) solution of (1.1) if for any other solution (Y, Z, K) of (1.1), we have Y ≤ Y (resp.Y ≤ Y ). We consider the following assumptions: Assume that the coefficients f : Ω × [0, T ] × R × Rd → R, g : Ω × [0, T ] × R × Rd → R , the terminal value ξ : Ω → R and the obstacle S : Ω × [0, T ] → R satisfy the following conditions: (H1) (i) ξ is a FT -measurable random variable such that, E(|ξ|2 ) < ∞, (ii) S ∈ S 2 ([0, T ]) such that, ST ≤ ξ a.s.; ∞

(H2) for any (t, y, z) ∈ [0, T ] × R × Rd , f (t, y, z), {gj (t, y, z)}j=1 are Ft -measurable such that

Z E 0

T

|f (s, 0, 0)|2 ds+

Z T ∞ X E |gj (s, 0, 0)|2 ds < +∞; j=1

0

(H3) for all t ∈ [0, T ] and (y1 , z1 ), (y2 , z2 ) ∈ R × Rd ,

| gj (t, y1 , z1 ) − gj (t, y2 , z2 ) |2 ≤ Cj | y1 − y2 |2 +αj || z1 − z2 ||2 where Cj > 0 and αj > 0 are constants with

∞ X

Cj < ∞ and α =

j=1

∞ X

αj < 1,

j=1

(H4) (i) for all t ∈ [0, T ] and (y1 , z1 ), (y2 , z2 ) ∈ R × Rd ,

| f (t, y1 , z1 ) − f (t, y2 , z2 ) |2 ≤ C(| y1 − y2 |2 + || z1 − z2 ||2 ), where C > 0 is a nonnogative constant; (ii) for all (t, y, z) ∈ [0, T ] × R × Rd , |f (t, y, z)| ≤ M (1 + |y| + ||z||), where M > 0 is a nonnegative constant, (H5) (i) for a.e. (t, ω) ∈ [0, T ] × Ω, the map (y, z) 7→ f (t, y, z) is continuous, (ii) for all (t, y, z) ∈ [0, T ] × R × Rd , |f (t, y, z)| ≤ ϕ(t) + M (|y| + ||z||), where ϕ ∈ M2 (0, T, R) is a positive process and M > 0 is a nonnegative constant. Lemma 2.3 (Pengju Duan et al. [10]). Under assumptions (H1) − (H4), there exists a unique solution (Y, Z, K) ∈ E 2 (0, T ) of (1.1). Remark 2.4. The above result still holds when assumption (H4)(ii) is replaced by (H5)(ii) ECP 20 (2015), paper 26.

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3

The main results

In this section, our principal aim, is to prove an existence result for reflected BSDEs driven by countable Brownian motions under general continuous conditions. More precisely, we will derive the existence of a minimal and a maximal solution to equation (1.1) under assumptions (H1)-(H3) and (H5). To this end, we establish first, the following comparison theorem, which extend the comparison theorem due to Aman and Owo [2] with RBSDEs driven by finite Brownian motions. For i = 1, 2, let us consider the following RBSDEs driven by countable Brownian motions:

                    

(i) Yti = ξ i +

T

Z

f i (s, Ysi , Zsi )ds +

t

∞ Z X j=1

T

←− gj (s, Ysi , Zsi )dB js +

Z

t

T

dKsi −

t

Z

T

Zsi dWs ,

t

(3.1)

(ii) Yti ≥ Sti , T

Z

(Yti − Sti )dKti = 0.

(iii) 0

Theorem 3.1 (Comparison Theorem). Let assumptions (H1) − (H3) hold. Assume that RBDSDEs (3.1) (i = 1, 2) have solutions (Y 1 , Z 1 , K 1 ) and (Y 2 , Z 2 , K 2 ), respectively. Assume moreover that: (i) ξ 1 ≤ ξ 2 a.s., (ii) St1 ≤ St2 a.s., for all t ∈ [0, T ] (iii) f 1 satisfies (H4)(i) such that f 1 (t, Y 2 , Z 2 ) ≤ f 2 (t, Y 2 , Z 2 ) a.s. (resp. f 2 satisfies (H4)(i) such that f 1 (t, Y 1 , Z 1 ) ≤ f 2 (t, Y 1 , Z 1 ) a.s.). Then, Yt1 ≤ Yt2 a.s., for all t ∈ [0, T ]. Proof. Applying Itô’s formula to |(Yt1 − Yt2 )+ |2 , we have

E|(Yt1 − Yt2 )+ |2 + E

Z

T

1{Ys1 >Ys2 } kZs1 − Zs2 k2 ds

t

=

1

2 + 2

Z

E|(ξ − ξ ) | + 2E

T

(Ys1 − Ys2 )+ f 1 (s, Ys1 , Zs1 ) − f 2 (s, Ys2 , Zs2 ) ds

t

Z

T

+2E

(Ys1 − Ys2 )+ (dKs1 − dKs2 )

t

+

∞ X

Z E

j=1

t

T

1{Ys1 >Ys2 } |gj (s, Ys1 , Zs1 ) − g(s, Ys2 , Zs2 )|2 ds.

From (i), E|(ξ 1 − ξ 2 )+ |2 = 0 and from (iii), we have

f 1 (s, Ys1 , Zs1 ) − f 2 (s, Ys2 , Zs2 ) ≤ f 1 (s, Ys1 , Zs1 ) − f 1 (s, Ys2 , Zs2 ). Therefore, from Young inequality, and the fact that f 1 satisfies (H4)(i) and g verifies ECP 20 (2015), paper 26.

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(H3), we get

Z T 1{Ys1 >Ys2 } kZs1 − Zs2 k2 ds E|(Yt1 − Yt2 )+ |2 + E t   Z T ∞ X 1 ≤  +βC + |(Ys1 − Ys2 )+ |2 ds Cj  E β t j=1 +(βC +

∞ X

Z αj )E t

j=1

Z

T

1{Ys1 >Ys2 } kZs1 − Zs2 k2 ds

T

(Ys1 − Ys2 )+ (dKs1 − dKs2 ).

+2E t

Since Ys1 > Ss2 ≥ Ss1 on the set {Yss > Ys2 } we derive that

Z E

T

(Ys1 − Ys2 )+ (dKs1 − dKs2 ) = −E

t

Z

T

(Ys1 − Ys2 )+ dKs2 ≤ 0.

t

Hence,

Z T E|(Yt1 − Yt2 )+ |2 + E 1{Ys1 >Ys2 } kZs1 − Zs2 k2 ds t   Z T ∞ X 1 Cj  E |(Ys1 − Ys2 )+ |2 ds ≤  + βC + β t j=1 Z T +(βC + α)E 1{Ys1 >Ys2 } kZs1 − Zs2 k2 ds. t

Consequently, choosing 0 < β
M and t ∈ [0, T ], (y, z), (yi , zi ) ∈ R × Rd , i = 1, 2, we have (i) −ϕ(t) − M (|y| + |z|) ≤ f n (t, y, z) ≤ f (t, y, z) ≤ f¯n (t, y, z) ≤ ϕ(t) + M (|y| + |z|); (ii) f n (t, y, z) is non-decreasing in n and f n (t, y, z) is non-increasing in n ; (iii) |ψ(t, y1 , z1 ) − ψ(t, y2 , z2 )| ≤ n(|y1 − y2 | + |z1 − z2 |), with ψ ∈

ECP 20 (2015), paper 26.

n o f n, f n ; ecp.ejpecp.org

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(iv) If (yn , zn ) → (y, z), then ψ(t, yn , zn ) → f (t, y, z) as n → +∞, with ψ ∈

o n f n, f n .

Now, we are ready to establish the main result of this paper which is the following theorem. Theorem 3.3. Under assumptions (H1) − (H3) and (H5), there exists a minimal (resp. a maximal) solution (Y , Z, K) ∈ E 2 (0, T ) (resp. (Y , Z, K) ∈ E 2 (0, T ) of (1.1). Proof. We only prove that (1.1) has a minimal solution. The other case can be proved similarly. For fixed (t, ω) ∈ [0, T ] × Ω, it follows from (H5), that (y, z) 7→ f (t, y, z) is continuous and with linear growth. Then, by Lemma 3.2, the associated sequences of functions f n and f n are Lipschitz functions. Therefore, since assumptions (H1) − (H4)(i) and (H5)(ii) hold, we get from Lemma 2.3, that there exists a unique solution (Y n , Z n , K n ) ∈ E 2 (0, T ) of the following RBDSDEs,

                    

(i) Ytn = ξ +

Z

T

f n (s, Ysn , Zsn )ds +

t

∞ Z X

T

←− gj (s, Ysn , Zsn )dB js +

Z

t

j=1

T

dKsn −

t

Z

T

Zsn dWs ,

t

(3.2)

(ii) Ytn ≥ St , Z (iii)

T

(Ytn − Stn )dKtn = 0.

0

Since by Lemma 3.2, f n ≤ f n+1 , for all n ≥ M , it follows from the comparison theorem (Theorem 3.1) that for every n ≥ M

Y n ≤ Y n+1 , dt ⊗ dP-a.s.

(3.3)

To complete the proof, it suffices to show that the sequence (Y n , Z n , K n ) converges to a process (Y , Z, K) which is the minimal solution of the RBDSDEs (1.1). To this end, we will sketch the proof in two steps: Step 1: A priori estimates There exists a constant C > 0 independent of n such that

sup E n≥M

sup |Ytn |2 +

0≤t≤T

Z

!

T

kZtn k2 dt + |KTn |2

≤ C.

(3.4)

0

Indeed, applying Itô’s formula to |Ytn |2 , we have

E|Ytn |2 + E

T

Z

kZsn k2 ds

t

= E|ξ|2 + 2E

Z

T

t

Ysn f n (s, Ysn , Zsn )ds + 2E

Z

T

Ysn dKsn +

t

∞ X j=1

Z E

T

|gj (s, Ysn , Zsn )|2 ds

t

From assumption (H3), the proprieties of f n and Young’s inequality, for any θ > 0, we have

2Ysn f n (s, Ysn , Zsn ) ≤ (1 + 2M +

M2 2 2 ) |Ysn | + θ|Zsn |2 + |ϕ(s)| , θ

1 2 2 2 2 |gj (s, Ysn , Zsn )| ≤ (1 + θ)Cj |Ysn | + (1 + θ)αj |Zsn | + (1 + ) |gj (s, 0, 0)| . θ ECP 20 (2015), paper 26.

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Using again Young inequality, we have for any β > 0, T

Z

Ysn dKsn = 2E

2E

T

Z

t

Ss dKsn ≤

t

1 2 E sup |Ss |2 + βE (KTn − Ktn ) . β 0≤s≤T

Since

KTn

−

Ktn

=

T

Z

Ytn

f (s, Ysn , Zsn )ds

−ξ− t

−

∞ Z X j=1

T

←− gj (s, Ysn , Zsn )dB js +

t

Z

T

Zsn dWs , t ∈ [0, T ],

t

we have, for any t ∈ [0, T ], 2

E (KTn − Ktn )  ≤

 2 2 2 ∞ Z Z Z T T T X ←−   2 Zsn dWs  gj (s, Ysn , Zsn )dB js + f n (s, Ysn , Zsn )ds + 5E |Ytn | + |ξ|2 + t t t j=1

≤

2 |Ytn |

5E

Z

2

+ |ξ| + 3T

T

M

2

2 |Ysn | +M 2

2 2 |Zsn | +|ϕ(s)|

! ds

t

 ∞ Z X +5E 

T

t

j=1

1 2 2 2 (1 + θ)Cj |Ysn | + (1 + θ)αj |Zsn | + (1 + ) |gj (s, 0, 0)| ds + θ

Z

T

 2 |Zsn | ds ,

t

Therefore,

E|Ytn |2

Z

T

kZsn k2 ds Z T 1 M2 2 n 2 n 2 2 ) |Ys | + θ|Zs | + |ϕ(s)| ds + E sup |Ss |2 ≤ E|ξ| + E (1 + 2M + θ β 0≤s≤T t ! Z T 2 2 2 2 5βE |Ytn | + |ξ|2 + 3T M 2 |Ysn | +M 2 |Zsn | +|ϕ(s)| ds +E

t

t ∞ X

Z T 1 2 2 2 2 |Zsn | ds (1 + θ)Cj |Ysn | + (1 + θ)αj |Zsn | + (1 + ) |gj (s, 0, 0)| ds + 5βE θ t t j=1 Z T ∞ X 1 2 n 2 n 2 + E (1 + θ)Cj |Ys | + (1 + θ)αj |Zs | + (1 + ) |gj (s, 0, 0)| ds. θ t j=1

+5β

Z

T

E

Consequently,

1 − 5β

2 E |Ytn |

i Z + 1 − θ − (1 + θ)α − 5β(3T M + (1 + θ)α + 1) E h

2

T

2

|Zsn | ds

t

≤

∞ h i Z T X M2 2 2 1 + 5β E |ξ| + (1 + 2M + ) + (5β + 1)(1 + θ) Cj + 15βT M 2 E |Ysn | ds θ t j=1 Z T Z ∞ T X 1 1 2 2 +(1 + 15βT )E |ϕ(s)| ds + (1 + )(1 + 5β) E |gj (s, 0, 0)| ds + E sup |Ss |2 . θ β 0≤s≤T t t j=1

ECP 20 (2015), paper 26.

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Choosing β, θ > 0 such that, β < 2

E |Ytn |

1−α 5(3T M 2 +α+1) , T

Z

2

≤

θ≤

1−α−5β(3T M 2 +α+1) , 1+α+5βα

2

|Ysn | ds + c3 E

c1 E |ξ| + c2 E

t

+c4

T

Z

2

|gj (s, 0, 0)| ds + c5 E sup |Ss |2 ,

E

0≤s≤T

t

j=1 2

where c1 =

c5 =

1+5β 1−5β , c2

=

2

|ϕ(s)| ds

t ∞ X

T

Z

we derive that

(1+2M + Mθ )+(5β+1)(1+θ) 1−5β

P∞

j=1

Cj +15βT M 2

, c3 =

1+15βT 1−5β

, c4 =

(1+ θ1 )(1+5β) , 1−5β

1 β(1−5β) .

Applying Gronwall’s inequality, we get 2 E |Ytn |

Z

2

≤

T

2

c1 E |ξ| + c3 E

|ϕ(s)| ds t

∞ X

+c4



T

Z

2

|gj (s, 0, 0)| ds + c5 E sup |Ss |2  ec2 T .

E

0≤s≤T

t

j=1

Therefore, we have the existence of a constant c = c(T, M, α) such that

E

2 |Ytn |

!

T

Z

kZsn k2 ds + |KTn |2

+ t



T

Z

2

2

≤ cE |ξ| +

|ϕ(s)| ds + t

∞ Z X



T

2

|gj (s, 0, 0)| ds + sup |Ss |2  , 0≤s≤T

t

j=1

which by Burkhölder-Davis-Gundy’s inequality provides

E

sup 0≤t≤T

!

T

Z

2 |Ytn |

kZsn k2 ds

+

+

|KTn |2

0



T

Z

2

2

≤ cE |ξ| +

|ϕ(s)| ds + 0

∞ Z X j=1



T

2

|gj (s, 0, 0)| ds + sup |Ss |2  < ∞. 0≤s≤T

0

Step 2: Convergence to the minimal solution We have from (3.3) and (3.4) the existence of a process Y such that Ytn % Y t a.s. for all t ∈ [0, T ]. Hence, it follows from Fatou’s lemma together with the dominated convergence theorem that 2

E |Y t |

Z

≤ C and

lim E

n→∞

!

T

|Ysn

2

− Y s | ds

=0

(3.5)

0

On the other hand, for all n > m ≥ M , applying Itô’s formula to |Ytn − Ytm |2 and noting

Z that

T

(Ysn − Ysm ) (dKsn − dKsm ) ≤ 0, we have

t

E|Ytn − Ytm |2 +

Z

T

kZsn − Zsm k2 ds

t

Z

T

(Ysn

≤ 2E t

+

∞ X j=1

Z E

− Ysm ) f n (s, Ysn , Zsn ) − f m (s, Ysm , Zsm ) ds

T

|gj (s, Ysn , Zsn ) − gj (s, Ysm , Zsm )|2 ds.

(3.6)

t

ECP 20 (2015), paper 26.

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Then, from Hölder’s inequality, the uniform linear growth condition on the sequence f n and the assumption (H3), we have,

E|Y0n − Y0m |2 + (1 − α)E

Z

T

kZsn − Zsm k2 ds

0

≤

Z T 21 Z T 12 n m 2 6E 2|ϕ(s)|2 + M 2 (|Ysn |2 + |Ysm |2 + kZsn k2 + kZsm k2 ) ds 2 E |Ys − Ys | ds 0  0  Z ∞ T X + Cj  E |Ysn − Ysm |2 ds. 0

j=1

Therefore, by virtue of inequality (3.4) and the fact that ϕ ∈ M2 (0, T, R), we have the existence of a constant C such that, for all n > m ≥ M ,

Z (1 − α)E

T

Z n m 2 kZs − Zs k ds ≤ C E

0

T

  Z ∞ 12 X n m 2   |Ys − Ys | ds + Cj E

0

T

|Ysn − Ysm |2 ds.

0

j=1

Thus from (3.5), {Z n } is a Cauchy sequence in the Banach space M2 (0, T, Rd ), and there exists an Ft -jointly measurable process Z such that {Z n } converges to Z as n → ∞. Similarly, by Itô’s formula together with Burkholder-Davis-Gundy inequality, it follows that

E sup |Ytn − Ytm |2 → 0, as n, m → ∞, 0≤t≤T

n from which we deduce that P-almost surely, Y converges uniformly to Y which is

continuous, such that E

sup |Y t |2

≤ C.

0≤t≤T

On the other hand, since Z n → Z in M2 (0, T, Rd ), along a subsequence which we still denote Z n , Z n → Z, dt ⊗ dP a.e., and there exists Π ∈ M2 (0, T, R) such that ∀n ≥ M, |Z n | < Π, dt ⊗ dP a.e.. Therefore, by Lemma 3.2, we have

f n (t, Ytn , Ztn ) −→

f (t, Y t , Z t ) dt ⊗ dP a.e., as n → ∞.

Then, by virtue of (H5)(ii) and (3.4), it follows from the dominated convergence theorem that

Z E t

T

2 f n (s, Ysn , Zsn ) − f (s, Y s , Z s ) ds −→ 0, as n → ∞.

(3.7)

Further, in view of Burkholder-Davis-Gundy inequality, (H3) and (3.5), we obtain

! X Z T Z T ∞ ← − ← − E sup gj (s, Ysn , Zsn )dB js − gj (s, Y s , Z s )dB js −→ 0, 0≤t≤T j=1 t t

as n → ∞,(3.8)

and

Z Z T T n Zs dWs − Z s dWs E sup 0≤t≤T t t

−→ 0,

ECP 20 (2015), paper 26.

as n → ∞.

(3.9)

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We get also, from Hölder’s inequality,

E sup |Ktn − Ktm |

2

0≤t≤T

≤

2

5E|Y0n − Y0m |2 + 5E sup |Ytn − Ytm | + 5T E 0≤t≤T

Z

T

0

2 f n (s, Ysn , Ysn ) − f m (s, Ysm , Zsm ) ds

2 ∞ Z t X ←− (gj (s, Ysn , Ysn ) − gj (s, Ysm , Zsm )) dB js +5E sup 0≤t≤T j=1 0 2 Z t +5E sup (Zsm+n − Zsn )dWs , 0≤t≤T

0

which, together with (3.7), (3.8), (3.9), provides 2

E sup |Ktn − Ktm | −→ 0,

as n, m → ∞.

0≤t≤T

Consequently, there exists a Ft -measurable process K with value in R+ such that 2

E sup |Ktn − K t | −→ 0,

as n → ∞.

0≤t≤T

Obviously, K 0 = 0 and {K t ; 0 ≤ t ≤ T } is a non-decreasing and continuous process. On the other hand, from the result of Saisho [11] (see p. 465), we have

Z

T

(Ysn

−

Ss )dKsn

0

Z

T

(Y s − Ss )dK s P − a.s.,

−→

as n → ∞.

0

Finally, passing to the limit in (3.2), we conclude that (Y , Z, K) is a solution of (1.1). 2 Now, let (Y, Z, K) ∈ Em (0, T ) be any solution of (1.1). By virtue of Theorem 3.1, we have Y n ≤ Y, ∀n ≥ M . Therefore, due to the above step and taking the limit, we have Y ≤ Y . The proof is complete.

References [1] Aman, A., Lp -solution of reflected generalized BSDEs with non-Lipschitz coefficients. Random Oper. Stoch. Equ., 17 no. 3: 201-219 (2009). MR-2574101 [2] A. Aman and J.M. Owo, Reflected backward doubly stochastic differential equations with discontinuous generator, Random Oper. Stoch. Equ. 20, 119-134 (2012).doi:10.1515/rose2012-0005. MR-2980358 [3] K. Bahlali, M. Hassani, B. Mansouri, and N. Mrhardy, One barrier reflected backward doubly stochastic differential equations with continuous generator, C.R. Acad. Sci. Paris, Ser. I 347, 1201–1206 (2009). MR-2567003 [4] K. Bahlali, S. Hamadène, B. Mezerdi, Backward stochastic differential equations with two reflecting barriers and continuous with quadratic growth coefficient, Stoch. Process. Appl. 115, 1107-1129 (2005). [5] Bally, V. and Matoussi, A., Weak solutions for SPDEs and backward doubly stochastic differential equations.J. Theoret. Probab., 14 no.1: 125-164 (2001). MR-1822898 [6] Boualem Djehiche, Modeste N’zi and Jean-Marc Owo, Stochastic viscosity solutions for SPDEs with continuous coefficients. J. Math. Anal. Appl. 384, 63-69 (2011) MR-2822850 [7] Boufoussi, B., Jan Van Casteren, and Mrhardy, N., Generalized backward doubly stochastic differential equations and SPDEs with nonlinear neumann boundary conditions. Bernoulli, 13 no. 2: 423-446 (2007) MR-2331258

ECP 20 (2015), paper 26.

ecp.ejpecp.org

Page 10/11


[8] J.P. Lepeltier, J. San Martin, Backward stochastic differential equations with continuous coefficients, Statist. Probab. Lett. 32, 425-430 (1997). MR-1602231 [9] E. Pardoux, S. Peng, Backward doubly stochastic differential equations and systèmes of quasilinear SPDEs. Proba. Theory Related Fields 98, 209–227 (1994). [10] Pengju Duan, Min Ren, and Shilong Fei, "Reflected Backward Stochastic Differential Equations Driven by Countable Brownian Motions" Journal of Applied Mathematics, Vol. 2013, Article ID 729636, 7 pages, (2013). doi:10.1155/2013/729636 [11] Y. Saisho, SDE for multidimensional domains with reflecting boundary, Probab. Theory Related Fields, 74, 455–477 (1987). MR-0873889

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