Existence of solution for Mean-field Reflected Discontinuous Backward Doubly Stochastic Differential Equation

Mostapha Abdelouahab Saouli 1 , 2
  • 1 Laboratory of Applied Mathematics, University of Biskra, Algeria
  • 2 Department of Mathematics, University of Kasdi Merbah Ouargla, Algeria
Mostapha Abdelouahab Saouli
  • Corresponding author
  • Laboratory of Applied Mathematics, University of Biskra, POB 145, Algeria
  • Department of Mathematics, University of Kasdi Merbah Ouargla, Algeria
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Abstract

In this paper we prove the existence of a solution for mean-field reflected backward doubly stochastic differential equations (MF-RBDSDEs) with one continuous barrier and discontinuous generator (left-continuous). By a comparison theorem establish here for MF-RBDSDEs, we provide a minimal or a maximal solution to MF-RBDSDEs.

1 Introduction

The theory of nonlinear backward stochastic differential equations (BSDEs in short) have been first introduced by Pardoux and Peng [6] (1990). They proved the existence and uniqueness of the adapted processes (Y,Z), solution of the following equation:
Yt=ξ+tTf(s,Ys,Zs)dstTZsdWs,0tT,
where the terminal value ξ is square integrable and the coefficient f is uniformly Lipschitz in (y, z), several authors interested in weakening this assumption; In [5] (1997), the authers prove the existence of a solution for one dimensional backward stochastic differential equations where the coefficient is continuous and it has a linear growth, they also obtain the existence of a minimal solution. In [3] (2008) the author prove the existence of the solution to BSDEs whose coefficient may be discontinuous in y and continuous in z.
A new kind of backward stochastic differential equations was introduced by Pardoux and Peng [7] in (1994) which is a class of backward doubly stochastic differential equation (BDSDE for short) of the form:
Yt=ξ+tTf(s,Ys,Zs)ds+tTg(s,Ys,Zs)dBstTZsdWs,0tT,
with two different directions of stochastic integrals, i.e., the equation involves both a standard (forward) stochastic integral dWt and a backward stochastic integral dBt and ξ is a random variable termed the terminal condition.

After the authors have proved an existence and unique solution when f and g are uniform Lipschitz, several authors interested to weakening this assumption, see [4]. In [9](2005) the authors obtained the existence of the solution of BDSDE under continuous assumption and gave the comparison theorem for one dimensional BDSDE.

On the other hand Bahlali et al [1] (2009) introduced a special class of reflected BDSDEs (RBDSDEs in short) which is a BDSDE but the solution is forced to stay above a lower barrier. In particular, a solution of such equation is a triplet of processes (Y,Z,K) satisfying
Yt=ξ+tTf(s,Ys,Zs)ds+tTg(s,Ys,Zs)dBs+tTdKstTZsdWs,t[0,T],
and YtSt a.s. for any t ∈ [0, T ]. The role of the nondecreasing continuous process (Kt)t∈ [0, T] is to puch upward the process Y in order to keep it above S, it satisfies the skorohod condition
0T(YsSs)dKs=0.
In this paper, motivated by the above results and by the result introduced by Xu, R. (2012) [10], we establish the existence of the a minimal solution to the following reflected MF-BDSDE,
Yt=ξ+tTE'(f(s,ω,ω',Ys,(Ys)',Zs,(Zs)'))ds+tTdKs+tTE'(g(s,ω,ω',Ys,(Ys)',Zs,(Zs)'))dBstTZsdWs,0tT,
whose coefficient may be discontinuous in y and continuous in z.
  • In Section 2, we give some preliminaries about MF-BDSDE with one continuous barrier.
  • In Section 3, under certain assumptions, we obtain the existence for a minimal solution to the Mean-field backward doubly stochastic differential equation with one continuous barrier and discontinuous generator (left-continuous).

2 Framework

Let (Ω, , P) be a complete probability space. For T > 0, let {Wt, 0 ≤ tT} and {Bt, 0 ≤ tT} be two independent standard Brownian motion defined on (Ω, , P) with values in ℝd and ℝ, respectively.

Let tW:=σ(Ws;0st) , and t,TB:=σ(BsBt;tsT) , completed with P-null sets. We put,
t:=tWt,TB.
It should be noted that (t) is not an increasing family of sub σ–fields, and hence it is not a filtration.

Let (Ω¯,¯,P¯)=(Ω×Ω,tt,PP) be the (non-completed) product of (Ω , P) with itself. We denote the filtration of this product space by ¯={¯t=tt,0tT} .

A random variable ξL0 (Ω, , P;ℝn) originally defined on Ω is extended canonically to Ω¯:ξ´(ω´,ω)=ξ(ω´),(ω´,ω)Ω¯=Ω×Ω. .

For every θL1(Ω¯,¯,P¯) , the variable θ (·, ω) : Ω → ℝ belongs to L1(Ω¯,¯,P¯) , P ()−a.s,. We denote its expectation by É(θ(,ω))=Ωθ(ω´,ω)P(dω´)

Notice that
{É(θ)=É(θ(,ω))L1(Ω,,P)andE¯(θ)=Ω¯θdP¯=ΩÉ(θ(,ω))P(dω)=E(É(θ)).

We consider the following spaces of processus:

  • Let ℳ2 (0, T, ℝd) denote the set of d– dimensional, t– progressively measurable processes {φt;t ∈ [0, T ]}, such that 𝔼0T|φt|2dt< .
  • We denote by 𝒮2 (0, T, ℝd), the set of t– adapted cádlág processes {φt; t ∈ [0, T]}, which satisfy 𝔼(sup0 ≤ t ≤ T|φt|2) < ∞.
  • 𝒜2 set of continuous, increasing, t-adapted process K: [0, T] × Ω → [0, +∞) with K0 = 0 and 𝔼(KT)2 < +∞.
  • 𝕃2 set of T- measurable random variables ξ :Ω → ℝ with 𝔼 |ξ|2 < +∞.

Definition 1
A solution of equation (2) is a triple (Y, Z, K) which belongs to the space 𝒮2 (0, T, ℝd) × ℳ2 (0, T, ℝd) × 𝒜2 and satisfies (2) such that:
{StYt,0tT,0T(YsLs)dKs=0.
Remark 1

In the case where S = −∞ (i.e., MF-BDSDEs without lower barrier), the process K has no effect i.e., K ≡ 0.

Remark 2

In the setup of system (2) the process S (·) play the role of reflecting barrier.

Remark 3

The state process Y (·) is forced to stay above the lower barrier S (·), thanks to the action of the increasing reflection process K (·).

The coefficient of mean-field Reflected BDSDE is a function. We assume that f and g satisfy the following assumptions on the data (ξ, f, g, S):

  • (H.1) The terminal value ξ be a given random variable in 𝕃2.
  • (H.2) (St)t ≥ 0, is a continuous progressively measurable real valued process satisfying
    𝔼(sup0tT(St+)2)<+,whereSt+:=max(St,0).
  • (H.3) For t ∈ [0, T], STξ, ℙ-almost surely.
  • (H.4)f : Ω × [0, T] × ℝ × ℝ × ℝd × ℝd → ℝ; g : Ω × [0, T] × ℝ × ℝ × ℝd × ℝd → ℝk be jointly measurable such that for any (y, y, z, z) ∈ ℝ × ℝ × ℝd × ℝd,
    {f(,ω,y,y',z,z')2(0,T,d),andg(,ω,y,y',z,z')2(0,T,d).
  • (H.5) There exist constant C ≥ 0 and a constant 0α12 such that for every (ω, t) ∈ Ω × [0, T ] and (y, y) ∈ ℝ2, (z, z) ∈ ℝd × ℝd,
    {(i)|f(t,y1,y1',z1,z1')f(t,y2,y2',z2,z2')|2C{|y1y2|2+|y1'y2'|2+|z1z2|2+|z1'z2'|2},(ii)|g(t,y1,y1',z1,z1')g(t,y2,y2',z2,z2')|2C{|y1y2|2+|y1'y2'|2}+α{|z1z2|2+|z1'z2'|2}.
  • (H.6) (i) For a.e (t, ω) the mapping (y, y, z, z) → f (t, y, y, z, z) is a cotinuous. (ii) There exist constant C ≥ 0 and a constant 0α12 such that for every (ω, t) ∈ Ω × [0, T] and (y, y) ∈ ℝ2, (z, z) ∈ ℝd × ℝd,
    {|f(t,y,y',z,ź)|C(1+|y|+|y'|+|z|+|ź|),gsatisfies(H.2)(ii).

We recall the following existence results.

Proposition 1

[2] (2014). Under the assumptions (H.1)–(H.5) the reflected BDSDE (2) has a unique solution (Y, Z, K) ∈ 𝒮2 (0, T, ℝd) × ℳ2 (0, T, ℝd) × 𝒜2.

3 Existence result

In this section we are interested in weakening the conditions on f. We assume that f and g satisfy the following assumptions:

  • (H.7) Linear growth: There esists a nonnegative process ft ∈ 𝕄2 (0, T, ℝd) such that
    (t,y,y',z)[0,T]×2×d,|f(t,y,y',z)|ft(ω)+C(|y|+|y'|+|z|).
  • (H.8)f (t, ·, y, z): ℝ → ℝ is a left continuous and f (t, y, ·,·) is a cotinuous.
  • (H.9) There exists a continuous fonction π : [0, T ] × (ℝ)2 × ℝd satisfying for y1y2, (y1',y2')()2 , (z1, z2) ∈ (ℝd)2
    {|π(t,y,y',z)|C(|y|+|y'|+|z|),f(t,ω,y1,y1',z1)f(t,ω,y2,y2',z2)π(t,y1y2,y1'y2',z1z2).
  • (H.10) Monotonicity in y: ∀ (y, y, z), f (t, y, y, z) is increasing in y.
  • (H.11)g satisfies (H.5)(ii) and g(t, 0, 0, 0) ≡ 0.

Hence, we only consider the following type of Mean-field reflected BDSDE:
Yt=ξ+tTE'(f(s,ω,ω',Ys,(Y˜s)',Zs))ds+tTdKs+tTE'(g(s,ω,ω',Ys,(Y˜s)',Zs))dBstTZsdWs,0tT.
Proposition 2

[2] (2014). Under the assumption (H.1)–(H.4) and (H.6), and for any random variable ξ ∈ 𝕃2the mean-field RBDSDE (3) a has an adapted solution (Y, Z, K) ∈ 𝒮2 (0, T, ℝd) × ℳ2 (0, T, ℝd) × 𝒜2, which is a minimal one, in the sense that, if (Y*, Z*, K*) is any other solution we Y ≤ Y*, P – a.s.

Now we prove a technical Lemma before we introduce the main theorem.

Lemma 3
Let π (t, y, y, z) satisfies (H.9), g satisfies (H.11) and h belongs in2 (0, T, ℝd). For a continuous function of finite variation K˜ belong in 𝒜2we consider the processes(Y˜,Z˜)𝒮2(0,T,)×2(0,T,d)such that:
{(i)Y˜t=ξ+tTE'(π(s,ω,ω',Y˜s,(Y˜s)',Z˜s)+h(s))ds+tTdK˜s+tTE'(g(s,ω,ω',Y˜s,(Y˜s)',Z˜s))dBstTZ˜sdWs,0tT,(ii)0TY˜sdK˜s0.
Then we have
  1. (i)The MF-RBDSDE (4) has a least one solution(Y˜,Z˜,K˜)𝒮2(0,T,d)×2(0,T,d)×𝒜2
  2. (ii)if h(t) ≥ 0 and ξ ≥ 0, we haveY˜t0 , dℙ × dt – a.s.
Proof
(i) See [2], (2014). (ii) Applying Tanaka's formula to |Y˜t|2 , we have
𝔼|Y˜t|2+𝔼tT1{Y˜s<0}|Z˜s|2ds=𝔼|ξ|22𝔼tTY˜sE'(π(s,Y˜s,(Y˜s)',Z˜s)+h(s))ds2𝔼tTY˜sdK˜s+𝔼tT1{Y˜s<0}||E'(g(s,Y˜s,(Y˜s)',Z˜s))||2ds.
Since 2𝔼tTY˜sdK˜s0 , h(s) ≥ 0 and ξ ≥ 0, we get
𝔼|Y˜t|2+𝔼tT1{Y˜s<0}|Z˜s|2ds2𝔼tTY˜sE'(π(s,Y˜s,(Y˜s)',Z˜s))ds+𝔼tT1{Y˜s<0}||E'(g(s,Y˜s,(Y˜s)',Z˜s))||2ds
By (H.9), we get |π(s,Y˜s,(Y˜s)',Z˜s)|C(|Y˜s|+|(Y˜s)'|+|Z˜s|) and by assumption (H.11) for g, we have
𝔼|Y˜t|2+𝔼tT1{Y˜s<0}|Z˜s|2ds(4C2+C2β+2C)𝔼tT|Y˜s|2ds+(α+β)𝔼tT1{Y˜s<0}|Z˜s|2ds.

Therefore, choosing 0 ≤ β ≤ 1 – α and using Gronwall inequality, we have Y˜t=0 , ℙ – a.s., ∀t ∈ [0, T], which implies that Y˜t0 ℙ – a.s., ∀t ∈ [0, T].

Before we prove the main result, we construct a sequence of MF-RBDSDEs as follows:
{Y¯t0=ξ+tTE'(C(|Y¯s0|+(Y¯s0)'+|Z¯s0|)fs)ds+tTE'(g(s,Y¯s0+(Y¯s0)'+Z¯s0))dBs+tTdK¯s0tTZ¯s0dWs,0tT,(ii)Y¯t0St,(iii)0T(Y¯s0Ss)dK¯s0=0.
{(i)Y¯tn=ξ+tTE'(f(s,Y¯sn1,(Y¯sn1)',Z¯sn1)+π(s,δY¯sn,δ(Y¯sn)',δZ¯tn))ds+tTE'(g(s,Y¯sn,(Y¯sn)',Z¯sn))dBs+tTdK¯sntTZ¯sndWs,0tT,(ii)Y¯tnSt,(iii)0T(Y¯snSs)dk˜sn=0.
{(i)Yt0=ξ+tTE'(C(|Ys0|+(Ys0)'+|Zs0|)+fs)ds+tTdKs0+tTE'(g(s,Ys0,(Ys0)'+Zs0))dBstTZs0dWs,0tT,(ii)Yt0St,(iii)0T(Ys0Ss)dKs0=0.

For these solutions above, we get some properties as follows:

Lemma 4
Under the assumptions (H.1) – (H.4) and (H.7) – (H.11), we have for any n ≥ 1 and t ∈ [0, T]
Y¯t0Y¯tnY¯tn+1Yt0.
Proof
We will prove Y¯t0Y¯tn at first. By Eqs. (5), and (6), we have
Y¯t1Y¯t0=tTE'(π(s,δY¯s1,δ(Y¯s1)',δZ¯s1)+Λs1)ds+tTE'(g(s,Y¯s1,(Y¯s1)',Z¯s1)g(s,Y¯s0+(Y¯s0)'+Z¯s0))dBs+tT(dK¯s1dK¯s0)tTδZ¯s1dWs,
where Λs1=f(s,Y¯s0,(Y¯s0)',Z¯s0)+C(|Y¯s0|+(Y¯s0)'+|Z¯s0|)+fs . By hypothesis (H.7) we have Λs10 , because (Y¯t0,Z¯t0) is the solution of Eq. (5), we get Λs12(0,T,d) . Therefore, from Lemma 3 we get Y¯t1Y¯t0 . Now we want to prove Y¯tnY¯tn+1 , for any n ≥ 0. We set
{δρsn+1=ρsn+1ρsn,Δψn+1(s,δY¯sn+1,δ(Y¯sn+1)',δZ¯sn+1)=ψ(s,δY¯sn+1+Y¯sn,δ(Y¯sn+1)'+(Y¯sn)',δZ¯sn+1+Z¯sn)ψ(s,Y¯sn,(Y¯sn)',Z¯sn).
Using Eq. (6), we have
δY¯tn+1=tTE'(π(s,δY¯sn+1,δ(Y¯sn+1)',δZ¯sn+1)+θsn+1)dstTδZ¯sn+1dWs+tTE'(Δgn+1(s,δY¯sn+1,δ(Y¯sn+1)',δZ¯sn+1))dBs+tTd(δK¯sn+1),
where θsn+1=Δfn(s,δY¯sn,δ(Y¯sn)',δZ¯sn)π(s,δY¯sn,δ(Y¯sn)',δZ¯sn) and θs0=Λs1 , ∀n ≥ 0. According to it definition, one cas show that θs0 and Δgn+1, ∀n ≥ 0 satisfy all assumption of Lemma 3. Moreover, since K¯tn is a continuous and increasing process, for all n ≥ 0, δK¯sn+1 is a contiuous process of finite variation and, using the same argument as one appear in [2], on can show that
0T(Y¯sn+1Y¯sn)d(δK¯sn+1)=0T(Y¯sn+1Y¯sn)dK¯sn+10T(Y¯sn+1Y¯sn)dK¯sn=0T(Y¯sn+1Y¯sn)dK¯sn+10,
by Lemma 3, we deduce that δY¯tn+10 , i.e. Y¯tn+1Y¯tnt ∈ [0, T], we have
Y¯tn+1Y¯tnY¯t0.
Now we shall prove that Y¯tn+1Yt0n ≥ 0, by Eqs.(3) and (7)
Yt0Y¯tn+1=tTE'(C(|Ys0Y¯s+1|+|(Ys0)'(Y¯sn+1)'|+|Zs0Z¯sn+1|)+Λsn+1)ds+tTE'(g(s,Ys0,(Ys0)'+Zs0)g(s,s,Y¯sn,(Y¯sn)',Z¯sn))dBs+tT(dKs0dK¯sn+1)+tT(Zs0Z¯sn+1)dWs,
where
Λsn+1=C(|Ys0Y¯s+1|+|(Ys0)'(Y¯sn+1)'|+|Zs0Z¯sn+1|+|Ys0|+(Ys0)'+|Zs0|)+fsf(s,Y¯sn,(Y¯sn)',Z¯sn)+π(s,δY¯sn+1,δ(Y¯sn+1)',δZ¯sn+1).
By Lemma 3, we deduce that Yt0Y¯tn+10 , i.e. Yt0Y¯tn+1 , for all t ∈ [0, T]. Thus we have for all n ≥ 0
Yt0Y¯tn+1Y¯tnY¯t0,d¯×dta.s.t[0,T].

The proof of Lemma 4 is complete.

Theorem 5
Let ξ ∈ 𝕃2 (T, ℝ) and t ∈ [0, T]. Under assumption (H.1) – (H.4) and (H.7) – (H.11), the reflected MF-BDSDEs (2) has a minimal solution
(Yt,Zt,Kt)0tT𝒮2(0,T,)×2(0,T,d)×𝒜2.
Proof
From Lemma 4, we know (Y¯tn)n0 is increasing and bounded in ℳ2 (0, T, ℝd). Since |Y˜tn|max(Y˜t0,Yt0)|Y˜t0|+|Yt0| for all t ∈ [0, T], we have
supn𝔼(sup0tT|Y¯tn|2)𝔼(sup0tT|Y¯t0|2)+𝔼(sup0tT|Yt0|2)<,
then according to the Lebesgue's dominated convergence theorem, we deduce that (Y¯tn)n0 converges in 𝒮2 (0, T, ℝ). We denote by Y¯ the limit of (Y¯tn)n0 .
On the other hand from Eq. (6), we deduce that
Y¯0n+1=Y¯Tn+1+0TE'(f(s,Y¯sn,(Y¯sn)',Z¯sn)+π(s,δY¯sn+1,δ(Y¯sn+1)',δZ¯sn+1))ds+tTE'(g(s,Y¯sn+1,(Y¯sn+1)',Z¯sn+1))dBs+tTdK¯sn+1tTZ¯sn+1dWs.
Applying Itô's formula, we obtain
𝔼|Y¯0n+1|2+𝔼0T|Z¯sn+1|2ds𝔼|Y¯Tn+1|2+2𝔼0TY¯sn+1dK¯sn+1+𝔼0T||E'(g(s,Y¯sn+1,(Y¯sn+1)',Z¯sn+1))||2ds+2𝔼0TY¯sn+1E'(f(s,Y¯sn,(Y¯sn)',Z¯sn)+π(s,δY¯sn+1,δ(Y¯sn+1)',δZ¯sn+1))ds.
From assumption (H.7) and Yöung's inequality, we get
2𝔼0TY¯tn+1E'[f(s,Y¯sn,(Y¯sn)',Z¯sn)]ds2𝔼0TY¯sn+1E'[fs(ω)+C(1+|Y¯sn|+|(Y¯sn)'|+|Z¯sn|)]ds,4C2𝔼0T|Y¯sn+1|2ds+(4C2+1)𝔼0T|Y¯sn+1|2ds+𝔼0T|Y¯sn|2ds+16C2𝔼0T|Y¯sn+1|2ds+116𝔼0T|Z¯sn|2ds+𝔼0T|fs(ω)|2ds,=(24C2+1)𝔼0T|Y¯sn+1|2ds+𝔼0T|Y¯sn|2ds+116𝔼0T|Z¯sn|2ds+𝔼0T|fs(ω)|2ds,
and from hypothesis(H.9) we get
2𝔼0TY¯sn+1E'(π(s,δY¯sn+1,δ(Y¯sn+1)',δZ¯sn+1))ds2𝔼0TY¯sn+1E'(C(|δY¯sn+1|+|(δY¯sn+1)'|+|δZ¯sn+1|))ds,4C𝔼0T|Y¯sn+1|2ds+4C2𝔼0T|Y¯sn+1|2ds+𝔼0T|Y¯sn|2ds+8C2𝔼0T|Y¯sn+1|2ds+18𝔼0T|Z¯sn+1|2ds+16C2𝔼0T|Y¯sn+1|2ds+116𝔼0T|Z¯sn|2ds,=(4C+28C2)𝔼0T|Y¯sn+1|2ds+𝔼0T|Y¯sn|2ds+18𝔼0T|Z¯sn+1|2ds+116𝔼0T|Z¯sn|2ds.
Using the two inequalities (8) and (9), we obtain
2𝔼0TY¯sn+1E'(f(s,Y¯sn,(Y¯sn)',Z¯sn)+π(s,δY¯sn+1,δ(Y¯sn+1)',δZ¯sn+1))ds(52C2+4C+1)𝔼0T|Y¯sn+1|2ds+2𝔼0T|Y¯sn|2ds+18𝔼0T(|Z¯sn+1|2+|Z¯sn|2)ds+𝔼0T|fs(ω)|2ds.
Then, we get
𝔼0T|Z¯sn+1|2ds𝔼|ξ|2+𝔼0T||E'(g(s,Y¯sn+1,(Y¯sn+1)',Z¯sn+1))||2ds+C+2𝔼0TY¯sn+1,dK¯sn+1+18𝔼0T(|Z¯sn+1|2+|Z¯sn|2)ds,
where C=2𝔼0T|Y¯sn|ds+(52C+4C+1)0T|Y¯sn+1|2ds+𝔼0T|fs(ω)|2ds .
Applying hypothesis (H. 11), we have
𝔼0T||E'(g(s,Y¯sn+1,(Y¯sn+1)',Z¯sn+1))||2ds4C𝔼0T|Y¯sn+1|2ds+2α𝔼0T|Z¯sn+1|2ds+2𝔼0T||g(s,0,0,0)||2ds.
Using Yöung's inequality, we obtain
2𝔼0TY¯sn+1dK¯sn+12𝔼0TSsdK¯sn+11θ𝔼(sup0tT|St|2)+θ𝔼|K¯Tn+1|2.
Therefore, there exists a constant Cθ depending on α, ξ, C and θ, we derive
𝔼0T|Z¯sn+1|2dsCθ+(18+2α)𝔼0T|Z¯sn+1|2ds+18𝔼0T|Z¯sn|2ds+θ𝔼|K¯Tn+1|2,
where Cθ=C+𝔼|ξ|2+4C0T|Y¯sn+1|2ds+1θ𝔼(sup0tT|St|2)+2𝔼0T||g(s,0,0,0)||2ds .
Chossing α such that 0<18+2α<1 , we obtain
𝔼0T|Z¯sn+1|2dsCθ+18𝔼0T|Z¯sn|2ds+θ𝔼|K¯Tn+1|2.
Moreover, since
K¯Tn+1=Y¯0n+1ξ0TE'(f(s,Y¯sn,(Y¯sn)',Z¯sn)+π(s,δY¯sn+1,δ(Y¯sn+1)',δZ¯sn+1))ds0TE'(g(s,Y¯sn+1,(Y¯sn+1)',Z¯sn+1))dBs+tTZ¯sn+1dWs,
by the Hölder inequality and B-D-G inequality, 𝔼 (X)2 ≤ 𝔼 (X2) and the properties on f, g, π that there exists two constants C1 and C2 depending on α, ξ and C of n such that
𝔼|K¯Tn+1|2C1+C2(𝔼0T|Z¯sn+1|2+|Z¯sn|2ds).
Return to inequality (10), we get
𝔼0T|Z¯sn+1|2dsCθ+θC1+(18+θC2)𝔼0T|Z¯sn|2ds+θC2𝔼0T|Z¯sn+1|2ds,
we chosing θ, such that θC2 ≤ 1, we have
𝔼0T|Z¯sn+1|2dsCθ+θC1+(18+θC2)𝔼0T|Z¯sn|2ds(Cθ+θC1)i=0i=n1(18+θC2)i+(18+θC2)n𝔼0T|Z¯s0|2ds.
Now chossing θ such that 18+θC2<1 and notting 𝔼0T|Z¯s0|2ds< . Obtain
supn𝔼0T|Z¯sn+1|2ds<,
consequently, we deduce
𝔼|K¯Tn+1|2<.
Now we shall prove that (Z¯n,K¯n) is a Cauchy sequence in ℳ2 (0, T, ℝd) × 𝒜2.
Applying Itô's formula to |δY˜sn,m|2=|Y˜snY˜sm|2 , we have
𝔼|Y¯tnY¯tm|2+𝔼0T|Z¯snZ¯sm|2ds=2𝔼0T(Y¯snY¯sm)E'(ΓsnΓsm)ds+20TY¯sn+1(dK¯sndK¯sm)+0T||E'(g(s,Y¯sn,(Y¯sn)',Z¯sn)g(s,Y¯sm,(Y¯sm)',Z¯sm))||2ds.
where Γsn=f(s,Y¯sn1,(Y¯sn1)',Z¯sn1)+π(s,δY¯sn,δ(Y¯sn)',δZ¯sn) . Since 0TY¯sn+1(dK¯sndK¯sm)0 , we obtain
𝔼0T|Z¯snZ¯sm|2ds2𝔼0T(Y¯snY¯sm)E'(ΓsnΓsm)ds+0T||E'(g(s,Y¯sn,(Y¯sn)',Z¯sn)g(s,Y¯sm,(Y¯sm)',Z¯sm))||2ds.
By the Hölder inequality and hypothesis (H.11), we deduce that
(1α)𝔼0T|Z¯snZ¯sm|2ds2𝔼(0T|Y¯snY¯sm|2ds)12𝔼(0T|E'(ΓsnΓsm)|2ds)12+2C𝔼0T|Y¯snY¯sm|2ds.
The boundedness of the sequence (Y¯n,Z¯n,K¯n) , we deduce that the Λ=supn[𝔼0TE'|Γsn|2ds]< , this yields that
(1α)𝔼0T|Z¯snZ¯sm|2ds4Λ𝔼(0T|Y¯snY¯sm|2ds)12+2C𝔼0T|Y¯snY¯sm|2ds,
which yields that (Z¯n)n0 is a Cauchy sequence in ℳ2 (0, T, ℝd). Then there exists Z ∈ ℳ2 (ℝd) such that
𝔼0T|Z¯snZs|2ds0asn.
On the other hand, by Burkhölder-Davis-Gundy inequality, we get
{𝔼sup0tT|tTZ¯sndWstTZsdWs|2𝔼tT|Z¯snZs|2ds0,asn,𝔼sup0tT|tTE'(g(s,Y¯sn,(Y¯sn)',Z¯sn))E'(g(s,Ys,(Ys)',Zs))|22C𝔼0T|Y¯snYs|2ds+α𝔼0T|Z¯snZs|2ds0,asn.
Therefore, from the properieties of f and π
Γsn=f(s,Y¯sn1,(Y¯sn1)',Z¯sn1)+π(s,δY¯sn,δ(Y¯sn)',δZ¯sn)f(s,Ys,(Ys)',Zs),
¯a.s., for all t ∈ [0, T] as n → ∞. Then follows by Lebesgue's dominated convergence theorem that
𝔼0T|E'(Γsnf(s,Ys,(Ys)',Zs))|2ds0,n
Since (Y˜s,Z˜s,Γsn) converges in 𝒮2 (0, T, ℝ) × ℳ2 (0, T, ℝd) × ℳ2 (0, T, ℝ2) and
𝔼(sup0tT|K¯tnK¯tm|2)𝔼|Y¯0nY¯0m|2+𝔼sup0tT|Y¯tnY¯tm|2+𝔼0T|E'(ΓsnΓsm)|2ds+𝔼sup0tT|0tE'(g(s,Y¯sn,(Y¯sn)',Z¯sn)g(s,Y¯sm,(Y¯sm)',Z¯sm))dBs|2+𝔼sup0tT|0t(Z¯snZ¯sm)dWs|2
for any n ≥ 0, we deduce from Bukhölder-Davis-Gundy inequality that
𝔼(sup0tT|K¯tnK¯tm|2)0,
as n → ∞. Consequently, there exists a t–mesurable process K wich value in ℝ such that
𝔼(sup0tT|K¯tnKt|2)0,
as n → ∞. Obviously, K0 = 0 and {Kt; 0 ≤ tT} is a increasing and continuous process. From Eq. (6), we have for all n ≥ 0, Y¯tnSt , ∀t ∈ [0, T], then YtSt, ∀t ∈ [0, T]. On the other hand, from the result of Saisho [8] (in 1987, p. 465), we have
0T(Y¯snSs)dK¯sn0T(YsSs)dKs,

¯a.s., as n → ∞. Using the identite 0T(Y¯snSs)dK¯sn=0 , for all n ≥ 0 we conclude that 0T(YsSs)dKs0 . Letting n → +∞ in Eq. (3), we prove that (Y, Z, K) is solution to Eq. (3). Let (Y*, Z*, K*) be any solution of the MF-RBDSDE (3), we have Y¯nY* , for all n ≥ 0 and therefore, Y. ≤ Y* i.e., Y is the minimal solution.

References

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    Bahlali. K, Hassani. M, Mansouri. B, Mrhardy. N, (2009), One barrier reflected backward doubly stochastic differential equations with continuous generator, Comptes Rendus Mathematique, Volume 347, Issue 19, Pages 1201–1206. http://doi.org/10.1016/j.crma.2009.08.001.

  • [2]

    Chaouchkhouan. N, Labed. Boubakeur & Badreddine. Mansouri, (2014), Mean-field Reflected Backward Doubly Stochastic DE With Continuous Coefficients*, Journal of Numerical Mathematics and Stochastics,. 6. 62–72.

  • [3]

    Jia. G, (2008), A class of backward stochastic differential equations with discontinuous coefficients, Statist. Probab. Lett. 78, 231–237. http://doi.org/10.1016/j.spl.2007.05.028

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    Lin. Q, (2011), Backward doubly stochastic differential equations with weak assumptions on the coefficients, Applied Mathematics and Computation, 217, 9322–9333. https://doi.org/10.1016/j.amc.2011.04.016.

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    Lepeltier. J.P., San Martin. J, (1997), Backward stochastic differential equations with continuous coefficients, Stat. Probab. Lett. 32, 425–430. https://doi.org/10.1016/S0167-7152(96)00103-4

  • [6]

    Pardoux. E, Peng. S, (1990), Adapted solution of a backward stochastic differential equation, Systems Control Lett. 4, 55–61. https://doi.org/10.1016/0167-6911(90)90082-6.

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    Pardoux. E, Peng, S, (1994), Backward doubly stochastic differential equations and systems of quasili near SPDEs, Probability Theory and Related Fields, 98, 209–227. https://doi.org/10.1007/BF01192514.

  • [8]

    Saisho. Y, (1987), Stochastic differential equations for multi-dimensional domain with reflecting boundary, Probability Theory and Related Fields, 74(3), 455–477. https://doi.org/10.1007/BF00699100.

  • [9]

    Shi. Y, Gu. Y, Liu. K, (2005), Comparison theorems of backward doubly stochastic differential equations and applications, Stochastic Analysis and Application, 23, 97–110. https://doi.org/10.1081/SAP-200044444.

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    Xu. R, (2012), Mean-field backward doubly stochastic differential equations and related SPDEs, Boundary Value Problems, 2012(1), 114. https://doi.org/10.1186/1687-2770-2012-114.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Bahlali. K, Hassani. M, Mansouri. B, Mrhardy. N, (2009), One barrier reflected backward doubly stochastic differential equations with continuous generator, Comptes Rendus Mathematique, Volume 347, Issue 19, Pages 1201–1206. http://doi.org/10.1016/j.crma.2009.08.001.

  • [2]

    Chaouchkhouan. N, Labed. Boubakeur & Badreddine. Mansouri, (2014), Mean-field Reflected Backward Doubly Stochastic DE With Continuous Coefficients*, Journal of Numerical Mathematics and Stochastics,. 6. 62–72.

  • [3]

    Jia. G, (2008), A class of backward stochastic differential equations with discontinuous coefficients, Statist. Probab. Lett. 78, 231–237. http://doi.org/10.1016/j.spl.2007.05.028

  • [4]

    Lin. Q, (2011), Backward doubly stochastic differential equations with weak assumptions on the coefficients, Applied Mathematics and Computation, 217, 9322–9333. https://doi.org/10.1016/j.amc.2011.04.016.

  • [5]

    Lepeltier. J.P., San Martin. J, (1997), Backward stochastic differential equations with continuous coefficients, Stat. Probab. Lett. 32, 425–430. https://doi.org/10.1016/S0167-7152(96)00103-4

  • [6]

    Pardoux. E, Peng. S, (1990), Adapted solution of a backward stochastic differential equation, Systems Control Lett. 4, 55–61. https://doi.org/10.1016/0167-6911(90)90082-6.

  • [7]

    Pardoux. E, Peng, S, (1994), Backward doubly stochastic differential equations and systems of quasili near SPDEs, Probability Theory and Related Fields, 98, 209–227. https://doi.org/10.1007/BF01192514.

  • [8]

    Saisho. Y, (1987), Stochastic differential equations for multi-dimensional domain with reflecting boundary, Probability Theory and Related Fields, 74(3), 455–477. https://doi.org/10.1007/BF00699100.

  • [9]

    Shi. Y, Gu. Y, Liu. K, (2005), Comparison theorems of backward doubly stochastic differential equations and applications, Stochastic Analysis and Application, 23, 97–110. https://doi.org/10.1081/SAP-200044444.

  • [10]

    Xu. R, (2012), Mean-field backward doubly stochastic differential equations and related SPDEs, Boundary Value Problems, 2012(1), 114. https://doi.org/10.1186/1687-2770-2012-114.

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