Zhi Li,Liping Xu

(School of Information and Math.,Yangtze University,Jingzhou 434023,Hubei, E-mail:lizhi_csu@126.com(Z.Li))

Jiaowan Luo

(School of Math.and Information Sciences,Guangzhou University,Guangzhou 510006)

LpSOLUTIONS TO BSDES WITH MONOTONIC AND UNIFORMLY CONTINUOUS GENERATORS∗†

Zhi Li,Liping Xu

(School of Information and Math.,Yangtze University,Jingzhou 434023,Hubei, E-mail:lizhi_csu@126.com(Z.Li))

Jiaowan Luo

(School of Math.and Information Sciences,Guangzhou University,Guangzhou 510006)

In this paper,we deal with Lp(p＞1)solutions to one dimensional backward stochastic differential equations(BSDEs)with discontinuous(left or right continuous) generators.We obtain an existence theorem of Lpsolutions to BSDEs whose generators are discontinuous,monotonic in y and uniformly continuous in z.

backward stochastic differential equation;existence theorem;Lpsolution;uniformly continuous condition

2000 Mathematics Subject Classification 60H15;60G15;60H05

Ann.of Appl.Math.

31:2(2015),175-181

1　Introduction

In this paper,we consider the following one-dimensional backward stochastic differential equations(BSDEs for short):

where T＞0 is a finite constant,ξ is a random variable called the terminal condition,the random function f:Ω×[0,T]×R×Rd→R is progressively measurable for each(y,z),called to be the generator of the BSDEs(1.1),and W is a d-dimensional Brownian motion.The solution(y.,z.)is a pair of adapted processes.The triple(f,ξ,T)is called the parameters of the BSDEs(1.1).

Such equations,in the nonlinear case,were firstly introduced in Pardoux and Peng[14], where the authors established an existence and uniqueness result for L2solutions to multidimensional BSDEs with square integrable parameters under the Lipschitzian assumption on the generator f.Since then,much attention has been paid to relaxing the Lipschitz hypothesis on the generator.For instance,Mao[12]established an existence and uniqueness result for L2solutions to the BSDEs(1.1)under non-Lipschitz condition;Lepeltier and San Martin[11]proved the existence of L2solutions to the BSDEs(1.1)when f is continuous and of linear growth in(y,z)and(f(t,0,0))t∈[0,T]is a bounded process;Fan and Jiang[7]extended it to the case that(f(t,0,0))t∈[0,T]is a square integral process;Briand and Hu[2] gave an existence result for L2solutions to the BSDEs(1.1)with quadratic growth in z and Jia[9]obtained the uniqueness of L2solutions to the BSDEs(1.1)where f does not depend on y and is uniformly continuous in z and(f(t,0))t∈[0,T]is a bounded process.

On the other hand,El Karoui et al.[6]first proved the existence and uniqueness for the Lp(p＞1)solution to the BSDEs(1.1)with p-integrable parameters under the Lipschitzian assumption on f.The Lipschitzian assumption on f in this result has also been relaxed by some researchers in recent years.For instance,Briand et al.[1]established an existence and uniqueness result to the BSDEs(1.1)with p-integrable parameters where f is monotonic in y and Lipschitzian in z.Briand et al.[3]obtained an existence result for the Lp(p＞1) solution to the BSDEs(1.1)with p-integrable parameters where f is monotonic in y and has a linear growth in z.Under the conditions that ξ is p-integrable and(f(t,0,0))t∈[0,T]is a bounded process,Chen[5]proved the existence of the Lp(1＜p≤2)solution to the BSDEs(1.1)with continuous and linear growth generators,and established the existence and uniqueness of the Lp(1＜p≤2)solution to the BSDEs(1.1)where f satisfies the Osgood condition in y and the uniformly continuous condition in z.Ma et al.[13]obtained the existence and uniqueness of the Lp(1＜p≤2)solution to the BSDEs(1.1)where f is monotonic in y and the uniformly continuous in z.

Motivated by these results,in this paper,we explore Lp(p＞1)solutions to the BSDEs (1.1)with discontinuous generators.We devote to an existence theorem of Lp(p＞ 1) solutions to the BSDEs(1.1)whose generators satisfy a kind of discontinuous(left continuous or right continuous)conditions in y,and is uniformly continuous in z.

The rest of this paper is organized as follows.In Section 2,we present some necessary preliminaries and lemmas.In Section 3,we state and prove our main result.

2　Preliminaries

Let(Ω,F,P)be a complete probability space of carrying d-dimensional standard Brownian motion(Wt)t≥0.Let{Ft}t≥0denote the natural filtration generated by(Wt)t≥0,augmented by P-null sets of F.For any integer n,if z∈Rn,let|z|denote its Euclid norm.

In what follows,fix two positive numbers T＞0 and p＞1.Let Spbe the set of all continuous and adapted processes θ=(θt)t∈[0,T]with values in R such that

And we denote Mpbe the set of all{Ft}-progressively measurably processes θ=(θt)t∈[0,T]with values in Rdsuch that

We shall use the following assumptions:

(H2)The function f(t,y,z)is left continuous and non-decreasing in y.

(H3)There exists a constant M＞0 such that for all(t,y,z)∈[0,T]×R×Rd,|f(t,y,z)|≤M.

(H4)f is uniformly continuous in z uniformly with respect to(ω,t,y),that is,there existsa continuous,non-decreasing function φ(·)from R+to itself with linear growth,which satisfies φ(0)=0 such that dP×dt-a.e.,

Here and henceforth we denote the constant of linear growth for φ by A,that is,0≤φ(x)≤A(x+1)for all x∈R+.

(H5)f(t,y,z)is Lipschitzian continuous,that is for all(t,yi,zi)∈[0,T]×R×Rd, i=1,2,

Next we list some useful lemmas from[8].

Lemma 2.1[8]Let the generator f and terminal condition ξ satisfy assumptions(H1) and(H5).Then,the BSDEs(1.1)has a unique solution(y.,z.)in Sp×Mp.

Lemma 2.2[8]Let f and f′be two generators of the BSDEs(1.1),assume that(y.,z.) and(y′.,z′.)are,respectively,Lpsolutions to the BSDEs(1.1)with paramaters(f,ξ,T)and (f′,ξ′,T).If ξ≤ξ′P-a.s.,f(resp.f′)satisfies(H1)and(H5)and(resp.f(t,yt,zt)≤f′(t,yt,zt))P-a.s.,then we have P-a.s.,for any t∈[0,T],

3　Main Result

In this section,we get the existence of a Lp(p＞1)solution to the BSDEs(1.1)such that the coefficient f satisfies(H1)-(H4).The result is obtained via the approximation of the generator by increasing sequences of Lipschitzian function.

We shall use the following approximation lemma to construct the minimal Lp(p＞1) solution to the BSDEs(1.1).

Lemma 3.1[4]Let b:[0,T]×R→R be a bounded measurable function such that for all t∈[0,T],b(t,·)is a non-decreasing and left continuous function.Then there exists a family of measurable functions

such that

(i)for any sequence xnincreasing to x∈R,we have,a.s.;

(ii)x→bn(t,x)is non-decreasing,for all n≥1,t∈[0,T];

(iii)n→bn(t,x)is non-decreasing,for all x∈R,t∈[0,T];

(iv)|bn(t,x)-bn(t,y)|≤2nM|x-y|;

(vi)for any t∈[0,T],n≥1,bn(t,x)≤b(t,x).

Corollary 3.1[10]Let b:[0,T]×R→R be a bounded measurable function such that for all t∈[0,T],b(t,·)is a non-decreasing and right continuous function.Then there exists a family of measurable functions

such that

(i)for any sequence xndecreasing to x∈R,we have,a.s.;

(ii)x→bn(t,x)is non-decreasing,for all n≥1,t∈[0,T];

(iii)n→bn(t,x)is non-increasing,for all x∈R,t∈[0,T];

(iv)|bn(t,x)-bn(t,y)|≤2nM|x-y|;

(vi)for any t∈[0,T],n≥1,bn(t,x)≥b(t,x).

Now let us state and prove the main result of this section.

Theorem 3.1Assume that(H1)-(H4)hold.Then,the BSDEs(1.1)has a solution (y.,z.)in Sp×Mp.Moreover,(y.,z.)is the minimal solution to the BSDEs(1.1),that is, for any other solution(y′,z′)∈Sp×Mpto the BSDEs(1.1),we have y≤y′.

Proof We construct the sequence(fn)n≥1with the function f(t,y,z)associated by Lemma 3.1 for all(t,z)∈[0,T]×Rd.It is easy to check that the sequence(fn)n≥1satisfies the following properties:

(i)for any sequence ynincreasing to y∈R,we havefor any z∈Rd,a.s.;

(ii)y→fn(t,y,z)is non-decreasing,for all n≥1,t∈[0,T]and z∈Rd;

(iii)n→fn(t,y,z)is non-decreasing,for all y∈R,t∈[0,T]and z∈Rd;

(iv)|fn(t,x,z)-fn(t,y,z)|≤2nM|x-y|for any x,y∈R and z∈Rd;

(v)|fn(t,y,z1)-fn(t,y,z2)|≤φ(|z1-z2|)for any y∈R and z1,z2∈Rd;

(vii)for any t∈[0,T],y∈R,z∈Rdand n≥1,fn(t,y,z)≤f(t,y,z).

By virtue of Lemma 2.1,let the processesand the sequence processes (yn,zn)n≥1respectively be the Lp(p＞1)solution to the following BSDE:for all t∈[0,T],

It follows from Lemma 2.2 that for all n≥1,,a.s.,for all t∈[0,T],it follows thatconverges to a limit y in Spand

On the other hand,applying Itˆo’s formula to,we can obtain

In view of the properties(vi)of fn,choosing a suitable constant K,we deduce that

But by the B-D-G’s inequality,we get

According to(3.5)and(3.6),for all n≥1 we obtain

In fact,for any n,m≥1,applying Itˆo’s formula to,we can get

Then,using the standard estimate method in Lemma 3.1 of Briand et al[1],we can obtain

P-a.s.,for all t∈[0,T]as n→∞.Then,it follows by the dominated convergence theorem that

as n→∞.On the other hand,by B-D-G’s inequality,

Choosing a subsequence if necessary,we can get that for almost all ω,

Hence,(y,z)∈Sp×Mpis a solution to the BSDEs(1.1).

Let(y′,z′)∈Sp×Mpbe any solution to the BSDEs(1.1).By virtue of Lemma 2.2, we have yn≤y′,for all n≥1,and therefore,y≤y′that is,y is the minimal solution.The proof is completed.

Remark 3.1 Similar to the proof of Theorem 3.1,we can prove that the BSDEs(1.1) has a maximal solution using Corollary 3.1 when the coefficient f is right continuous,nondecreasing and bounded.

Remark 3.2 We consider the following BSDE:

where sgn{ys}=1,if y＞0,otherwise sgn{ys}=-1.We can check that the above equation satisfies(H1)-(H4).By Theorem 3.1,the above BSDE has a minimal solution.

References

[1]Ph.Briand,B.Delyon,Y.Hu,E.Pardoux and L.Stoica,Lpsolutions of backward stochastic differential equations,Stochastic Process.Appl.,108(2003),109-129.

[2]Ph.Briand and Y.Hu,BSDE with quadratic growth and unbounded terminal value,Probab. Theory Relat.Fields.,136:4(2006),604-618.

[3]Ph.Briand,J.P.Lepeltier and J.San Martin,One-dimensional backward stochastic differential equations whose coefficient is monotonic in y and non-Lipschitz in z,Bernoulli.,3:1(2007),80-91.

[4]B.Boufoussi and Y.Ouknine,On a SDE driven by a fractional Brownian motion and with monotone drift,Electron.Comm.Probab.,8(2003),122-134.

[5]S.Chen,Lpsolutions of one-dimensional backward stochastic differential equations with continuous coefficients,Stoch.Anal.Appl.,28(2010),820-841.

[6]N.El Karoui,S.Peng,and M.C.Quenez,Backward stochastic differential equations in finance, Math.Finance.,7:1(1997),1-71.

[7]S.Fan,L.Jiang,Existence and uniqueness result for a backward stochastic differential equation whose generator is Lipschitz in y and uniformly continuous in z,J.Appl.Math.Comput., 36(2011),1-10.

[8]S.Fan,L.Jiang,Lp(p＞1)solutions for one-dimensional BSDEs with linear-growth generators,J.Appl.Math.Comput.,38(2012),295-304.

[9]G.Jia,A uniqueness theorem for the solution of backward stochastic differential equations, C.R.Acad.Sci.Pair,Ser I.,346(2008),439-444.

[10]Z.Li and J.W.Luo,One barrier reflected backward doubly stochastic differential equations with discontinuous monotone coefficients,Statist.Probab.Lett.,82(2012),1841-1848.

[11]J.P.Leprltier and J.San Martin,Backward stochastic differential equations with continuous coefficients,Statist.Probab.Lett.,32:4(1997),425-430.

[12]X.Mao,Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients,Stochastic Process.Appl.,58(1995),1525-1534.

[13]M.Ma,S.Fan and X.Song,Lp(p＞1)solutions of backward stochastic differential equations with monotonic and uniformly continuous generators,Bull.Sci.Math.,137(2013),97-106.

[14]E.Pardoux and S.Peng,Adapted solution of a backward stochastic differential equations, Systems Control Letter,14(1990),55-61.

(edited by Liangwei Huang)

∗This research was partially supported by the NNSF of China(No.11271093),the Science Research Project of Hubei Provincial Department of Education(No.Q20141306)and the Cultivation Project of Yangtze University for the NSF of China(No.2013cjp09).

†Manuscript October 15,2013