Lp Solutions for Multidimensional BDSDEs with Locally Weak Monotonicity Coefficients*

Dejian TIAN Runyu ZHU

Abstract In this paper， the authors establish the existence and uniqueness theorem of Lp (1 ＜p ≤2) solutions for multidimensional backward doubly stochastic differential equations (BDSDEs for short)under the p-order globally (locally) weak monotonicity conditions.Comparison theorem of Lp solutions for one-dimensional BDSDEs is also proved.These conclusions unify and generalize some known results.

Keywords Backward doubly stochastic differential equation， Locally monotonicity condition， Lp solution

1 Introduction

The following backward stochastic differential equation (BSDE for short) was first introduced by Pardoux and Peng [13]:

The existence and uniqueness result of L2solutions was proved under the Lipschitz condition.Since then， BSDEs have been developed rapidly and connected with other related fields， such as stochastic control and partial differential equations etc.Researchers have obtained several results under the weaker conditions on coefficients， such as [4–9， 15， 17， 19] etc.

For studying a probabilistic representation of certain quasilinear stochastic partial differential equations (SPDEs for short)， Pardoux and Peng [14] first proposed the BDSDE and got the existence and uniqueness result of L2solutions under the Lipschitz condition.The BDSDE is given as bellow:

where{Bt}t≥0and {Wt}t≥0are mutually independent standard Brownian motions with values in Rland Rdrespectively.The integral with respect to W·is a standard forward It?o integral，while the integral with respect to B·is a backward one.These two kinds of integrals are special cases of It?o-Skorohod integral.For each (y，z)∈Rk×Rk×d， coefficients

are jointly measurable.BDSDE (f，g，T，ξ) denotes the BDSDE with parameters (f，g，T，ξ).The following lemma comes from Section 3 in Bihari [3] and definitions will be used in the remaining of the paper.

Lemma 1.1?0 ≤t ≤T，suppose thatD(t)andG(t)are continuous and positive functions.?u ≥0，q(u)is a non-negative and non-decreasing continuous function.Letδ ≥0，M ≥0.If

and?u0＞0， denoteQ(u)=then

whereQ-1(·)is the inverse function ofQ(·)， and the boundary furnished by(1.2)is independent ofu0.

Definition 1.1A solution for BDSDE(1.1)is an(Ft)-measurable process(Yt，Zt)t∈[0，T]with values inRk× Rk×dsuch thatdP-a.s.，t →Ytcontinuous，t →Zt∈L2(0，T)，t →f(t，Yt，Zt) ∈L1(0，T)，t →g(t，Yt，Zt) ∈L2(0，T).And for allt ∈[0，T]， the solution satisfies BDSDE(1.1).

Definition 1.2Let(Yt，Zt)t∈[0，T]be a solution for BDSDE(1.1)andp ＞1，and(Yt，Zt)t∈[0，T]∈Sp×Mp.Then(Yt，Zt)t∈[0，T]is anLpsolution for BDSDE(1.1).

There have been various extensions of the BDSDEs to non-Lipschitz condition on coefficients or to Lp(1 ＜p ≤2) solutions， and we refer to some references， Shi， Gu and Liu [16] for linear growth condition， Lin [10] and Lin and Wu [11] for left-Lipschitz or uniformly continuous conditions， Owo [12] for stochastic Lipschitz condition.It is important for studying BDSDEs with weaker conditions，because BDSDEs have the closely connection to the theory of stochastic partial differential equations (SPDEs for short).For the relationship between BDSDEs and SPDEs， the readers can refer to [1–2， 14， 18， 20–21] etc.

Here， we would like to mention the following several results on multidimensional BDSDEs，which is related closely to our result.First of all， Wu and Zhang [20] investigated BDSDEs with locally monotone coefficients by adding the assumption that f satisfies γ-growth condition in (y，z).Second， Zong and Hu [24] considered the Lpsolution to BDSDEs for the monotone coefficient f with linear growth condition on z in the infinite time horizon.Furthermore， in BSDEs， Fan [6] studied the existence and uniqueness of Lp(p ＞1) solutions under the p-order weak monotonicity conditions.

Motivated by Wu and Zhang [20] and Fan [6]， this paper is devoted to the results of them to the BDSDEs with p-order globally (locally) weak monotonicity condition.We will prove the existence and uniqueness result of BDSDEs in this weak assumption.The methods in this paper should be explained in two aspects: (i) Compared with BSDEs’ situation， BDSDEs have two Brownian motions with two kinds of stochastic integrals.Conditional mathematical expectation can not make the integrals term w.r.t Brownian motions disappear simultaneously，and some standard approaches to deal with classical BSDEs can not be adapted effortless to the framework of BDSDEs， such as stopping time method.(ii)The existence and uniqueness result for solutions in p-order locally weak monotonicity situation is proved by the similar methods of Wu and Zhang [20] with a additional assumption ρN(x) := μN(yùn)·x+ρ(x).This assumption combines locally monotone and weak monotonicity conditions.

We present two main results in this paper.Theorem 4.1 deals with the existence and uniqueness result of Lpsolutions for multidimensional BDSDEs under the p-order globally weak monotonicity conditions.A priori estimate and a truncation method are integrated together to derive the results.Theorem 5.1 investigates BDSDEs with p-order locally weak monotonicity conditions.Comparison theorem of Lpsolutions for one-dimensional BDSDEs is also put forward and proved.As a byproduct of Theorem 5.1， Remark 5.1 extends the results of Zhu and Tian [22–23].

This paper is organized as follows.Some preliminaries are introduced in Section 2.We establish two priori estimates in Section 3.In Section 4， we give the existence and uniqueness theorem of Lp(1 ＜p ＜2) solutions for BDSDEs with p-order globally weak monotonicity coefficients.In Section 5， we present the existence and uniqueness theorem of Lp(1 ＜p ≤2)solutions for BDSDEs with p-order locally weak monotonicity coefficients.In Section 6，we give the comparison theorem.

2 Preliminaries

Suppose that k and d are two positive integers.|y| denotes the Euclidean norm of a vector y ∈Rk.〈x，y〉 denotes the inner product of vectors x，y ∈Rk.For a k×d matrix z， |z| :=where zTis the transpose of z.S denotes the set of all nondecreasing and concave functions ρ(·):R+→R+with ρ(0)=0; ?x ＞0， ρ(x)＞0 and

Let (Ω，F(xiàn)，P) be a complete probability space.(Bt)t≥0and (Wt)t≥0are two mutually independent Brownian motions in this space.Suppose that N is the class of P-null sets of F.?t ∈[0，T]，F(xiàn)or a process η，

Especially， {Ft，0 ≤t ≤T} is neither increasing nor decreasing， and it can not be a filtration.In this paper， a given real number T ＞0 is the terminal time; random vector ξ ∈Rkis FTmeasurable; f，g are (Ft)-measurable.Lp(FT;Rk) (or Lp) denotes the set of all Rk-valued，F(xiàn)T-measurable random vectors ξ such that E[|ξ|p] ＜∞.Sp(0，T;Rk) (or Sp) denotes the set of all Rk-valued， (Ft)-adapted and continuous processes (Yt)t∈[0，T]such that

Mp(0，T;Rk×d) (or Mp) denotes the set of all Rk×d-valued， (Ft)-progressively measurable processes (Zt)t∈[0，T]such that

In the sequel， we introduce the following hypotheses.

(H1) (g(t，0，0))t∈[0，T]∈M2.And there exist constants K ＞0 and 0 ＜α ＜1 such that for all (yi，zi)∈Rk×Rk×d， i=1，2， dP ×dt-a.e.，

(H2) For any given (ω，t)， f(ω，t，·，·) is continuous.

(H3)p

(H4) f satisfies γ-growth condition in (y，z)， i.e.， ?K ＞0， γ ∈[0，1)， such that ?(y，z) ∈Rk×Rk×d，

Remark 2.1According to inequality|x|γ≤1+|x|，γ ∈[0，1)，(H4)implies that|f(t，y，z)|≤K(3+|y|+|z|).

3 Priori Estimates

In this section， we establish two useful priori estimates.We first introduce the following assumptions.

(A1) dP ×dt-a.e.， ?(y，z)∈Rk×Rk×d，

where θ and λ are two non-negative constants， (ft)t∈[0，T]and (υt)t∈[0，T]are two non-negative，(Ft)-progressively measurable processes with

(A2) dP ×dt-a.e.， ?(y，z)∈Rk×Rk×d，

where λ is a non-negative constant， ?(·) ∈S， (ft)t∈[0，T]is a non-negative (Ft)-progressively measurable process and satisfies

In the same way and steps of Proposition 3.1 in[23]，it is not difficult to obtain the following proposition.

Proposition 3.1Suppose that1 ＜p ≤2， and(A1)，(H1)and(H3)phold.Let(Y，Z)be a solution of BDSDE(1.1)such thatY ∈Sp.ThenZ ∈Mp.Moreover，?t ∈[0，T]，

wheredθ，λ，K，p，T，αis a non-negative constant depending on(θ，λ，K，p，T，α)， dα，pis another non-negative constant depending on(α，p).

The proof of the following estimate is similar to that of Proposition 3.2 in [23]， so we omit its partial proof.

Proposition 3.2Suppose that1 ＜p ≤2， and(A2)，(H1)and(H3)phold.Let(Y，Z)be anLpsolution of BDSDE(1.1).Then there exists a non-negative constantdp，λ，α，K，Tdepending on(p，λ，α，K，T)such that?t ∈[0，T]，

ProofAccording to the proof of Proposition 3.2 in [23]， we can easily get

where constant hp，K，αdepends on (p，K，α)and

Applying Gronwall’s inequality in the previous inequality， the proof of Proposition 3.2 can be completed.

4 BDSDEs with p-Order Globally Weak Monotonicity Coefficients

In this section， we study the BDSDEs with p-order globally weak monotonicity coefficients.The following assumptions will be used.

(H5)f is Lipschitz continuous in z，i.e.，there exists a constant K ＞0 such that dP×dt-a.e.，?y ∈Rk，z1，z2∈Rk×d，

(H6)pf satisfies p-order globally weak monotonicity condition in y， i.e.， there exists a function ρ(·)∈S s.t.， dP ×dt-a.e.， ?y1，y2∈Rk， z ∈Rk×d，

Remark 4.1According to [6， Proposition 1]， we can know that for any 1 ＜p ≤2， (H6)p?(H6)2.

Under the assumptions (H1)–(H2)， (H3)2， (H4)–(H5) and (H6)2， Zhu and Tian [22] showed that BDSDE has a unique L2solution.The following theorem generalizes the result to Lpsituation.

Theorem 4.1Let1 ＜p ＜2.under the conditions(H1)–(H2)，(H3)p，(H4)–(H5)and(H6)p， BDSDE(1.1)has a uniqueLpsolution.

ProofWe divide the proof into two steps.

Step 1Let 1 ＜p ＜2.Under the conditions (H1)–(H2)， (H3)p， (H4)–(H5) and (H6)p， we prove the existence of Lpsolutions for BDSDE (1.1).

For any n ≥1， x ∈Rk， let qn(x):=

Obviously， ξn，fnsatisfy (H2)， (H3)2， (H4)–(H5) and (H6)p.Furthermore， by Remark 4.1，it implies that fnalso satisfies (H6)2.Then by [22， Theorem 3.2]， for each n ≥1， BDSDE(ξn，fn，g，T) has a unique L2solution (Yn，Zn).Therefore， (Yn，Zn)∈Sp×Mp.

On the other hand， by the definitions of ξnand fn(t，0，0) and the assumption (H3)p， we know that

In the sequel，we will prove thatis a Cauchy sequence in Sp×Mp.For any integrals n，m ≥1， let (Ymn，Zmn) be a solution of the following BDSDE:

Assumptions (H5) and (H6)pyield that

Proposition 3.2 yields that there exists a constant C ＞0 independent of n，m such that

Noting that gmn(s，0，0)= 0.Using (4.1)， the fact that ρ(·) is of linear growth and Gronwall’s inequality yields

Thus，by taking limsup in(4.3)with respect to m and n，and by virtue of(4.1)，F(xiàn)atou’s lemma，the continuity and monotonicity of ρ(·) and Bihari’s inequality， we obtain

On the other hand， fnsatisfies (H6)2and ρ(·) is of linear growth.Then for any k ≥1， we have

Here， we use the estimate (see [6]): ρ(x) ≤(k +2A)x+for any x ≥0 and k ≥1，where A is the constant such that ρ(x)≤A(1+x)， ?x ≥0.

By Proposition 3.1， there exist constants Ck，K，p，T，αand Cα，pdepending on (k，K，p，T，α)and (α，p) respectively such that

In the previous inequality， first letting n，m →∞and then k →∞， it implies

Step 2We prove the uniqueness.Let (Yi，Zi) be Lpsolutions of BDSDE (1.1)， i = 1，2.(H5) and (H6)pimply that

According to Proposition 3.2， there exists a constant Cp，K，α，Tdepending on (p，K，α，T) such that

We get Y1=Y2by Bihari’s inequality.

On the other hand， f satisfies (H6)2， then

Proposition 3.1 yields that there exist constants Cα，pand Cp，K，α，T，kdepending on (α，p) and(p，K，α，T，k) respectively such that

Letting k →∞in previous inequality， we obtain the uniqueness of the solutions.

Now， we give an example of BDSDEs which satisfies the assumptions in Theorem 4.1.

Example 4.1Assume that k = 1， 1 ＜p ≤2， ξ ∈Lp， N ∈N.Let g(t，y，z) := y+0.5z，f(t，y，z):=sin|z|+h(|y|)， where

Obviously， f，g satisfy (H1)–(H2) and (H3)p.?y，z， |f(t，y，z)| ≤3， then f satisfies 0-growth condition in (y，z); |f(t，y，z1)-f(t，y，z2)|≤|z1-z2|， then f satisfies Lipschitz condition in z;

and h(·)∈S.Then f satisfies p-order globally weak monotonicity condition in y.Theorem 4.1 yields that BDSDE (1.1) with the above f，g，ξ has a unique Lpsolution.

5 BDSDEs with p-Order Locally Weak Monotonicity Coefficients

In this section， we will extend the globally weak monotonicity condition to locally weak monotonicity condition.Let g(t，0，0) ≡0 for all t ∈[0，T] throughout this section.We first introduce the following assumptions.

(H5′) f satisfies locally Lipschitz condition in z， i.e.， ?N ∈N， ?LN＞0 such that for any y，z1，z2with |z1|，|z2|，|y|≤N，

(H6′)pf satisfies p-order locally weak monotonicity condition in y，i.e.，for any N ∈N，there exist functions ρN(·) such that for any z， y1， y2with |z|，|y1|，|y2|≤N，

where ?x ≥0， ρN(x):=μN(yùn)·x+ρ(x) with μN(yùn)∈R， ρ(·)∈S.

The following lemma can be proved in a similar way as [20， Lemma 3.3]， so we omit its proof.

Lemma 5.1?1 ＜p ≤2， under the conditions(H2)，(H3)p，(H4)，(H5′)and(H6′)p， there exists a sequencesuch that

(i)for any givenn，ω，t，fn(t，·，·)continuous;

(ii) ?n，|fn(t，y，z)|≤|f(t，y，z)|≤K(1+|y|γ+|z|γ);

(iii) ?N，n →∞， then(fn-f)→0， where

(iv) ?n，fnsatisfiesp-order globally weak monotonicity condition iny;moreover， for anyn，Nwithn ≥N，

whereρN(·)is the same as the concave function in(H6′)pandy1，y2，zsatisfy|y1|，|y2|，|z|≤N;

(v) ?n，fnsatisfies globally Lipschitz condition inz;moreover， for anyn，Nwithn ≥N，

whereLN＞0andy，z1，z2satisfy|y|，|z1|，|z2|≤N.

The following two estimates are useful for dealing with the locally weak monotonicity coefficients.

Proposition 5.1Let1 ＜p ≤2.Letfi，gsatisfy(H1)–(H2)，(H3)p，(H4)，(H5′)，(H6′)p.Suppose that(Yi，Zi)isLpsolution ofBDSDE (ξ，fi，g，T)，i = 1，2.Then there exist nonnegative constantsd1，d2independent ofNsuch that

where

with(x)=ρ(x)+x， x ≥0and0 ＜ε ＜1-α.

ProofNote that |fi(t，y，z)| ≤K(3+|y|+|z|) and g(t，0，0) ≡0.By the similar methods in Propositions 3.1–3.2， we can obtain that there exists a non-negative constant Cp，K，α，Tdepending on (p，K，α，T) such that

Set

It?o’s formula yields that

where

Applying H¨older’s inequality and Young’s inequality deduces

where C ＞0 depends on p，K，α，T，γ，E[|ξ|p].The above inequality comes from

and (5.2)， where CK，pand Cp，γare two positive constants related with (K，p) and (p，γ)respectively.By (H6′)p， we obtain

From (H5′)， there exists a constant ε ＞0 such that

In view of H¨older’s inequality and Young’s inequality again， it implies

Adding up the last five inequalities， we finally obtain that

where

Therefore，by the method in Proposition 3.2，we obtain that there exists a non-negative constant Cα，p，ε，Kdepending on (α，p，ε，K) such that

Proposition 5.2Let1 ＜p ≤2.Letfi，gsatisfy(H1)–(H2)，(H3)p，(H4)，(H5′)，(H6′)p.Suppose that(Yi，Zi)isLpsolutions of BDSDE(ξ，fi，g，T)，i=1，2.Then for anyk ≥1， there exists a constantA ＞0such that

where non-negative constantd3is independent ofNandk，d4is independent ofNbut depends onk.

ProofLetbe an Lpsolution of (5.3).SetThen (5.2) still holds.It?o’s formula and (H1) yield

(H6′)2holds for f1according to Remark 4.1.By the method similar to Proposition 5.1， we have

Note that ρ(·) is of linear growth.Then for any k ≥1， there exists a constant A ＞0 such that

where Cα，p，K，T，ε，k＞0 depends on (α，p，K，T，ε，k).

In the sequel， motivated by Wu and Zhang [20， Theorem 3.2]， we prove the following existence and uniqueness theorem by assuming a technical assumption (5.6).

Theorem 5.1Let1 ＜p ≤2，(H1)–(H2)，(H3)p，(H4)，(H5′)，(H6′)phold and

whereQ(u)=u0＞0，u ≥0withd1，d2，ρ(·)andΠ2in(5.1).Then BDSDE(1.1)has a unique solution inSp×Mp.

ProofLet (Yi，Zi)， i = 1，2 are two solutions of BDSDE (1.1).By Lemma 1.1 and (5.1)，we get

where C1＞0 is independent of N and d1，d2，Π2，Q(·) are defined in (5.6).Letting N →∞in previous inequality yields Y1=Y2.Furthermore，can be obtained by (5.4).Letting first N →∞and then k →∞in above inequality yields Z1=Z2.The proof of uniqueness is completed.

In the sequel， we prove the existence.Let {fn}n≥1be the approximation sequence in Lemma 5.1.Then for every n， fnsatisfies p-order globally weak monotonicity condition in y and globally Lipschitz condition in z.From Theorem 4.1，we have that BDSDE(ξ，fn，g，T)has an Lpsolution (Yn，Zn).For any n ≥1， fnsatisfies (iv) and (v) in Lemma 5.1.According to Proposition 5.1 and Lemma 1.1， we can know that there exists a constant C2＞0 independent of m，n，N such that

where d1，d2，Q(·)，Π2are defined in(5.6).First letting m，n →∞and then N →∞in previous inequality， by virtue of (5.6)， we obtain

Furthermore， combined with (5.4)， we have

Letting m，n →∞， N →∞， k →∞r(nóng)espectively， we get

In the sequel， we only need to prove that in Lp(Ω)， as n →∞，

where

with a constant C ＞0 independent of n，N.Note thatis a Cauchy sequence in Sp×Mp， then，dP ×dt-a.s.As n →∞， the continuity of f and Lebesgue’s dominated convergence theorem yield

Hence， letting n，N →∞in (5.7)， it yields that

Therefore，the existence of Lpsolutions can be obtained by taking limit in BDSDE(ξ，fn，g，T).

Remark 5.1In Theorem 5.1，ρN(x)=μN(yùn)·x+ρ(x)， x ≥0， 1 ＜p ≤2.Assume that(H1)–(H2)， (H3)p， (H4)， (H5′)， (H6′)phold.The existence and uniqueness result of Lpsolutions in the following two special cases are corollaries of Theorem 5.1.

Case 1ρN(·)=ρ(·)， μN(yùn)=0， LN=K.

In this case， BDSDE (f，g，T，ξ) satisfies (H1)–(H2)， (H3)p， (H4)–(H5) and (H6)p.Assumption (5.6) is reduced to

where Q1(u)=(5.6) holds.

This case is reduced to Theorem 4.1， and it generalizes the result of Zhu and Tian [22] to Lpsituation.

Case 2ρN(x)=μN(yùn)·x， ?x ＞0; ρ(·)=0.

Assumption (5.6) is

where Q2(u)=Then

Then (5.6) becomes

In this situation， we can choose d2×Π2=in Proposition 5.1.Taking， we equivalently have that

In fact， our condition is slightly weaker than [23， Theorem 5.1] since we only focus on Y in Proposition 5.1.

6 Comparison Theorem

We establish the comparison theorem of Lpsolutions for one-dimensional BDSDE (1.1).

Theorem 6.1Let1 ＜p ＜2，ξ1≤ξ2， and(Yi，Zi)beLpsolutions forBDSDE(ξi，fi，g，T)，i=1，2.Assume that one of the following conditions holds:

(i) f1satisfies(H5)，(H6)p， andf1(t，Y2t ，)≤f2(t，Y2t ，)，dP ×dt-a.e.;

(ii) f2satisfies(H5)，(H6)p， andf1(t，Y1t ，)≤f2(t，Y1t ，)，dP ×dt-a.e.

Then， for anyt ∈[0，T]，Y1t≤Y2t，dP-a.s.

ProofThe comparison theorems under the above two conditions are proved in a same way，so we only give the proof under (i).SetUsing It?o’s formula and Tanaka’s formula yield

Note that f1satisfies (H5) and (H6)p.Then

we have

By using the concavity of ρ(·) and Jensen’s inequality， we obtain that there exists a constant Cp，K，αdepending on (p，K，α) such that

AcknowledgementThe authors would like to thank the editors and the anonymous reviewers for their insightful comments and valuable suggestions， which have helped them to improve the paper.