BSDEs with monotone generator driven by time changed Lévy noises

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R E S E A R C H

Open Access

BSDEs with monotone generator driven

by time-changed Lévy noises

Xiaohui Shen and Long Jiang

*

*_{Correspondence:}

[email protected] School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, P.R. China

Abstract

In this paper, we establish an existence and uniqueness theorem for solutions to backward stochastic diﬀerential equations driven by time-changed Lévy noises, in which the generator is monotonic and general growth with respect toy. Our

conclusion extends the corresponding result in Di Nunno and Sjursen (Stoch. Process. Appl. 124:1679-1709, 2014).

MSC: 60H10

Keywords: backward stochastic diﬀerential equation; time-changed Lévy noise; monotone generator; existence and uniqueness

1 Introduction and preliminary

In , Pardoux and Peng [] firstly put forward the theory of nonlinear backward stochastic differential equations (BSDEs for short) and proved an existence and unique-ness result under the Lipschitz assumption. Since then, many scholars have paid more attention to the existence and uniqueness of solutions for BSDEs under weaker assump-tions, such as Lepeltier and San Martin [], Mao [], Jia [] and Fan []. In particular, Pardoux [] investigated the existence and uniqueness ofL_{solutions for BSDEs when the} generatorgsatisfies monotonicity, continuity and general growth conditions with respect toy.

On the other hand, many scholars were devoted their work to investigating BSDEs driven by diﬀerent noises and its applications such as Cohen and Szpruch [], Nualart and Schoutens [], Zhou [] and their references therein. In particular, Di Nunno and Sjursen [] introduced a new class of BSDEs driven by time-changed Lévy noises. In fact, it derives from a conditional Brownian motion and a doubly stochastic Poisson random ﬁeld of the following type:

yt=ξ+

T t

g(s,λs,ys,zs)ds–

T t

R

zs(x)μ(ds,dx), t∈[,T], (.)

where the terminal timeT > , the terminal condition ξ ∈L₍_,_F

T,P), the generator g(w,t,λ,y,z) :×[,T]×[, +∞)_×_R_×_→_R_is_G

t-measurable for allt∈[,T] andμ

is the mixture of a conditional Brownian measureBon [,T]× {}and a centered doubly

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stochastic Poisson measureH˜ on [,T]×Rwith

μ() :=B∩[,T]× {}+H˜∩[,T]×R

, ⊂[,T]×R.

They proved the existence and uniqueness of solution under the following conditions (L)-(L):

(L)

g(w,t,λ,y,z) –gw,t,λ,y,z≤Cy–y,

(L)

g(w,t,λ,y,z) –gw,t,λ,y,z≤Cz–z_λ,

whereC> is a constant and

z–z_λ:=z() –z()λB+

R

z(x) –z(x)q(dx)λH.

Moreover, they also obtained a suﬃcient maximum principle for a general optimal control problem.

Motivated by the above work, our paper aims to discuss the existence and uniqueness of solutions for BSDEs driven by time-changed Lévy noises when the generatorgsatisﬁes monotonicity, continuity and general growth conditions with respect toy, which general-ize the existence and uniqueness result of []. Moreover, we introduce the stability result for the ﬁrst time under this structure.

In the sequel, we shall introduce the preliminary to establish our desired result. For more details, please refer to [] and the references therein.

Let (,F,P) be a complete probability space andλ:= (λB,λH)∈L(×[,T]; (R+)) be a pair of stochastic process whose components are nonnegative and continuous in prob-ability. LetX= ([,T]× {})∪([,T]×R), whereR=R\ {}andBXdenotes the Borel σ-algebra onX. Deﬁne the random measureonXby

() :=

T



{(t,)∈}(t)λBt dt+

T



R

(t,x)q(dx)λH_t dt

:=B∩[,T]× {}+H∩[,T]×R

,

as the mixture of measures on disjoint sets,⊆X. Here,qis a deterministic andσ-ﬁnite measure on the Borel sets ofRsatisfying

Rxq(dx) < +∞.

Here,F_{represents the}_σ_{-algebra which is generated by the values of}_._B_and_H_are called signed random measures on the Borel sets of [,T]× {}and [,T]×R, respec-tively. And they satisfy Deﬁnition . of []. LetH˜ :=H–H_{be a signed random measure}

given by

˜

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Deﬁnition ([]) We deﬁne the signed random measureμon the Borel subsets ofXby

μ() :=B∩[,T]× {}+H˜∩[,T]×R

, ⊆X.

Based on the deﬁnitions ofBandH,μis the mixture of a conditional Brownian measure

Bon [,T]× {}and a centered doubly stochastic Poisson measureH˜ on [,T]×R. Moreover, we know that

Eμ()|F = , EB()|F =B(),

EH˜()|F =H(), Eμ()|F =(), and

Eμ()μ()|F =E

μ()|F E

μ()|F = , if∩=φ.

Here the random measuresBandHare speciﬁc types of time-changed Brownian motion and a pure jump Lévy process.

LetF={Ft,t∈[,T]}, whereFt:=

r>tF

μ

r andFtμ=FtB∨FtH∨Ft. LetG={Gt,t∈

[,T]}whereGt=F

μ

t ∨F. Remark thatGT=FTandG=F. Next, we introduce some deﬁnitions of spaces.

LetS

Gbe the space ofG-adapted càdlàg processesψsuch that

E

sup

t∈[,T]

|ψt|

< +∞;

letL

Gbe a subspace ofL(×[,T]×R,F⊗BX,P⊗) of the random ﬁelds admitting

aG-predictable modiﬁcation and its norm be

ψ L_G:=E

T



ψs()λBsds+

T



R

ψs(x)q(dx)λHs ds

< +∞;

letbe the space of functionsz:R→Rsuch that|z()|₊

R|z(x)|q(dx) < +∞.

Deﬁnition  A solution to the BSDE (.) is a pair of stochastic processes (yt,zt)t∈[,T] satisfying (.) such that (yt,zt)t∈[,T]∈S_G×L_G.

2 Main result

In this section, we shall state the existence and uniqueness result for solution of BSDE (.). Let us start with introducing the following assumptions (H)-(H):

(H) dP×dt-a.e., for eachz∈, the mappingy→g(t,λ,y,z)is continuous. Moreover, there exists a constantα∈Rsuch that

y–yg(w,t,λ,y,z) –gw,t,λ,y,z≤αy–y;

(H) there exists a continuous increasing functionφ:R+_→_R+_{such that}_dP_×_dt_-a.e.,

for eachy∈R,

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(H) there exists a constantC> such thatdP×dt-a.e., for eachy∈Randz,z∈

g(w,t,λ,y,z) –gw,t,λ,y,z≤Cz–z_λ;

(H) E[_T|g(w,t,λt, , )|dt] < +∞.

Theorem  is the main result of this paper.

Theorem  Under assumptions(H)-(H),for each ξ ∈L₍_,_F

T,P),BSDE(.) has a unique solution(yt,zt)t∈[,T].

Remark  It is not hard to check that (L)⇒(H) and (H). Thus, our main result extends the corresponding conclusion of [].

Before giving the proof of Theorem , we establish the following two propositions (see Propositions -). Proposition  and Proposition  are, respectively, the prior estimate and stability of the solutions to BSDEs, which play an important role in the proof of our main result. For convenience, we always assume thatα=  in (H). Indeed if (yt,zt)t∈[,T] is a solution of (.), then (y˜t,z˜t)t∈[,T]withy˜t=eαtytandz˜t=eαtztsatisﬁes an analogous BSDE

with terminal conditionξ˜=eαT_ξ _{and generator}

˜

g(t,λ,y,z) =eαt_g_t_,_λ_,_e–αt_y_,_e–αt_z_–_αy_. ˜

g satisﬁes assumptions (H)-(H) withα= . Hence in the rest of this section, we will suppose thatα= .

Proposition  Let g satisfy(H)and(H);let(yt,zt)t∈[,T]be a solution of BSDE(.).Then

there exists a constant A> depending on C and T such that,for each≤u≤t≤T,

E

sup

r∈[t,T]

|yr|Gu

+E

T t

R

_zs(x)(ds,dx)Gu

≤A

E|ξ||Gu +E

T t

g(s,λs, , )dsGu

.

Proof From Deﬁnition , BSDE (.) is equivalent to the following BSDE:

yt=ξ+

T t

g(s,λs,ys,zs)ds–

T t

zs()dBs–

T t

R

zs(x)H˜(ds,dx), t∈[,T].

Applying the Itô formula to|ys|_{, we get}

d|ys|= –ysg(s,λs,ys,zs)ds+ ys

zs()dBs+

R

zs(x)H˜(ds,dx)

+

zs()dBs+

R

zs(x)H˜(ds,dx)



= –ysg(s,λs,ys,zs)ds+ ys

R

zs(x)μ(ds,dx) +

R

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Integrating both sides fromttoT, we deduce formly integrable martingale. In fact, there exists a constantp>  such that

E

inequality and (.), we can obtain



Furthermore, it follows from the BDG inequality that there exists a positive constant d

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By virtue of (.) and (.), it follows from (.) that

Thus, from the above inequality and (.), we have

Esup

Finally, Gronwall’s inequality yields, for eacht∈[,T],

ht≤e(+d)(C+)(T–t) + dE|ξ||Gu +

which completes the proof of Proposition .

Proposition  is a stability result. For eachn≥, let (yt,zt)t∈[,T] and (ynt,znt)t∈[,T] be, respectively, a solution of the BSDE (.) and the following BSDE depending on the pa-rametern:

Furthermore, we introduce the following assumptions (B) and (B):

(B) ξn_∈_L₍_,_F

T,P)and allgnsatisfy (H) and (H);

(B) limn→∞E[|ξn–ξ|+T|gn(s,λs,ys,zs) –g(s,λs,ys,zs)|ds] = .

Proposition  Under assumptions(B)and(B),we have

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Applying Itô’s formula to|ˆyn

Taking the mathematical expectation on both sides of (.), by Gronwall’s inequality, we can obtain

Then by (.) witht=  and the previous inequality, we can get

E

Finally, taking the supremum overt∈[,T] on both sides of (.) and the mathematical expectation, and applying the BDG inequality to the supremum of the martingale on the right-hand side, we have

E sup

Note that the aboveKis a positive constant depending onCandT. Thus, in view of (B),

we complete the proof of Proposition .

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Proof of Theorem  The uniqueness part of Theorem  is established immediately by Proposition . Hence, we just need to prove the existence part, which can be divided into three steps.

Step. We shall prove that, for eachξ ∈L₍_,_F

T,P), if there exists a constantM> 

such that

dP-a.s., |ξ| ≤M and dP×dt-a.e., g(t,λ, , )≤M, (.) andgsatisfy (L) and (H), then there exists a unique solution to BSDE (.).

For (ut,vt)t∈[,T]∈SG×LG, deﬁne

Moreover, by the martingale representation theorem

yt=E

G×LGequipped with the following norm:

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Thus, (yt,zt)t∈[,T]is a unique solution of BSDE (.).

Step. We shall prove that, for eachξ∈L₍_,_F

T,P) andV∈LG, ifgsatisfy (H)-(H), then there exists a unique solution to the following BSDE:

yt=ξ+

T t

g(s,λs,ys,Vs)ds–

T t

R

zs(x)μ(ds,dx). (.)

Firstly, we assume that (.) holds. For givenV,g(s,λ,y,Vs) will be viewed asg(s,λ,y). By similar proof as Proposition . in [], we can construct of smooth approximations (gn,n∈N) ofg,

gn(t,λ,y) :=

ρn∗g(t,λ,·)

(y),

whereρn:R→R+is a sequence of smooth functions with compact support which

approx-imate the Dirac measure at  and satisfyρn(x)dx= . For eachn≥,gnis smooth and

monotone iny, and thus locally Lipschitz inyuniformly with respect to (w,t,λ). Hence, we need to introduce a truncation functionqpingn:

gn,p(t,λ,y) :=gn

t,λ,qp(y),qp(y) := py

|y| ∨p.

By Step , (ynt,p,ztn,p)t∈[,T]is a unique solution for BSDE (.) with generatorgn,p. The

se-quence (ynt,p)t∈[,T]is bounded and its boundedness does not depend onp. In fact, since

gn,psatisﬁes globally Lipschitz iny, applying the Itô formula to|ysn,p|on [t,T], we can get

ynt,p

_≤

E|ξ||Gt +E

T t

gn,p(s,λs, )dsGt

+ (C+ )

T t

Eyn_s,p|Gt ds.

By (.) and Gronwall’s inequality, for eachr∈[t,T], we can obtain

Eyn_r,p|Gt ≤M( +T)e(C+)(T–r).

Takingr=t, we obtain

ynt,p



≤M( +T)e(C+)T.

Therefore, forplarge enough, (yn·,p,z·n,p) turns into (yn·,zn·). From the above facts,gnsatisﬁes

conditions of Proposition  with relative constant independent ofn. So, (yn

t,unt,znt)t∈[,T]is bounded,i.e.,

sup

n∈N E

T



yn_sds+

T



un_sds+

T



R

zn_s(dx,ds)

≤ ¯M,

whereun_t =gn(t,λt,ynt). Thus, we can ﬁnd a subsequence (named again (ynt,unt,znt)t∈[,T]) converging weakly to (yt,ut,zt)t∈[,T]. Moreover, the martingale

T t

Rzns(x)μ(ds,dx)

con-verges weakly to_tT_Rzs(x)μ(ds,dx) inL₍_×_[,_T_{]). In fact, let}_η_∈_L₍_,_F

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can be written asη=E[η|F_{] +}T

Lastly, condition (.) can be taken away by a truncation procedure. For eachx∈Rand

m≥, letqm₍_x_{) =}_xm_/(_|_x_{| ∨}_m_{). Set} unique solution to the following BSDE:

ym_t =ξm+

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Clearly, we can ﬁnd a mappingˆ:S

G×LG→SG×LGby Step , whereˆ((U·,V·)) =

(y·,z·) with solution (y·,z·)∈SG×LGsatisfying BSDE (.). And it is a contractive mapping with the norm · βwith suitableβby the same arguments as Step . Thus, it has a ﬁxed point (yt,zt)t∈[,T]which is a solution of the BSDE (.).

Acknowledgements

The authors would like to express great thanks to the anonymous referee for his/her careful reading and helpful suggestions. This research was supported by the National Natural Science Foundation of China (11371362, 11601509) and Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYLX16_0520).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally in this article. All authors read and approved the ﬁnal manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 22 January 2017 Accepted: 13 July 2017

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