Backward stochastic differential equations driven by fractional noise with non-Lipschitz coefficients

Journal Pre-proof Backward stochastic differential equations driven by fractional noise with non-Lipschitz coefficients Xianye Yu, Mingbo Zhang

PII: DOI: Reference:

S0167-7152(19)30327-X https://doi.org/10.1016/j.spl.2019.108681 STAPRO 108681

To appear in:

Statistics and Probability Letters

Received date : 10 September 2019 Accepted date : 2 December 2019 Please cite this article as: X. Yu and M. Zhang, Backward stochastic differential equations driven by fractional noise with non-Lipschitz coefficients. Statistics and Probability Letters (2019), doi: https://doi.org/10.1016/j.spl.2019.108681. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Elsevier B.V. All rights reserved.

Journal Pre-proof

BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY FRACTIONAL NOISE WITH NON-LIPSCHITZ COEFFICIENTS†

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XIANYE YU1 AND MINGBO ZHANG2,∗

1

p ro

School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, P.R. China; 2 School of Statistics, Jiangxi University of Finance and Economics, Nanchang, P.R. China Emails: X.Yu: [email protected], M. Zhang: [email protected]

Pr e-

Abstract. In this paper, we study the backward stochastic differential equations driven by fractional Brownian motion with Hurst parameter H greater than 1/2, where the coefficient is non-Lipschitz continuous and the stochastic integral is the Skorohod integral.

1. Introduction

Given H ∈ (0, 1), a fractional Brownian motion (fBm, for short) B H = {BtH , t ≥ 0} with Hurst parameter H defined on (Ω, F H , P ) is a mean zero Gaussian process such that B0 = 0 and    1  2H t + s2H − |t − s|2H E BtH BsH = 2

Jo

urn

al

for all t, s ≥ 0. The fBm displays the long memory for 21 < H < 1, which meanP H − BnH ), we have ∞ s that for r(n) := EB1H (Bn+1 n=1 r(n) = ∞. For H < 1/2, we P∞ have n=1 r(n) < ∞ and we say that the fBm has the property of short memory. Moreover, the fBm admits the self-similarity, stationary increments and H¨older’s continuity. Owing to these compact properties and its applications in various scientific areas including telecommunications, turbulence and finance, there has been considerable interest in studying fBm as a driving noise in models to characterize some real world phenomena. For H = 1/2, B H coincides with the standard Brownian motion. However, since B H is neither a semimartingale nor a Markov process unless H = 1/2, many of the powerful techniques from classical stochastic analysis are not available when dealing with B H . But, as a Gaussian process, one can construct the stochastic calculus of variations with respect to B H and establish the Skorohod integral. Some surveys about the Skorohod integral with respect to fBm could be found in Duncan et al. [4], Hu [5], Nualart [10] and references therein. ∗

Corresponding author. 2010 Mathematics Subject Classification. 60H10, 60H07, 60G22. Key words and phrases. Backward stochastic differential equation; Fractional Brownian motion; Non-Lipschitz. 1

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X. YU AND M. ZHANG

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The purpose of this paper is to study the existence of the solution for backward stochastic differential equations driven by fractional Brownian motion under the nonLipschitz condition, where the stochastic integral is the Skorohod integral. Consider the following backward stochastic differential equations: ( −dY (t) = f (t, ηt , Y (t), Z(t))dt − Z(t)dBtH , t ∈ [0, T ], (1.1) Y (T ) = h(ηT ),

Jo

urn

al

Pr e-

p ro

Rt Rt with T > 0, where ηt = η0 + 0 bs ds + 0 σs dBsH and the stochastic integral is Skorohod integral. As we know, the general backward stochastic differential equations driven by Brownian motion were first introduced by Pardoux and Peng [11]. Since the work of Pardoux and Peng, many authors considered the backward stochastic differential equations under different conditions. Lepeltier and Martin [8] established the existence of a solution for one dimensional backward stochastic differential equation where the coefficient is continuous. Pardoux and Zhang [12] studied a new class of backward stochastic differential equations and gave the probalilistic formula for solutions of semilinear partial equations with Neumann boundary condition. When Brownian motion is replaced by fractional Brownian motion, Bender [1] solved a class of linear backward stochastic differential equations by using the solution of a partial differential equation and, Hu and Peng [7] proved the existence and uniqueness for the general backward stochastic differential equations via quasi-conditional expectation when the coefficient in (1.1) is uniformly globally Lipschitzian. Maticiuc and Nie [9] revised the condition in [7] and obtained the existence and uniqueness for such equations. They also first considered the fractional backward variational inequalities. Borkowska [3] studied the generalized backward differential equations with fractional noise and established the connection between this solution and the solution for parabolic partial differential equations with Neumann boundary condition, which can be regarded as an extension of the results in [12]. In [2], Bender transferred the backward differential equations driven by fractional noise to the backward stochastic differential equations driven by Brownian motion to obtain the existence and uniqueness results for such equations driven by fractional Brownian motion with arbitrary Hurst parameter H ∈ (0, 1). Motivated by these works, we are interested in considering the backward differential equations driven by fractional Brownian motion under the weaker condition than Lipschitz one, and our main result can be regarded as an extension of the result in Hu and Peng [7]. The plan of this paper is as follows. Section 2 contains some necessary preliminaries on the fractional Brownian motion, including the Skorohod integral with respect to it and fractional Itˆo formula. In Section 3, we formulate and prove a main theorem of this paper. 2. Preliminaries In this section, we briefly recall some basic results of fractional Brownian motion with 1/2 < H < 1, which are used frequently in this paper. In this paper we let C stand for a positive constant depending only on some determinate parameters and its value may be different in different appearance.

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FRACTIONAL BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

3

Let HT be the completion of the linear space E generated by the indicator functions 1[0,t] , t ∈ [0, T ] with respect to the inner product h1[0,s] , 1[0,t] iT =

 1  2H t + s2H − |t − s|2H . 2

of

The application ϕ ∈ E → B H (ϕ) is an isometry from E to the Gaussian space generated by B H and it can be extended to HT . The elements in HT may be distributions (see [13]). Denote by S the set of smooth functionals of the form F = f (B H (ϕ1 ), B H (ϕ2 ), . . . , B H (ϕn )),

DsH F

p ro

where f is a polynomial function of n variables and ϕi ∈ HT . The derivative operator DH (the Malliavin derivative) of a functional F of the form above is defined as n X ∂f = (B H (ϕ1 ), B H (ϕ2 ), . . . , B H (ϕn ))ϕj (s), ∂x j j=1

s ∈ [0, T ].

Pr e-

The derivative operator DH is then a closable operator from L2 (Ω) into L2 (Ω; HT ). We denote by D1,2 the closure of S with respect to the norm q kF k1,2 := E|F |2 + EkDH F k2T .

The divergence integral δ H is the adjoint of derivative operator DH . That is, we say that a random variable u in L2 (Ω; HT ) belongs to the domain of the divergence operator δ H , denoted by Dom(δ H ), if E hDH F, uiT ≤ CkF kL2 (Ω)

al

for every F ∈ S. In this case δ H (u) is defined by the duality relationship   (2.1) E F δ H (u) = EhDH F, uiT

urn

for any F ∈ S. We have S ⊂ D1,2 ⊂ Dom(δ H ). We will use the notation Z T H δ (u) = us dBsH 0

Jo

to express theR Skorohod integral of a process u, and the indefinite Skorohod integral t is defined as 0 us dBsH = δ H (u1[0,t] ). For details about the Skorohord integral associated with fractional Brownian mtotion, we refer to Hu [5], Nualart [10] and references therein. Now we introduce another derivative Z T H Dt F = φ(t − s)DsH F ds 0

RtRt where φ(·) := H(2H − 1)| · | . We put hξ, ζit = 0 0 φ(u − v)ξ(u)ζ(v)dudv and kξk2t = hξ, ξit for continuous functions ξ, ζ on [0, T ]. Let Ht be the completion of the continuous functions under this Hilbert norm, and Ht contains the distributions. Moreover, we have the continuous embedding L2 ([0, T ]) ⊂ L1/H ([0, T ]) ⊂ HT . The following results are well-known (see Duncan et al [4], Hu [5]), which are useful in the sequel. 2H−2

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4

X. YU AND M. ZHANG

Proposition 2.1. Let F be a stochastic process such that   Z TZ T H 2 2 |Ds Ft | dsdt < ∞. E kF kT + 0

0

of

RT Then the Itˆo-Skorohod type stochastic integral 0 Fs dBsH exists in L2 (Ω, F H , P ). Moreover, we have Z T  H E Fs dBs = 0, 0

E

Z

0

T

Fs dBsH

2

 Z 2 = E kF kT +

T

Z

T

p ro

and

0

H DH s Ft Dt Fs dsdt

0



.

Proposition 2.2. Let f, g : [0, T ] → R be deterministic continuous functions. If Z t Z t Xt = X 0 + gs ds + fs dBsH , t ∈ [0, T ], 0

Pr e-

0

where X0 is a constant and F ∈ C 1,2 ([0, T ] × R), then for any t ∈ [0, T ], Z t Z t ∂F ∂F F (t, Xt ) =F (0, X0 ) + (s, Xs )ds + (s, Xs )dXs 0 ∂s 0 ∂x   Z 1 t ∂ 2F d 2 + (s, Xs ) kf ks ds, 2 0 ∂x2 ds

where

Z sZ 0

Z

al

d d kf k2s = ds ds

= 2fs

0

0

s

φ(v − u)fv fu dudv

s

φ(v − s)fv dv.

urn

Proposition 2.3. Let fi (s), gi (s) be in D1,2 such that Z T (|fi (t)|2 + |gi (t)|2 )dt < ∞, E 0

Then

Jo

where i = 1, 2. Assume also that DtH fi (s) is continuously differentiable with respect to RT RT 2 (s, t) ∈ [0, T ] for almost all ω ∈ Ω, and E 0 0 |DH t fi (s)| dsdt < ∞. Denote Z t Z t Xi (t) = gi (s)ds + fi (s)dBsH , t ∈ [0, T ]. X1 (t)X2 (t) =

0

Z

0

t

X1 (s)g2 (s)ds +

0

Z

Z

t

0

t

Z

X1 (s)f2 (s)dBsH t

X2 (s)f1 (s)dBsH 0 0 Z t Z t H + Ds X1 (s)g2 (s)ds + DH s X2 (s)g1 (s)ds.

+

0

X2 (s)g1 (s)ds +

0

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FRACTIONAL BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

5

3. Backward stochastic differential equations driven by fBm In this section we consider the backward stochastic differential equations with respect to fractional Brownian motion. Throughout this paper we let Z t Z t bs ds + σs dBsH , ηt = η0 + 0

0

0

and we have

0

p ro

of

where η0 is a given constant, b and σ are two deterministic continuous functions such that σt 6= 0 for all t ∈ [0, T ]. Recall that Z tZ t 2 |u − v|2H−2 σu σv dudv, kσkt = H(2H − 1)

Pr e-

d (kσk2t ) = 2ˆ σt σt > 0, t ∈ (0, T ], dt Rt where σ ˆt = 0 φ(u − t)σu du. The objective of this section is to study the backward stochastic differential equations driven by fractional Brownian motion: ( −dY (t) = f (t, ηt , Y (t), Z(t))dt + Z(t)dBtH , t ∈ [0, T ], (3.1) Y (T ) = h(ηT ), where the stochastic integral is understood as the Skorohod integral. We assume that: (H1 ) h : R → R is a differentiable function with polynomial growth; (H2 ) There exists a constant L0 > 0 such that inf

0≤t≤T

σ ˆt ≥ L0 ; σt

al

(H3 ) f : [0, T ] × R3 → R is a continuous function and there exists a constant L > 0 such that for 0 ≤ t ≤ T , (x, y, z) ∈ R3 , |f (t, x, y, z)| ≤ L(1 + |y| + |z|).

(3.2)

urn

In order to give the definition of the solution for the above backward stochastic differential equations, we introduce the sets (see Hu and Peng [7]): Ct ={ψ(t, ηt ), ψ(t, x) is a continuously differentiable with respect to t and twice continuously differentiable with respect to x}

and

Jo

V[0,T ] ={Y = ψ(·, η· ) | ψ(t, ηt ) ∈ Ct for all t ∈ [0, T ]}.

e[0,T ] the completion of V[0,T ] under the following norm Denote by V Z T 1/2 βt 2 kY kβ := e E|Y (t)| dt . 0

Definition 3.1. A pair (Y, Z) is called a solution of (3.1), if the following conditions are satisfied: e[0,T ] × V e[0,T ] ; (1) (Y, Z) ∈ V RT RT (2) Y (t) = h(ηT ) + t f (s, ηs , Y (s), Z(s))ds − t Z(s)dBsH , t ∈ (0, T ].

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6

X. YU AND M. ZHANG

Now let us state the existence of the solution for backward stochastic differential equation (3.1). Theorem 3.1. Let the assumptions (H1 ), (H2 ) and (H3 ) be satisfied. Then the backe[0,T ] × V e[0,T ] . ward stochastic differential equation (3.1) has a solution (Y, Z) ∈ V

of

Before proving the above theorem, we need the following lemma which gives the approximation of continuous functions by the Lipschitz functions and it was presented by Lepeltier and Martin [8].

p ro

Lemma 3.1. Let f : Rp → R be a continuous function and there exists a constant C < ∞ such that for all x ∈ Rp |f (x)| ≤ C(1 + |x|). Then the sequence of functions

fn (x) = infp {f (y) + n|x − y|}

Pr e-

y∈Q

is well defined for n ≥ C and it satisfies (i) (ii) (iii) (iv)

linear growth: for any x ∈ Rp , |fn (x)| ≤ C(1 + |x|); monotonicity in n: for any x ∈ Rp , fn (x) ↑; Lipschitz condition: for any x, y ∈ Rp , |fn (x) − fn (y)| ≤ n|x − y|; n→∞ n→∞ strong convengence: if xn −→ x, then fn (xn ) −→ f (x).

Remark 1. It is obvious that fn (x) ≤ f (x) for any x ∈ Rp and any n ≥ C. Indeed, f (x) − fn (x) = f (x) − infp {f (y) + n|x − y|}

al

y∈Q

= f (x) + maxp {−f (y) − n|x − y|} y∈Q

urn

≥ f (x) + (−f (x) − 0) = 0.

Consider, for fixed (t, x) , the sequence fn (t, x, y, z) associated with f in (3.1). By Lemma 3.1, fn are Lipschitz functions in (y, z) and we also define the function g(t, x, y, z) := L(1 + |y| + |z|) and Z T Z T n n n (3.3) Y (t) = h(ηT ) + fn (s, ηs , Y (s), Z (s))ds − Z n (s)dBsH , n ≥ L, (3.4)

Jo

t

Ye (t) = h(ηT ) +

Z

t

T

t

e g(s, ηs , Ye (s), Z(s))ds −

Z

t

T

H e Z(s)dB s .

Note that the function g is also Lipschitz function. From Hu and Peng [7], we get that the above backward stochastic differential equations (3.3) and (3.4) have a unique e[0,T ] × V e[0,T ] under the assumptions (H1 ) and (H2 ). Then from Theorem solution in V 12.3 of Hu et al. [6], we deduce that for any m ≥ n ≥ L

(3.5)

Y n (t) ≤ Y m (t) ≤ Ye (t) a.s.

e β ≤ C. Moreover, it is clear that kYe kβ ≤ C and kZk

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FRACTIONAL BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

7

Lemma 3.2. For the backward stochastic differential equation (3.3), we have the following estimate:    2  Z T eβT LT βs n 2 βT 2 e |Y (t)| ds ≤ Ee h(ηT ) + exp (T − t) 3 L0 t

of

for any n ≥ L, where t ∈ [0, T ].

t

t

p ro

Proof. Applying Itˆo formula, we obtain Z T Z T βs n 2 βt n 2 βT 2 eβs Y n (s)fn (s, ηs , Y n (s), Z n (s))ds e |Y (s)| ds + 2 e |Y (t)| = e h(ηT ) − β t t Z T Z T n n −2 eβs Y n (s)Z n (s)dBsH − 2 eβs DH s Y (s)Z (s)ds. n DH s Y (s)

Pr e-

From Hu and Peng [7], we know that = σσˆss Z n (s). By the assumption (H2 ) and |fn (t, ηt , y, z)| ≤ L(1 + |y| + |z|), it is obvious that   Z T Z T βt n 2 βs n 2 βs n 2 E e |Y (t)| + β e |Y (s)| ds + 2L0 e |Z (s)| ds t

≤ EeβT h(ηT )2 + 2E βT

2

Z

t

T

eβs Y n (s)fn (s, ηs , Y n (s), Z n (s))ds

t

≤ Ee h(ηT ) + 2LE

Z

T

t

eβs |Y n (s)|(1 + |Y n (s)| + |Z n (s)|)ds.

Together with inequality 2ab ≤ εa2 + 1ε b2 , where ε > 0, thus   Z T Z T βs n 2 βs n 2 βt n 2 e |Y (s)| ds + 2L0 e |Z (s)| ds E e |Y (t)| + β Z

t

T

al

t

βT

2

βs

urn

≤ Ee h(ηT ) + LE e ds + L(3 + ε)E t Z T L + E eβs |Z n (s)|2 ds. ε t

Choosing ε = L/L0 and β ≥ 3L + 1, we have  Z T Z βt n 2 βs n 2 E e |Y (t)| + e |Y (s)| ds + L0 t

t

Z

T

t

eβs |Y n (s)|2 ds

T βs

n

2

e |Z (s)| ds



Jo

Z T L(eβT − eβt ) L2 ≤ Ee h(ηT ) + + E eβs |Y n (s)|2 ds β L0 t Z T 2 βT βt (e − e ) L βs n ≤ EeβT h(ηT )2 + +E e |Y (s)|2 ds. 3 L0 t βT

2

By Gronwall’s inequality,

  Z T 2  (eβT − eβt ) L βT 2 Ee |Y (t)| ≤ Ee h(ηT ) + exp ds 3 L0 t    2  eβT LT βT 2 ≤ Ee h(ηT ) + exp 3 L0 βt

n

2

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8

X. YU AND M. ZHANG

which leads to Z T t

   2  eβT LT βT 2 e |Y (s)| ds ≤ Ee h(ηT ) + exp (T − t) 3 L0 βs

n

2

for any n ≥ L, where t ∈ [0, T ).



kY n kβ ≤ C,

of

Lemma 3.3. There exists a constant C > 0 depending on L0 , L, T and Eh(ηT )2 such that kYe kβ ≤ C,

kZ n kβ ≤ C,

p ro

for all n ≥ L.

e β ≤ C. kZk

e β ≤ C. Thus, it is enough to prove that Proof. It is obvious that kYe kβ ≤ C and kZk there exists a constant C > 0, which does not depend on n, such that kY n kβ ≤ C,

kZ n kβ ≤ C.

and

 Z βt n 2 E e |Y (t)| +

t

Pr e-

It follows form Lemma 3.2 that for any n ≥ L    2  eβT LT n βT 2 kY kβ ≤ Ee h(ηT ) + exp T, 3 L0 T

βs

n

2

e |Y (s)| ds + L0

(eβT − eβt ) ≤ Ee h(ηT ) + +E 3 βT

2

al

which implies that Z T E eβs |Z n (s)|2 ds t

urn

EeβT h(ηT )2 eβT L2 ≤ + + 2 L0 3L0 L0



Z

t

T

Z

T

t 2

for any n ≥ L and t ∈ [0, T ). Thus we obtain

2

n

2

e |Z (s)| ds

L βs n e |Y (s)|2 ds, L0

eβT Ee h(ηT ) + 3 βT

βs



exp



L2 T L0





(T − t),

kZ n kβ ≤ C

where the constant C is not depending on n, and we complete the proof.



Jo

Now, we present the proof of Theorem 3.1. Proof of Theorem 3.1. Firstly, we show that the sequence {(Y n (·), Z n (·))}n≥L is a Cauchy e[0,T ] × V e[0,T ] . By Lemma 3.2, then sequence in V    2  LT eβT n βT 2 exp T, kY kβ ≤ Ee h(ηT ) + 3 L0 and {Y n } is increasing. It follows from the Dominated convergence theorem that Y n e[0,T ] . The limit of {Y n } is denoted by Y . We deduce from Scheff´e’s converges in V lemma that Z T

lim E

n→∞

0

eβt |Y n (t) − Y (t)|2 dt = 0,

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FRACTIONAL BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS

9

e[0,T ] . Then, let us define ∆Y n,m (s) := which means that {Y n } is Cauchy sequence in V Y n (s) − Y m (s), ∆Z n,m (s) := Z n (s) − Z m (s) and ∆f (n,m) (s) := fn (s, ηs , Y n (s), Z n (s)) − fm (s, ηs , Y m (s), Z m (s)).

≤ 2E

t

t

T

eβs ∆Y n,m (s)∆f (n,m) (s)ds

t

 Z ≤2 E

T βs

e |∆Y

t

n,m

βs

e |∆Z

1/2  Z (s)| ds E

p ro

(3.6)

Z

T

2

T

βs

e |∆f

t

(n,m)

 Z E

T

βs

e |∆Y

0

 12  Z (s)| ds E

Pr e-

1 ≤ L0

n,m

T

2

βs

e |∆f

0

2

(s)| ds

where we also have used the Cauchy-Schwarz inequality. Hence Z T E eβs |∆Z n,m (s)|2 ds 0

n,m

 (s)| ds 2

of

Again by using Itˆo formula we obtain for m, n ≥ L  Z T Z βs n,m 2 βt n,m 2 e |∆Y (s)| ds + 2L0 E e |∆Y (t)| + β

(n,m)

1/2

.

 21 (s)| ds . 2

urn

al

By |fn (s, ηs , y, z)| ≤ L(1 + |y| + |z|) and Lemma 3.3, thus Z T eβs |∆f (n,m) (s)|2 ds E 0 Z T Z T eβs |fm (s, ηs , Y m (s), Z m (s))|2 ds eβs |fn (s, ηs , Y n (s), Z n (s))|2 ds + 2E ≤ 2E 0 0 Z T Z T βs n n 2 2 2 eβs (1 + |Y m (s)| + |Z m (s)|)2 ds e (1 + |Y (s)| + |Z (s)|) ds + 2L E ≤ 2L E 0 0 Z T Z T Z T 2 βs 2 βs n 2 2 ≤ 12L e ds + 6L E e |Y (s)| ds + 6L E eβs |Z n (s)|2 ds 0 0 0 Z T Z T eβs |Y m (s)|2 ds + 6L2 E eβs |Z m (s)|2 ds ≤ C, + 6L2 E 0

T

βs

e |∆Z

Jo

which leads to  Z E

0

0

n,m

2

(s)| ds

2

C ≤ 2E L0

Z

T

0

eβs |∆Y n,m (s)|2 ds.

e[0,T ] , and we denote by Z the limit of {Z n }. Therefore, {Z n } is Cauchy sequence in V Secondly, arguing as in Theorem 22 in Maticiuc and Nie [9], one can prove that (Y, Z) satisfies Z T Z T t ∈ (0, T ]. Y (t) = h(ηT ) + f (s, ηs , Y (s), Z(s))ds − Z(s)dBsH , t

t

Indeed, we deduce from (3.6) that

lim E|Y n (t) − Y (t)|2 = 0,

n→∞

t ∈ [0, T ]

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10

X. YU AND M. ZHANG

and lim E

n→∞

Z

T

0

n

2

|Y (t) − Y (t)| dt + E

From (i) and (iv) of Lemma 3.1 we have

Z

T

0

|Z n (t) − Z(t)|2 dt = 0.

n→∞

of

fn (t, ηt , Y n (t), Z n (t)) −→ f (t, ηt , Y (t), Z(t)) a.e. and

|fn (t, ηt , Y n (t), Z n (t))|2 ≤ 3L + 3L|Y n (t)|2 + 3L|Z n (t)|2 .

= −Y (t) + h(ηT ) +

f (s, ηs , Y (s), Z(s))ds := θ(t),

t

2

in L2 (Ω, F H , P ),

and Z 1[t,T ] → Z1[t,T ] in L (Ω, F H , P ; HT ). By the duality relationship (2.1), (3.3), and (3.7), we conclude that for any F ∈ S

Pr e-

n

p ro

Hence, for arbitrary  > 0 and for all t ∈ [, T ],   Z T n n n fn (s, ηs , Y (s), Z (s))ds lim −Y (t) + h(ηT ) + n→∞ t (3.7) Z T

E(hD·H F, Z(·)1[t,T ] (·)iT ) = lim E(hD·H F, Z n (·)1[t,T ] (·)iT ) n→∞  Z T  n H Z (s)dBs = E(F θ(t)). = lim E F n→∞

t

By the definition of Skorohod integral, we know that Z1[t,T ] ∈ Dom(δ H ) and δ H (Z1[t,T ] ) = θ(t). Therefore, Z T Z T Z(s)dBsH , t ∈ [, T ], f (s, ηs , Y (s), Z(s))ds − Y (t) = h(ηT ) + t

t

al

and the theorem follows from that  is arbitrary.



urn

Remark 2. Maticiuc and Nie [9] revised the conditions in [7] and proved the existence and uniqueness of the solution of equation (1.1) under the Lipschitz case. Modifying the coefficient’s conditions in [9] to the non-Lipschitz case (H3 ), one can still obtain the existence for the equation (1.1) by our method. Acknowledgments

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This project was sponsored by NSFC (No. 11701589, 11871484, 11971432) and (GJJ190265). The authors would like to thank the referees for their careful reading of the paper and helpful comments. References

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