Generalized fractional BSDE with jumps andLipschitz coefficients

Statistics and Probability Letters 154 (2019) 108549

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Generalized fractional BSDE with jumps and Lipschitz coefficients Qun Shi School of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330200, PR China

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Article history: Received 30 November 2017 Received in revised form 3 December 2018 Accepted 25 June 2019 Available online 2 July 2019

a b s t r a c t In this work, we deal with a generalized BSDE driven by fractional Brownian motion with Hurst parameter H ∈ (1/2, 1) and a Poisson random measure. Moreover, we establish existence and uniqueness of solution in the case of Lipschitz coefficients. © 2019 Elsevier B.V. All rights reserved.

Keywords: Fractional Brownian motion Backward stochastic differential equation Poisson jumps Malliavin derivative

1. Introduction General backward stochastic differential equations driven by a Brownian motion were first studied by Pardoux and Peng (1992). Later Pardoux and Zhang (1998) introduced the generalized BSDEs, i.e. BSDEs with an additional term—an integral with respect to an increasing process. Backward stochastic differential equations driven by a fractional Brownian motion with H ∈ (1/2, 1) were first considered by Biagini et al. (2002), where they studied the stochastic maximal principle in the framework of a fractional Brownian motion. By adapting the four-step scheme introduced by Ma et al. (1994) and the so-called S-transform, Bender (2005) studied BSDEs driven by a fractional Brownian motion with H ∈ (0, 1). Indeed, throughout a backward parabolic PDE, he constructed an explicit solution of a kind of linear fractional BSDE. Hu and Peng (2009) were the first to study nonlinear BSDEs governed by a fractional Brownian motion. It is well known that backward stochastic differential equation provided stochastic representation of solution of some classes of partial differential equations of second order. With the help of backward stochastic differential equations with respect to a Brownian motion and a Poisson random measure, some authors generalized this result to integro-partial differential equations. The pioneer result on BSDEs, established by Pardoux and Peng (1990) requires Lipschitz condition on the drift of the equation. Sow study on BSDE with jumps, established by Sow (2014) requires non-Lipschitz coefficients and application to large deviations. Our aim in the present work is to extend result to general BSDEs with jumps driven by fractional Brownian motion. Let us recall that, for H ∈ (0, 1), a fBm (BH (t))t ⩾0 with Hurst parameter H is a continuous and centered Gaussian process with covariance [ ] 1 E BH (t)BH (s) = (t 2H + s2H − |t − s|2H ), t , s ⩾ 0. 2 For H = 1/2, the fBm is a standard Brownian motion. If H 1/2, then BH (t) has a long-range dependence, which means ∑> ∞ H H that for r(n) := cov (BH (1), BH (n + 1) − BH (n)), we have n=1 r(n) = ∞. Moreover, B is self-similar, i.e. B (at) has the E-mail address: [email protected]. https://doi.org/10.1016/j.spl.2019.06.025 0167-7152/© 2019 Elsevier B.V. All rights reserved.

2

Q. Shi / Statistics and Probability Letters 154 (2019) 108549

same law as aH BH (t) for any a > 0. Since there are many models of physical phenomena and finance which exploit the self-similarity and the long-range dependence, fBm is a very useful tool to characterize such type of problems. However, since fBm is not semimartingales nor Markov processes when H ̸ = 1/2, we cannot use the classical theory of stochastic calculus to define the fractional stochastic integral. In essence, two different integration theories with respect to fractional Brownian motion have been defined and studied. The first one, originally due to Young (1936), concerns the pathwise Riemann–Stieltjes integral which exists if the integrand has Hölder continuous paths of order α > 1 − H. But it turns out that this integral has the properties comparable to the Stratonovich integral, which leads to difficulties in applications. The second one concerns the divergence operator (Skorohod integral), defined as the adjoint of the derivative ¨ (1998). operator in the framework of the Malliavin calculus. This approach was introduced by Decreusefond and U¨stunel Concerning the study of BSDEs in the fractional framework, the major problem is the absence of a martingale representation type theorem with respect to fBm. For the first time, Hu and Peng (2009) overcomes this problem, in the case H > 1/2. For that, they used the notion of quasi-conditional expectation ˆ E. We now introduce a class of reflected diffusion processes with standard Brownian motion. Let G be an open connected subset of Rd , which is such that for some l ∈ C 2 (Rd ), G = {x : l(x) > 0}, ∂ G = {x : l(x) = 0} and |∇ l(x)| = 1 for x ∈ ∂ G. Note that at any boundary point x ∈ ∂ G, ∇ l(x) is a unit normal vector to the boundary, pointing towards to the interior of G. If drift coefficient and diffusion coefficient satisfy some Lipschitz, then it follows from the results in Lions and Sznitman (1984) (see also Saisho, 1987) that for each x ∈ ∂ G, there exists a unique pair of progressively measurable continuous processes (ηt , Λt ), such that

ηt = η0 +

t

∫ 0

Λt =

t

∫

σ (s)dBs +

b(s)ds + 0

t

∫

∇ l(ηs )dΛs , 0 ⩽ t ⩽ T , 0

t

∫

1ηs ∈∂ G dΛs , Λ. is a nondecreasing process. 0

The existence of such a problem driven by fBm was shown in Ferrante and Rovira (2013) and a set D = (0, +∞). In this paper we study the generalized BSDEs with jumps driven by fBm with Hurst parameter H > 1/2. We prove that kind of equation has a unique solution. The paper is organized as follows. In Section 2 we give some definitions and results about fractional stochastic integral which will be needed throughout the paper. Section 3 contains the definition of the generalized BSDEs with jumps driven by fBm and assumptions. Finally, Section 4 is devoted to prove the main theorem of the paper. 2. Fractional stochastic calculus Denote, for given H ∈ (1/2, 1), φ (x) = H(2H − 1)|x|2H −2 , x ∈ R. Let ξ and η be measurable functions on [0, T ]. Define

⟨ξ , η⟩t =

t

∫ t∫

φ (u − v )ξ (u)η(v )dudv 0

0

and ∥ξ ∥2t = ⟨ξ , ξ ⟩t . Note that , for any t ∈ [0, T ], ⟨ξ , η⟩t is a Hilbert scalar product. Let H be the completion of the measurable functions such that ∥ξ ∥2t < ∞. The elements of H may be distributions (refer to Pipiras and Taqqu, 2000). Let (ξn )n be a sequence in H such that ⟨ξi , ξj ⟩T = δij . By PT denote the set of all polynomials of fractional Brownian Motion in [0, T ], i.e. it contains all elements of the form F (ω ) = f

(∫

T

ξ1 (t)dBHt , . . . ,

0

T

∫

) ξk (t)dBHt ,

0

where f is a polynomial function of k variables. The Malliavin derivative operator DH s of an element F ∈ PT is defined as follows: DH s F

(∫ T ) ∫ T k ∑ ∂f H H = ξ1 (t)dBt , . . . , ξk (t)dBt · ξi (s), s ∈ [0, T ]. ∂ xi 0 0 i=1

Since the divergence operator DH is closable from L2 (Ω , F , P) to (Ω , F , H), By D1,2 denote the Banach space be the 2 completion of PT with the following norm: ∥F ∥21,2 = E |F |2 + E ∥DH s F ∥T . Now we also introduce another derivative

DHt F =

T

∫

φ (t − s)DHs Fds. 0

The following results are well known, refer to Duncan et al. (2000), Hu (2005) and Hu and Øksendal (2003).

Q. Shi / Statistics and Probability Letters 154 (2019) 108549

3

Theorem 2.1 (Hu, 2005, Proposition 6.25). Let F : (Ω , F , P) → H be a stochastic process such that T

( ∫ 2 E ∥F ∥T +

T

∫

2 DHs Ft dsdt

< ∞.

|

| 0

)

0

Then, the Itô-type stochastic integral denoted by

(∫

T

Fs dBH s

E

)2

T

( ∫ = E ∥F ∥2T +

0

2 Fs dBH s exists in L (Ω , F , P). Moreover, E

T

∫

0

0

∫T

DHs Ft DHt Fs dsdt

)

(∫

T 0

Fs dBH s

)

= 0 and

.

0

Theorem 2.2 (Hu, 2005, Proposition 10.3). Let f , g: [0, T ] → R be deterministic continuous functions. If

∫

t

Xt = X0 +

t

∫

f (s)dBH s , t ∈ [0, T ],

g(s)ds + 0

0

where X0 is a constant and F ∈ C 1,2 ([0, T ] × R), then F (t , Xt ) = F (0, X0 ) +

∂F (s, Xs )ds + ∂s

t

∫ 0

∫ 0

t

∂F 1 (s, Xs )dXs + ∂x 2

t

∫ 0

∂ 2F d (s, Xs ) (∥f ∥2s )ds, t ∈ [0, T ]. ∂ x2 ds

∫T

Theorem 2.3 (Hu, 2005, Proposition 11.1). Let fi (s), gi (s), i = 1, 2 are in D1,2 and E 0 (|fi (s)| + |gi (s)|)ds < ∞. Assume that DHt f1 (s) and DHt f2 (s) are continuously differentiable with respect to (s, t) ∈ [0, T ] × [0, T ] for almost all ω ∈ Ω . Suppose that T

(∫

T

∫

2

|DHt fi (s)| dsdt

E 0

)

< ∞.

0

For i = 1, 2, denote t

∫

t

∫

fi (s)dBH s , t ∈ [0, T ],

gi (s)ds +

Xi (t) =

0

0

Then t

∫ X1 (t)X2 (t) =

t

∫

X1 (t)f2 (s)dBH s +

X1 (s)g2 (s)ds + 0

DHs X1 (s)f2 (s)ds

+

t

∫

X2 (t)f1 (s)dBH s

X2 (s)g1 (s)ds +

0 t

∫

t

∫ 0

0

t

∫

DHs X2 (s)f1 (s)ds.

+ 0

0

3. Generalized fractional BSDE with jumps In the practical case we will always encounter some stochastic perturbation with jumps. So we need an Itô formula for stochastic process with jumps. Let (Ω , F , (Ft )0⩽t ⩽T , P), T > 0, be a complete stochastic basis such that F0 contains all P-null elements of F and Ft + = ∩δ>0 Ft +δ = Ft and suppose that the filtration is generated by two mutually independent processes: a d-dimensional fractional Brownian motion (BH t )0⩽t ⩽T and a Poisson random measure µ on V × R+ . The space V = R − {0} is equipped with its Borel field E with compensator ν (dv, dt) = dt λ(dv ) such that {˜ µ(A × [0, t ]) = (µ − ν )(A × [0, t ]×)} is a martingale for any A ∈ E satisfying λ(A) < ∞,where λ is a σ -finite measure on E and satisfies

∫

(1 ∧ |v|2 )λ(dv ) < ∞. V

Consider process

∫ t∫ Xt = x0 + At + Mt + V

0

f (s, v, ω)˜ µ(dv, ds),

where x0 is a constant, {At }t ⩾0 is a continuous finite variational process, {Mt }t ⩾0 is a continuous local square integrable martingale, then we have the following general Itô formula (for the proof see Situ, 2005 Theorem 93): If F ∈ C 2 (R), then t

∫

F ′ (Xs )dAs +

F (Xt ) − F (x0 ) = 0

t

∫

F ′ (Xs )dMs + 0

∫ t∫ + 0

V

∫ t∫

2

∫

t

F ′′ (Xs )d⟨M ⟩s

0

[F (Xs + f (s, v, ω)) − F (Xs )]˜ µ(dv, ds) [F (Xs + f (s, v, ω)) − F (Xs ) − F ′ (Xs )f (s, v, ω)]ν (dv, ds).

+ 0

1

V

4

Q. Shi / Statistics and Probability Letters 154 (2019) 108549

Assume that • η0 is a given constant, • b, σ : [0, T ] → R are continuous deterministic, σ is differentiable and σt ̸= 0, t ∈ [0, T ]. ∫t ∫t Note that, since ∥σ ∥2t = H(2H − 1) 0 0 |u − v|2H −2 σ (u)σ (v )dudv , we have d dt

(∥σ ∥

= 2σ (t)ˆ σ (t) > 0, w here ˆ σ (t) =

2 t)

t

∫

φ (t − v )σ (v )dv, 0 ⩽ t ⩽ T . 0

Remark 3.1 (Remark 6 in Maticiuc and Nie, 2015). There exists a suitable constant M > 0 which is only dependent on H such that t 2H −1 M

ˆ σ (t) ⩽ Mt 2H −1 , 0 ⩽ t ⩽ T . σ (t)

⩽

since

∫ t ∫ 1 φ (t − v )σ (v )dv = H(2H − 1) (t − v )2H −2 σ (v )dv = H(2H − 1) (t(1 − u))2H −2 σ (tu)tdu 0 0 0 ∫ 1 2H −2 2H −1 (1 − u) σ (tu)du, = H(2H − 1)t

ˆ σ (t) =

t

∫

0

then by continuity of σ , we get the remark. We now introduce a class of reflected diffusion processes. Let G be an open connected subset of Rd , which is such that for some l ∈ C 2 (Rd ), G = {x : l(x) > 0}, ∂ G = {x : l(x) = 0} and |∇ l(x)| = 1 for x ∈ ∂ G. Note that at any boundary point x ∈ ∂ G, ∇ l(x) is a unit normal vector to the boundary, pointing towards the interior of G. Let η0 ∈ G and (ηt , Λt ) be a solution of the following reflected SDE with respect to fractional Brownian motion

ηt = η0 +

t

∫

t

∫

σ (s)dBHs +

b(s)ds + 0

t

∫

∇ l(ηs )dΛs , 0 ⩽ t ⩽ T ,

(1)

0

0

By a solution of (1), we mean a pair of processes such that η. ∈ G, Λ is a nondecreasing process, Λ0 = 0, and ∫T (ηt − a)dΛs ⩽ 0 for any a ∈ G, 0

Λt =

t

∫

1ηs ∈∂ G dΛs . 0

The existence of such a problem was shown in Lions and Sznitman (1984) for a standard Brownian motion. Remark 3.2. This problem is solved in Ferrante and Rovira (2013) for a fractional Brownian motion and a set G = (0, ∞). Given a final time T > 0, a final condition ξ , which is a FT measurable real valued random variable and the functions f : Ω × [0, T ] × R × R × R × R → R, g : Ω × [0, T ] × R × R → R, we consider the following generalized BSDE with jumps with respect to fBm with parameters (ξ , f , g , Λ) (short name GFBSDEP) Yt = ξ +

T

∫

f (s, ηs , Ys , Zs , Us )ds + t

T

∫

g(s, ηs , Ys )dΛs − t

T

∫

Zs dBH s − t

T

∫ t

∫ V

Us (v )˜ µ(dv, ds), 0 ⩽ t ⩽ T .

(2)

in order to give a probabilistic formula for the solution of a system of elliptic PDEs, this requires the new term—an integral with respect to a increasing process in Eq. (2) which is independent of Zs , Us (v ), the local time of the diffusion on the boundary. Next we introduce the following sets: 1,2 • Cpol ([0, T ]×R) is the space of all C 1,2 functions over [0, T ]×R, which together with their derivatives are of polynomial growth,

{ } 1,2 ∂ψ • V[0,T ] = Y = ψ (·, η) : ψ ∈ Cpol ([0, T ] × R), ∂ t is bounded , t ∈ [0, T ] , •˜ V[H0,T ] the completion of V[0,T ] under the following norm (where β > 0) (∫ T )1/2 (∫ T )1/2 ∥Y ∥β = t 2H −1 E [eβ Λt |Yt |2 ]dt = t 2H −1 E [eβ Λt |ψ (t , ηt )|2 ]dt , 0

•

0

˜ µ, R) the space of mapping U : Ω × [0, T ] × V → R which are P ⊗ E measurable such that ( ∫ T∫ )1/2 ∥U ∥β = E eβ Λt |Ut (v )|2 λ(dv )dt < ∞.

L2[0,T ] (

0

V

Q. Shi / Statistics and Probability Letters 154 (2019) 108549

5

We assume that the coefficients f and g of the GFBSDEP are continuous functions and satisfy the following assumption (H1): (H 1.1): There exists K > 1 s.t for 0 ⩽ t ⩽ T , (y, y′ ) ∈ R2 , (z , z ′ ) ∈ R2 , (x, x′ ) ∈ R2 , (u, u′ ) ∈ R2

|f (t , x, y, z , u) − f (t , x′ , y′ , z ′ , u′ )| ⩽ K (|y − y′ | + |x − x′ | + |z − z ′ | + |u − u′ |), |g(t , x, y) − g(t , x, y′ )| ⩽ K |y − y′ |. (H 1.2): There exist β > 0 and a function ψ with bounded derivative s.t ξ = ψ (ηT ), E(eβ ΛT |ξ |2 ) < ∞ and the integrability condition holds T

(∫ E

eβ ΛT |f (s, 0, 0, 0, 0)|2 ds +

0

T

∫

eβ ΛT |g(s, ηs , 0)|2 dΛs

)

< ∞.

0

4. Existence and uniqueness of solution 1/2

Definition 4.1. A triplet of processes (Yt , Zt , Ut )0⩽t ⩽T is called a solution to (2), if (Yt , Zt , Ut ) ∈ ˜ V[0,T ] × ˜ V[H0,T ] × L2[0,T ] (˜ µ, R) and satisfies (2). Let us recall the following result (we give a brief proof since it is an adaptation of Lemma 4.2 in Hu and Peng (2009)) that will be used next. Before we prove it ,we need another basic fact: Lemma 4.2. Assume X is a mean nonzero Gaussian with nonzero covariance, if for two continuous functions k1 (x), k2 (x) such that k1 (X ) = k2 (X ), then k1 (x) = k2 (x) for all x ∈ R. Proof. Let fX (x) denote the density function of X , we have fX (x) = √

1 2π θ

2

e

− (x−µ2) 2θ

,

where µ denote mean, θ 2 denote variance. Since k1 (X ) = k2 (X ), take expectation on both sides of this equality, we get +∞

∫

(k1 (x) − k2 (x))fX (x)dx = 0, −∞

by density of C0∞ (R) in C (R) and fX (x) ⩾ 0 for all x ∈ R, consequently k1 (x) = k2 (x) for all x ∈ R.

□

0 ,1

Lemma 4.3. Assume that h1 , h2 and h3 ∈ Cpol ([0, T ] × R) such that t

∫

h1 (s, ηs )ds +

t

∫

h2 (s, ηs )dBH s +

h3 (s, ηs )dΛs = 0, 0 ⩽ t ⩽ T . 0

0

0

t

∫

Then we have h1 (s, x) = h2 (s, x) = h3 (s, x) = 0, 0 ⩽ s ⩽ T , x ∈ R. Proof. To simplify notation, we let η0 = b(t) = 0 for all t ∈ [0, T ] in (1). Similarly to Hu and Peng (2009) Theorem 12.3, we have

) ∂ pu,s (ηu − y)h1 (s, ηs )dy σ (u)dBH u 0 R ∂x ) ∫ s (∫ ∂ + pu,s (ηu − y)h1 (s, ηs )dy ∇ l(ηu )dΛu , ∂ 0 R x

h1 (s, ηs ) = Eh1 (s, ηs ) +

∫ s (∫

where pt (x) = √

1 2π t

x2

e− 2t

and pu,s (x) = p∥σ ∥s −∥σ ∥u (x). Thus, by stochastic Fubini theorem t

∫ 0

) ∂ pu,s (ηu − y)h1 (s, ηs )dy σ (u)dBH h1 (s, ηs )ds = Eh1 (s, ηs )ds + u ds 0 0 0 R ∂x ) ∫ t ∫ s (∫ ∂ + pu,s (ηu − y)h1 (s, ηs )dy ∇ l(ηu )dΛu ds 0 0 R ∂x ∫

t

∫ t ∫ s (∫

6

Q. Shi / Statistics and Probability Letters 154 (2019) 108549

) ∂ pu,s (ηu − y)h1 (s, ηs )dyds dBH u u R ∂x 0 (∫ t ∫0 ) ∫ t ∂ + ∇ l(ηu ) pu,s (ηu − y)h1 (s, ηs )dyds dΛu ∂ u R x 0 ) (∫ t ∫ ∫ t ∫ t ∂ = Eh1 (s, ηs )ds + [h2 (u, ηu ) + σ (u) pu,s (ηu − y)h1 (s, ηs )dyds ]dBH u u R ∂x 0 0 (∫ t ∫ ) ∫ t ∂ + [h3 (u, ηu ) + ∇ l(ηu ) pu,s (ηu − y)h1 (s, ηs )dyds ]dΛu u R ∂x 0 ∫ t ∫ t h3 (u, ηu )dΛu , − h2 (u, ηu )dBH u − ∫

t

Eh1 (s, ηs )ds +

=

∫

t

σ (u)

(∫ t ∫

0

0

Thus from assumption, we have t

∫

Eh1 (s, ηs )ds = 0, 0

] ∂ pu,s (ηu − y)h1 (s, ηs )dyds dBH u = 0, 0 u R ∂x ] ∫ t[ ∫ t∫ ∂ h3 (u, ηu ) + ∇ l(ηu ) pu,s (ηu − y)h1 (s, ηs )dyds dΛu = 0. 0 u R ∂x ∫t ∫ ∂ ∫t ∫ But h2 (u, ηu ) + σ (u) u R ∂ x pu,s (ηu − y)h1 (s, ηs )dyds and h3 (u, ηu ) + ∇ l(ηu ) u R ∂∂x pu,s (ηu − y)h1 (s, ηs )dyds are Fu adapted (since these are a function of ηu ). So from Theorem12.1 of Hu and Peng (2009), we see that ∫ t∫ ∂ h2 (u, ηu ) + σ (u) pu,s (ηu − y)h1 (s, ηs )dyds = 0, u R ∂x ∫ t∫ ∂ h3 (u, ηu ) + ∇ l(ηu ) pu,s (ηu − y)h1 (s, ηs )dyds = 0. ∂ u R x In our situation, (ηu , Λu ) is a solution of reflected SDE with respect to fractional Brownian motion ∫ s ∫ s H ∇ l(ηu )dΛu , 0 ⩽ u ⩽ s, σ (u)dBu + ηu = ∫ t[

h2 (u, ηu ) + σ (u)

∫ t∫

0

0

where Λ is a nondecreasing process, and

Λu =

∫

s

1ηu ∈∂ G dΛu .

0

Although ηu is not center Gaussian process, but by Lemma 4.2, we have h2 (u, z) + σ (u)

∫ t∫ u

h3 (u, z) + ∇ l(z)

R

∫ t∫ u

R

∂ pu,s (ηu − y)h1 (s, z)dyds = 0, ∂x

(3)

∂ pu,s (ηu − y)h1 (s, z)dyds = 0. ∂x

(4)

for all z ∈ R. Next, the step is same as Lemma 4.2 of Hu and Peng (2009), and consequently h1 (s, z) = 0 for all 0 ⩽ s ⩽ T , z ∈ R. Finally, bringing h1 (s, z) = 0 into the formulas (3) and (4), h2 (u, z) = 0, h3 (u, z) = 0 are then an immediate consequence for all 0 ⩽ s ⩽ T , z ∈ R. □ It is well known the following Lemma (refer to Hu and Peng, 2009). Lemma 4.4. Let (Yt , Zt )0⩽t ⩽T be a solution of the GFBSDE (2) with U = 0. Then we have the stochastic representation

DHt Yt =

ˆ σ (t) Zt , 0 ⩽ t ⩽ T , σ (t)

Now ,we give the main result of this paper: Theorem 4.5. Assume (H1) holds. Then there exists a unique solution of (2). Moreover, for all t ∈ [0, T ],

( E

eβ Λs |Yt |2 +

T

∫ t

eβ Λs s2H −1 |Zs |2 ds +

T

∫ t

eβ Λs |Ys |2 dΛs

)

T

∫

∫

eβ Λs |Us (v )|2 λ(dv )ds ⩽ C Θ (t , T ),

+E t

V

Q. Shi / Statistics and Probability Letters 154 (2019) 108549

7

where T

( ∫ Θ (t , T ) := E eβ ΛT |ξ |2 +

T

∫

eβ Λs |f (s, 0, 0, 0, 0)|2 ds +

t

T

∫

eβ Λs |ηs |2 ds +

t

eβ Λs |g(s, ηs , 0)|2 dΛs

)

.

t

Proof. First we will show the second part of the above theorem. Assume that (Y , Z , U) is a solution of (2). By C we will denote a constant which may vary from line to line. From the Itô formula T

∫

eβ Λt |Yt |2 + 2

eβ Λs (DH s Ys )Zs ds + β

t

β ΛT

eβ Λs |Ys |2 dΛs +

T

∫

t T

∫

T

∫

∫

t

∫

T

|ξ | + 2 e |Ys |f (s, ηs , Ys , Zs , Us )ds + 2 eβ Λs |Ys |g(s, ηs , Ys )dΛs t t ∫ T∫ ∫ T µ(dv, ds) + 2 eβ Λs |Ys |Us (v )˜ +2 eβ Λs |Ys |Zs dBH s 2

=e

β Λs

eβ Λs |Us (v )|2 ν (dv, ds) V

t

t

V

By Lipschitz continuity of f and g we have 2yf (t , η, y, z , u) ⩽ 2K |y|(|η| + |y| + |z | + |u|) + 2|y||f (t , 0, 0, 0, 0)| MK 2

⩽ (4K 2 + 2K +

1

1

2

M

+ 1)|y|2 + |η|2 + |u|2 +

s2H −1

s2H −1 |z |2 + |f (t , 0, 0, 0, 0)|2

2yg(t , η, y) ⩽ 2K |y|2 + 2|y||g(t , η, 0)| ⩽ (2K + 1)|y|2 + |g(t , η, 0)|2 There, we can write

( E

eβ Λt |Yt |2 +

M

⩽ E(eβ ΛT |ξ |2 ) + 2E ⩽ E(e

β ΛT

t T

∫

∫

2

T

∫

2

eβ Λs s2H −1 |Zs |2 ds + β

T

∫

T

∫

MK 2

2

(4K + 2K +

eβ Λs (|ηs |2 + |Us |2 )ds +

s2H −1

2

(4K + 2K + t

1

+

M

T

∫ E

∫

T

∫

eβ Λs |Us (v )|2 λ(dv )ds V

eβ Λs |Ys |g(s, ηs , Ys )dΛs

M

s2H −1 eβ Λs |Zs |2 ds +

t

1 2

+ 1)e

β Λs

T

∫

2

eβ Λs |Ys |2 dΛs

|Ys | ds + (2K + 1)E t

s2H −1 eβ Λs |Zs |2 ds

E t

∫

t T

T

∫

1

eβ Λs |f (s, 0, 0, 0, 0)|2 ds + E

⩽ Θ (t , T ) + E

T

∫ +E

t

t T

|ξ | ) + E

∫

)

t

eβ Λs |Ys |f (s, ηs , Ys , Zs , Us )ds + 2E

t

+E

eβ Λs |Ys |2 dΛs

t

t

+E

T

∫

MK

T

t 2

s2H −1 T

∫

1 eβ Λs |g(s, ηs , 0)|2 dΛs + + E 2

+ 1)e ∫

β Λs

t T

∫

2

T

∫

∫

eβ Λs |Us (v )|2 λ(dv )ds V

eβ Λs |Ys |2 dΛs

|Ys | ds + (2K + 1)E t

eβ Λs |Us (v )|2 λ(dv )ds

E t

V

Choosing β ⩾ (2K + 2), we get

(

β Λt

1

∫

T

β Λs 2H −1

∫

T

β Λs

)

1

|Yt | + e s |Zs | ds + e |Ys | dΛs + E M t 2 t ∫ T 2 MK ⩽ Θ (t , T ) + E (4K 2 + 2K + 2H −1 + 1)eβ Λs |Ys |2 ds. E

e

2

2

2

T

∫

∫

t

eβ Λs |Us (v )|2 λ(dv )ds V

s

t

By Gronwall’s inequality, β Λt

Ee

{

|Yt | ⩽ Θ (t , T ) exp (4K + 2K + 1)(T − t) + MK 2

2

2T

2H −1

− t 2H −1 2 − 2H

}

and also get T

(∫ E

eβ Λs s2H −1 |Zs |2 ds +

t

T

∫

eβ Λs |Ys |2 dΛs

t

)

T

∫

∫

eβ Λs |Us (v )|2 λ(dv )ds ⩽ C Θ (t , T ).

+E t

V

Now we will prove the existence and uniqueness of the solution of (1). The method used here is the fixed point 1/2 1/2 theorem. We will show that the mapping Γ : ˜ V[0,T ] × ˜ V[H0,T ] × L2[0,T ] (˜ µ, R) → ˜ V[0,T ] × ˜ V[H0,T ] × L2[0,T ] (˜ µ, R) given by

(X , W , ∆) → Γ (X , W , ∆) = (Y , Z , U) is a contraction, where (Y , Z , U) is a solution of the following generalized BSDE: Yt = ξ +

T

∫

f (s, ηs , Xs , Ws , ∆s )ds + t

T

∫

g(s, ηs , Xs )dΛs − t

T

∫

Zs dBH s − t

T

∫ t

∫ V

Us (v )˜ µ(dv, ds)

8

Q. Shi / Statistics and Probability Letters 154 (2019) 108549

1/2 , i = 1, . . . , k + 1. First we will show that Γ is a contraction on ˜ V[ t , T ] × ˜ V[Ht ,T ] × L2[t ,T ] (˜ µ, R). Take k k k H 2 X, X′ ∈ ˜ W, W′ ∈ ˜ V[t ,T ] and ∆, ∆′ ∈ L[t ,T ] (˜ µ , R), let Γ (X , W , ∆) = (Y , Z , U) and Γ (X ′ , W ′ , ∆′ ) = (Y ′ , Z ′ , U ′ ) and k k let Y = Y − Y ′ , Z = Z − Z ′ , U = U − U ′ , X = X − X ′ , W = W − W ′ , ∆ = ∆ − ∆′ . From Itô formula, for t ∈ [tk , T ], we have ( ) ∫ T ∫ T ∫ T∫ 2 2 2 β Λs E eβ Λt |Y t | + 2 eβ Λs (DH Y )Z ds + β e | Y | d Λ + E eβ Λs |U s (v )| µ(dv, ds) s s s s s t t t V ∫ T ′ ′ ′ β Λs = 2E e |Y s |(f (s, ηs , Xs , Ws , ∆s ) − f (s, ηs , Xs , Ws , ∆s ))ds t ∫ T + 2E eβ Λs |Y s |(g(s, ηs , Xs ) − g(s, ηs , Xs′ ))dΛs i−1 T k

Let k ∈ N and ti = 1/2 V[ t , T ] , k

t

Note that 2|ys |(f (s, ηs , xs , ws , ∆s ) − f (s, ηs , x′s , ws′ , ∆′s )) ⩽ 2K |ys |(|xs | + |w s | + |∆s |). K2

2|ys |(g(s, ηs , xs ) − g(s, ηs , x′s )) ⩽ 2K |ys ||xs | ⩽ Choose β =

|ys |2 + α|xs |2 for some α > 0.

α

K2

+ 1. Then by the Schwartz inequality we obtain ) ∫ T∫ ∫ T ∫ T 2 2 2 2 2 β Λt β Λs 2H −1 β Λs eβ Λs |U s (v )| µ(dv, ds) E e |Y t | + e s |Z s | ds + e |Y s | dΛs + E M t t V t ∫ T ∫ T 2 β Λs β Λs e |Y s |(|X s | + |W s | + |∆s |)ds + α E e |X s | dΛs = 2KE t t ∫ T( ∫ T ) )1/2 2 1/2 ( β Λ 2 Ee s (|X s | + |W s | + |∆s |)2 ds + α E eβ Λs |X s | dΛs . ⩽ 2K Eeβ Λs |Y s | α

(

t

t

) ∫T 2 1/2 2 Denote ϕ (t) = Eeβ Λs |Y s | and ψ (t) = α E t eβ Λs |X s | dΛs which is nonincreasing. Then by above (

ϕ (t)2 ⩽ 2K

T

∫

( )1/2 ϕ (t) Eeβ Λs (|X s | + |W s | + |∆s |)2 ds + ψ (t), t ∈ [tk , T ].

t

Applying Lemma 20 in Maticiuc and Nie (2015) to the above inequality we get

ϕ (t) ⩽

√

T

∫

Eeβ Λs (|X s | + |W s | + |∆s |)2

(

2K

)1/2

ds +

√ ψ (t), t ∈ [tk , T ].

t

and therefore for t ∈ [tk , T ] β Λs

Ee

2

|Y s | ⩽ 4K

2

T

(∫

(

Ee

β Λs

(|X s | + |W s | + |∆s |)

2 1/2

)

)2 ds

+ 2ψ (t),

t

Integrating both sides on [tk , T ] of above inequality, we can compute

∫

T

ϕ (s) ds ⩽ 2ψ (tk )(T − tk ) + 4K 2

2

∫

tk

T

T

(∫

tk

(

β Λs

Ee

2 1/2

(|X s | + |W s | + |∆s |)

t

⩽ 2ψ (tk )(T − tk ) + 12K 2 (T − tk )

T

(∫

2

(Eeβ Λs |X s | )1/2 ds

)2

tk 2

(∫

T

(

+ 12K (T − tk )

s2H −1

tk

+ 12K 2 (T − tk )

T

(∫

1

β Λs 2H −1

Ee

s

|W s |

2

)2

)1/2

ds

)2

2

(Eeβ Λs |∆s | )1/2 ds

tk

⩽ 2ψ (tk )(T − tk ) + 12K 2 (T − tk )2 E

∫

T

2

eβ Λs |X s | ds

tk

+ 12K 2 (T − tk )

T

∫

tk

+ 12K 2 (T − tk )2 E

∫

1 s2H −1 T

T

∫ dsE

tk 2

eβ Λs |∆s | ds

tk

˜ (tk , T ), := C · (T − tk )Θ

2

eβ Λs s2H −1 |W s | ds

)

)2 ds

dt

Q. Shi / Statistics and Probability Letters 154 (2019) 108549

9

and similarly T

∫

1

ϕ (s)2 ds ⩽

s2H −1

tk

C 2 − 2H

˜ (tk , T ), · (T 2−2H − tk2−2H ) · Θ

where

(∫

˜ (tk , T ) = E Θ

T

2

eβ Λs s2H −1 |W s | ds +

∫

tk

T

2

eβ Λs |X s | (ds + dΛs )

)

T

∫

∫

2

eβ Λs |∆s (v )| λ(v )ds.

+E

tk

tk

V

Using above inequalities, we deduce T

(∫

tk T

∫

2

eβ Λs |Y s | (ds + dΛs )

e

2

|Y s | ds + C α E

T

∫

e

tk T

∫

T

∫

∫

2

|X s | dΛs + CE

T

∫

β Λs

e tk

2

2

2

eβ Λs |U s (v )| λ(v )ds

+E tk

β Λs

tk

+ CE

)

tk

β Λs

⩽ E

T

∫

2

eβ Λs s2H −1 |Z s | ds +

E

1

α

V

1

2

|Y s | (2 +

s2H −1

)ds

2

eβ Λs α (|X s | + s2H −1 |W s | + |∆s | )ds

tk

˜ (tk , T ) + ⩽ C · (T − tk )Θ (

⩽ C

α + (2 +

1

α

C

T

∫

α

ϕ (s)(2 + tk

)(T − tk ) +

1

1 s2H −1

˜ (tk , T ) )ds + C α Θ

(T 2−2H − tk2−2H )

α

)

˜ (tk , T ) Θ

Choosing α such that C α ⩽ 1/4 and taking k large enough that C (α + 2)(T − tk )/α ⩽ 1/4 and C (T 2−2H − tk2−2H )/α ⩽ 1/4, we obtain T

(∫ E

2

eβ Λs s2H −1 |Z s | ds +

tk

⩽

3 4

T

∫

tk

k ,T ]

)

∫

T

∫

+E tk

2

eβ Λs |U s (v )| λ(v )ds

V

˜ (tk , T ) Θ 1/2 k ,T ] 1/2 V[ t , T ] k

Thus Γ is contraction operator in ˜ V[ t L2[t

2

eβ Λs |Y s | (ds + dΛs )

1/2 ט V[Ht ,T ] × L2[t ,T ] (˜ µ, R), and (Y n , Z n , U n ) is a Cauchy sequence in ˜ V[ t , T ] × ˜ V[Ht ,T ] × k k k k

(˜ µ, R), where (Y 0 , Z 0 , U 0 ) ∈ ˜ Ytn+1

ט V[Ht ,T ] × L2[t ,T ] (˜ µ, R), and for n ⩾ 0 k k ∫ T ∫ T g(s, ηs , Ysn )dΛs := ξ + f (s, ηs , Ysn , zsn , Usn )ds + t t ∫ T ∫ T∫ n+1 H − Zs dBs − Usn+1 (v )˜ µ(dv, ds) t

t

V

1/2 V[ t , T ] k

µ, R) being a limit of (Y n , Z n , U n ), i.e. ט V[Ht ,T ] × L2[t ,T ] (˜ k k ( ) ∫ T 2 2 2 eβ Λs (|Ysn − Ys | + s2H −1 |Zsn − Zs | )ds = 0, lim E eβ Λt |Ytn − Yt | +

Then there exists (Y , Z , U) ∈ ˜

n→+∞

tk

(∫

T

n→+∞

eβ Λs |Ysn − Ys | dΛs 2

lim E

= 0,

tk

(∫

T

∫

eβ Λs |Usn (v ) − Us (v )| λ(dv )ds 2

lim E

n→+∞

)

tk

)

= 0,

V

Therefore for any t ∈ [tk , T ],

( ) ∫ T ∫ T ∫ T∫ n+1 n n n n n+1 lim −Yt +ξ + f (s, ηs , Ys , Zs , Us )ds + g(s, ηs , Ys )dΛs − Us (v )˜ µ(dv, ds) n→∞ t t t V ∫ T ∫ T ∫ T∫ = − Yt + ξ + f (s, ηs , Ys , zs , Us )ds + g(s, ηs , Ys )dΛs − Us (v )˜ µ(dv, ds) t

t

t

V

in L2 (Ω , F , P) and Z n 1[t ,T ] → Z 1[t ,T ] in L2 (Ω , F , H). We show (Y , Z , U) that satisfies (1) on [tk , T ]. The next step is to solve the equation on [tk−1 , tk ]. With the same arguments, repeating the above technique we obtain a uniqueness of the solution of generalized BSDEP with respect to fBm on the whole interval [0, T ]. □

10

Q. Shi / Statistics and Probability Letters 154 (2019) 108549

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