Forward-Backward Doubly Stochastic Differential Equations with Brownian Motions and Poisson Process.doc

Forward-Backward Doubly Stochastic Differential Equations with Brownian Motions and Poisson Process ∗
Zhu Qingfeng
School of Statistics and Mathematics, Shandong University of Finance, Jinan 250014
Shi Yufeng
School of Mathematics and System Science, Shandong University, Jinan 250100
Abstract The existence and uniqueness for solution of backward doubly stochastic dif- ferential equations with Brownian motions and Poisson process and that of forward-backward doubly stochastic differential equations with Brownian motions and Poisson process can be obtained by means of a method of continuation. Furthermore the continuity of the solutions of FBDSDEP depending on parameters is also proved.
Key Words Forward-backward doubly stochastic differential equations; stochastic anal- ysis; random measure; Poisson process
1 Introduction
Let (Ω, F , P ) be a probability space, and [0, T ] be a fixed arbitrarily large time duration throughout this paper. We suppose {Ft }t≥0 is generated by the following three mutually independent processes:
(i) Let {Wt ; 0 ≤ t ≤ T } and {Bt ; 0 ≤ t ≤ T } be two standard Brownian motions defined on
(Ω, F , P ), with values respectively in Rd and in Rl .
(ii) Let N be a Poisson random measure, on R+ × X, where X ⊂ Rl is a nonempty
open set equipped with its Borel field B(X ), pensator Nb (dx, dt) = λ(dx)dt, such that
Ne (A × [0, t]) = (N − Nb )(A × [0, t])t≥0 is a martingale for all A ∈ B(X ) satisfying λ(A) < ∞. λ
is assumed to be a σ-finite measure on (X, B(X )) and is called the characteristic measure.
Let N denote the class of P -null elements of F . For each t ∈[0, T ], we define
.
Ft = F W ∨ F B ∨ F N
t t,T t
where for any process {ηt }, F η
= σ{ηr −ηt ; s ≤ r ≤ t} ∨ N , F η= F η. Note that the collec-
s,t
t 0,t
tion {Ft , t
∈[0, T
]} is neither increasing nor decreasing, and it does not constitute mon
filtration.
For any n ∈ N , let M 2 (0, T ; Rn ) denote the set of (classes of dP ⊗ dt . equal) n− dimen-
T 2