带跳的正向-倒向重随机微分方程的极大值原理.docVIP

 Maximum principles for forward-backward doubly stochastic differential equations with jumps# Xu Shuli, Jiang Jun* 5 10 (Hubei Province Key Laboratory of Systems Science in Metallurgical Process, Wuhan University of Science and Technology, WuHan 430081) Abstract: The forward-backward stochastic differential equations has received considerable research attention in a large of domains, especially in mathematical finance. The subject of stochastic maximum principles for forward-backward stochastic optimal control problems has been discussed by many authors, this paper researchs a stochastic system consisting of a forward-backward doubly stochastic differential equations with jump and obtains a genereal sufficient maximum principle for forward-backward doubly stochastic differential equations with jump. Keywords: Maximum principle; Forward-backward stochastic differential equations; Jump; Stockastic optimal control; Adjoint equation 15 0 Introduction We let (?, F , F,P) be a complete filtered probability space and T?? 0 be fixed throughout this paper, with F?? {Ft }t≥0 being its natural filtration augmented by all the P?? null sets. Let W (t ) : 0?≤ t?≤ T B(t ) : 0?≤ t?≤ T  be two mutually independent standard one- 20 dimensional Brownian motions, Let N denote the class of P?? null sets of G/!For each t?∈ [0, T ] , we define!Gu !Gux?∨ Gu-UC , where Gux?? N?∨ {W (r)?? W (0) : 0?≤ r?≤ t} ,! Gu-UC?? N?∨ {B(r )?? B(t ) : t?≤ r?≤ T } . Let?? (t )???? (t , w); t?≥ 0, w?∈?? be a Levy process on (?, F , F,P) . For simplicity we 2 25  t 0  ∫  0  sN (dm, ds);  t?≥ 0 , where a, b are constants,  0 ? \{0} and N (dt, ds)?? N (dt, ds)?? v(ds)dt is the compensation of the jump measure N (?,??) of?? (t ) , v being the Levy measure of?? (t ) . We consider a stochastic control problem, the system is governed by a forward-backward doubly stochastic differential equation driven by levy processes of the form: 30 ?dY (t)???? g (t, X (t), Y (t ), Z (t ), u