标题：OPTIMAL CONTROL PROBLEMS OF FULLY COUPLED FBSDEs AND VISCOSITY SOLUTIONS OF HAMILTON-JACOBI-BELLMAN EQUATIONS
作者：Li, Juan; Wei, Qingmeng
作者机构：[Li, Juan] Shandong Univ, Sch Math & Stat, Weihai 264209, Peoples R China.; [Wei, Qingmeng] NE Normal Univ, Sch Math & Stat, Changchun 130024, Peopl 更多
通讯作者地址：[Wei, QM]NE Normal Univ, Sch Math & Stat, Changchun 130024, Peoples R China.
来源：SIAM JOURNAL ON CONTROL AND OPTIMIZATION
关键词：fully coupled FBSDEs; value functions; stochastic backward semigroup;; dynamic programming principle; viscosity solution
摘要：In this paper we study stochastic optimal control problems of fully coupled forward-backward stochastic differential equations (FBSDEs). The recursive cost functionals are defined by controlled fully coupled FBSDEs. We use a new method to prove that the value functions are deterministic, satisfy the dynamic programming principle, and are viscosity solutions to the associated generalized Hamilton-Jacobi-Bellman (HJB) equations. For this we generalize the notion of stochastic backward semigroup introduced by Peng Topics on Stochastic Analysis, Science Press, Beijing, 1997, pp. 85-138. We emphasize that when sigma depends on the second component of the solution (Y, Z) of the BSDE it makes the stochastic control much more complicated and has as a consequence that the associated HJB equation is combined with an algebraic equation. We prove that the algebraic equation has a unique solution, and moreover, we also give the representation for this solution. On the other hand, we prove a new local existence and uniqueness result for fully coupled FBSDEs when the Lipschitz constant of sigma with respect to z is sufficiently small. We also establish a generalized comparison theorem for such fully coupled FBSDEs.