General Linear Quadratic Optimal Stochastic Control Problem Driven by a Brownian Motion and a Poisson Random Martingale Measure with Random Coefficients

The main purpose of this paper is to discuss detailed the stochastic LQ control problem with random coefficients where the linear system is a multidimensional stochastic differential equation driven by a multidimensional Brownian motion and a Poisson random martingale measure. In the paper, we will establish the connections of the multidimensional Backward stochastic Riccati equation with jumps (BSRDEJ in short form) to the stochastic LQ problem and to the associated Hamilton systems. By the connections, we show the optimal control have the state feedback representation. Moreover, we will show the existence and uniqueness result of the multidimensional BSRDEJ for the case where the generator is bounded linear dependence with respect to the unknowns martingale term

A Maximum Principle for Optimal Control of Stochastic Evolution Equations

A general maximum principle is proved for optimal controls of abstract semilinear stochastic evolution equations. The control variable, as well as linear unbounded operators, acts in both drift and diffusion terms, and the control set need not be convex.Comment: 20 page

A global maximum principle for optimal control of general mean-field forward-backward stochastic systems with jumps

In this paper we prove a necessary condition of the optimal control problem for a class of general mean-field forward-backward stochastic systems with jumps in the case where the diffusion coefficients depend on control, the control set does not need to be convex, the coefficients of jump terms are independent of control as well as the coefficients of mean-field backward stochastic differential equations depend on the joint law of $(X(t),Y(t))$. Two new adjoint equations are brought in as well as several new generic estimates of their solutions are investigated for analysing the higher terms, especially, those involving the expectation which come from the derivatives of the coefficients with respect to the measure. Utilizing these subtle estimates, the second-order expansion of the cost functional, which is the key point to analyse the necessary condition, is obtained, and whereafter the stochastic maximum principle.Comment: 32 page

Notes on the Cauchy Problem for Backward Stochastic Partial Differential Equations

Backward stochastic partial differential equations of parabolic type with variable coefficients are considered in the whole Euclidean space. Improved existence and uniqueness results are given in the Sobolev space $H^n$ ($=W^n_2$) under weaker assumptions than those used by X. Zhou [Journal of Functional Analysis 103, 275--293 (1992)]. As an application, a comparison theorem is obtained.Comment: 20 page