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.

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