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The main purpose of this paper is to discuss detailed the stochastic LQcontrol problem with random coefficients where the linear system is amultidimensional stochastic differential equation driven by a multidimensionalBrownian motion and a Poisson random martingale measure. In the paper, we willestablish the connections of the multidimensional Backward stochastic Riccatiequation with jumps (BSRDEJ in short form) to the stochastic LQ problem and tothe associated Hamilton systems. By the connections, we show the optimalcontrol have the state feedback representation. Moreover, we will show theexistence and uniqueness result of the multidimensional BSRDEJ for the casewhere the generator is bounded linear dependence with respect to the unknownsmartingale term.

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