General Linear Quadratic Optimal Stochastic Control Problem Driven by a Brownian Motion and a Poisson Random Martingale Measure with Random Coefficients
For researchers in stochastic control, this paper provides a theoretical foundation for solving LQ problems with jumps and random coefficients, but it is an incremental extension of existing theory.
This paper solves the general linear quadratic optimal stochastic control problem with random coefficients, where the system is driven by Brownian motion and a Poisson random martingale measure. It establishes connections between the multidimensional backward stochastic Riccati equation with jumps and the optimal control problem, proving the existence and uniqueness of the Riccati equation and showing that the optimal control has a state feedback representation.
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.