Backward Stochastic Riccati Equation with Jumps Associated with Stochastic Linear Quadratic Optimal Control with Jumps and Random Coefficients

Abstract

In this paper, we investigate the solvability of matrix valued backward stochastic Riccati equations with jumps (BSREJ), which are associated with a stochastic linear quadratic (SLQ) optimal control problem with random coefficients and driven by both Brownian motion and a Poisson process. By the dynamic programming principle, Doob--Meyer decomposition, and inverse flow technique, the existence and uniqueness of the solution for the BSREJ are established. The difficulties addressed in this issue not only are brought from the high nonlinearity of the generator of the BSREJ like the case driven only by Brownian motion, but also from that (i) the inverse flow of the controlled linear stochastic differential equation driven by Poisson process may not exist without additional technical conditions, and (ii) showing that the inverse matrix term involving a jump process in the generator is well-defined. Utilizing the structure of the optimal control problem, we overcome these difficulties and establish the existence of the solution. In addition, we show the construction of the optimal feedback control with the help of the Riccati equation and the relation between the solution of the Riccati equation and the value function of the SLQ problem, which also implies the uniqueness of BSREJ.