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

TL;DR

This paper studies the existence and uniqueness of solutions to backward stochastic Riccati equations with jumps, which are linked to stochastic linear quadratic control problems involving random coefficients, Brownian motion, and Poisson jumps.

Contribution

It establishes the solvability of matrix-valued BSREJs with jumps, addressing challenges from nonlinearity and jump processes, and provides a verification theorem for uniqueness.

Findings

Proved existence and uniqueness of solutions to BSREJ with jumps.

Developed techniques to handle inverse flow issues caused by jumps.

Provided a verification theorem ensuring solution uniqueness.

Abstract

In this paper, we investigate the solvability of matrix valued Backward stochastic Riccati equations with jumps (BSREJ), which is associated with a stochastic linear quadratic (SLQ) optimal control problem with random coefficients and driven by both Brownian motion and Poisson jumps. By dynamic programming principle, Doob-Meyer decomposition and inverse flow technique, the existence and uniqueness of the solution for the BSREJ is established. The difficulties addressed to 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 jumps may not exist without additional technical condition, and ii) how to show the inverse matrix term involving jump process in the generator is well-defined. Utilizing the structure of the optimal problem, we overcome these difficulties and establish the existence of the solution. In additional, a verification theorem for BSREJ is given which implies the uniqueness of the solution.