Comparison theorems for multidimensional BSDEs with jumps and applications to constrained stochastic linear-quadratic control

Abstract

In this paper, we, for the first time, establish two comparison theorems for multidimensional backward stochastic differential equations with jumps. Our approach is novel and completely different from the existing results for the one-dimensional case. Using these and other delicate tools, we then construct solutions to coupled two-dimensional stochastic Riccati equation with jumps in both standard and singular cases. In the end, these results are applied to solve a cone-constrained stochastic linear-quadratic control problem and a mean-variance portfolio selection problem with jumps. Different from no-jump problems, the optimal (relative) state processes may change their signs, which is of course due to the presence of jumps.