Indefinite Stochastic Linear-Quadratic Optimal Control Problems with Random Coefficients and Poisson Jumps: Closed-Loop Representation of Open-Loop Optimal Controls

This paper is concerned with stochastic linear-quadratic (SLQ) optimal control problems with random coefficients and Poisson jumps.

The weighting matrices are allowed to be random and indefinite.

Under the uniform convexity condition, the global fundamental matrix representation $P=\mathbf Y\mathbf X^{-1}$, used in the diffusion case, is generally unavailable because Poisson jumps may cause the optimal state fundamental matrix $\mathbf X$ to become singular.

We construct the process $P$ directly from the stochastic value flow and prove that the associated stochastic Riccati equation with jumps (SRE-J) admits a unique maximal strongly regular solution, which gives a closed-loop representation of the unique open-loop optimal control.

We also give sufficient conditions for uniform convexity and present indefinite SLQ examples with jumps.