Constrained zero-sum LQ differential games for jump-diffusion systems with random coefficients

TL;DR

This paper studies a constrained stochastic differential game with jump-diffusion systems, establishing saddle point solutions via advanced stochastic Riccati equations under regime switching and random coefficients.

Contribution

It introduces a novel approach to characterize saddle points in constrained stochastic differential games with jumps using extended Riccati equations.

Findings

Established open-loop saddle point characterization under UCC condition.

Derived a closed-loop saddle point representation using multidimensional indefinite stochastic Riccati equations.

Proved existence of solutions to these Riccati equations in a special case.

Abstract

This paper investigates a cone-constrained two-player zero-sum stochastic linear-quadratic (SLQ) differential game for stochastic differential equations (SDEs) with regime switching and random coefficients driven by a jump-diffusion process. Under the uniform convexity-concavity (UCC) condition, we establish the open-loop solvability of the game and characterize the open-loop saddle point via the forward-backward stochastic differential equations (FBSDEs). However, since both controls are constrained, the classical four-step scheme fails to provide an explicit expression for the saddle point. To overcome this, by employing Meyer's It\^o formula together with the method of completing the square, we derive a closed-loop representation for the open-loop saddle point based on solutions to a new kind of multidimensional indefinite extended stochastic Riccati equations with jumps (IESREJs). Furthermore, for a special case, we prove the existence of solutions to IESREJs.