Open-loop and closed-loop saddle points of infinite dimensional linear–quadratic stochastic differential games with Poisson jumps

This paper investigates linear–quadratic stochastic zero-sum differential games with Poisson jumps (SLQGP) in infinite-dimensional spaces, establishing necessary and sufficient conditions for the existence of both open-loop and closed-loop saddle points. By employing mild solutions to address unbounded operators and Yosida approximation techniques to derive state-dual process relations, we characterize open-loop saddle points through constrained linear forward–backward stochastic evolution equations with Poisson jumps and convexity–concavity conditions. Furthermore, closed-loop saddle points are shown to be linked to the regular solution of a Riccati differential equation. To illustrate the practical applicability of our theoretical framework, we analyze a controlled stochastic heat equation with Poisson jumps, providing a concrete example that demonstrates the effectiveness of our approach.