Infinite-Dimensional Backward Stochastic Linear-Quadratic Optimal Control Problem with Jumps

This paper studies a linear-quadratic optimal control problem for backward stochastic differential equations with jumps in an infinite-dimensional Hilbert space, driven by a Brownian motion and a compensated Poisson random measure. We first formulate strong and mild solutions and introduce the Yosida approximation to justify the use of the Itô formula in the presence of an unbounded generator. We then establish existence and uniqueness of the optimal control by a convex variational argument. By constructing the adjoint equation, we derive the stationarity condition and the associated infinite-dimensional stochastic Hamiltonian system. Finally, under solvability and invertibility assumptions for the operator Riccati equation, we obtain a feedback representation of the optimal control.