Linear–quadratic optimal control of infinite-dimensional stochastic evolution equation with jumps

This paper discusses a stochastic linear–quadratic optimal control problem with jumps in an infinite-dimensional Hilbert space. The state equation of this linear–quadratic optimal control problem is a stochastic evolution equation driven by a Poisson random martingale measure and a one dimensional Brownian motion. The cost functional is a quadratic generalized function consisting of a state process and a control process. In order to ensure the suitability and solvability of the problem, firstly, by using the infinite-dimensional stochastic analysis theory, two types of semilinear forward and backward stochastic evolution equations are investigated separately to prove the continuous dependence on the generating elements as well as the existence and uniqueness of the solutions. Secondly, through Yosida approximation theory, a new infinite-dimensional duality relation is constructed between the state equations and the adjoint equations, which is used to obtain the dual representation of the optimal control and the solvability of the infinite-dimensional stochastic Hamiltonian system. Here the stochastic Hamiltonian system consisting of state equations, adjoint equations and stationarity conditions is a infinite-dimension fully coupled forward backward stochastic evolution equations. Finally, an infinite-dimensional Riccati equation for the control system is introduced to decouple the stochastic Hamiltonian system, and the state feedback representation of the optimal control and the corresponding value function are derived.