The paper is concerned with a class of stochastic differential equations in infinite dimensional Hilbert space with random coefficients driven by Teugels martingales which are more general processes and the corresponding optimal control problems. Here Teugels martingales are a family of pairwise strongly orthonormal martingales associated with Lévy processes (see Nualart and Schoutens ^[21]). There are three major ingredients. The first is to prove the existence and uniqueness of the solutions by continuous dependence theorem of solutions combining with the parameter extension method. The second is to establish the stochastic maximum principle and verification theorem for our optimal control problem by the classic convex variation method and dual techniques. The third is to represent an example of a Cauchy problem for a controlled stochastic partial differential equation driven by Teugels martingales which our theoretical results can solve.

In this paper, a stochastic linear-quadratic (LQ, for short) optimal control problem with jumps in an infinite horizon is studied, where the state system is a controlled linear stochastic differential equation containing affine term driven by a one-dimensional Brownian motion and a Poisson stochastic martingale measure, and the cost functional with respect to the state process and control process is quadratic and contains cross terms. Firstly, in order to ensure the well-posedness of our stochastic optimal control of infinite horizon with jumps, the $L^2$-stabilizability of our control system with jump is introduced. Secondly, it is proved that the $L^2$-stabilizability of our control system with jump is equivalent to the non-emptiness of the admissible control set for all initial state and is also equivalent to the existence of a positive solution to some integral algebraic Riccati equation (ARE, for short). Thirdly, the equivalence of the open-loop and closed-loop solvability of our infinite horizon optimal control problem with jumps is systematically studied. The corresponding equivalence is established by the existence of a $stabilizingsolution$ of the associated generalized algebraic Riccati equation, which is different from the finite horizon case. Moreover, any open-loop optimal control for the initial state $x$ admiting a closed-loop representation is obatined.