This paper first develops a Lyapunov-type theorem to study global well-posedness (existence and uniqueness of the strong variational solution) and asymptotic stability in probabil-ity of nonlinear stochastic evolution systems (SESs) driven by a special class of Lévy processes, which consist of Wiener and com-pensated Poisson processes. This theorem is then utilized to devel-op an approach to solve an inverse optimal stabilization problem for SESs driven by Lévy processes. The inverse optimal control design achieves global well-posedness and global asymptotic sta-bility of the closed-loop system, and minimizes a meaningful cost functional that penalizes both states and control. The approach does not require to solve a Hamilton-Jacobi-Bellman equation (HJBE). An optimal stabilization of the evolution of the fre-quency of a certain genetic character from the population is in-cluded to illustrate the theoretical developments.