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As we know, the periodic functions are symmetric within a cycle time, and it is meaningful to generalize the periodicity into more general cases, such as almost periodicity or almost automorphy. In this work, we introduce the concept of Poisson Sγ2-pseudo almost automorphy (or Poisson generalized Stepanov-like pseudo almost automorphy) for stochastic processes, which are almost-symmetric within a suitable period, and establish some useful properties of such stochastic processes, including the composition theorems. In addition, we apply a Krasnoselskii–Schaefer type fixed point theorem to obtain the existence of pseudo almost automorphic solutions in distribution for some semilinear stochastic differential equations driven by Lévy noise under Sγ2-pseudo almost automorphic coefficients. In addition, then we establish optimal control results on the bounded interval. Finally, an example is provided to illustrate the theoretical results obtained in this paper.

Pseudo Almost Automorphic Solutions for Stochastic Differential Equations Driven by Lévy Noise and Its Optimal Control