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

Abstract

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.

Well-posedness for a stochastic Camassa–Holm type equation with higher order nonlinearities

“…However, it is difficult to achieve this for SPDEs with general nonlinear terms and nonlinear noise. For instance, the cut-off in our case will have to involve -Due to the above unsolved technical issue, the martingale approach is difficult to apply in our problem and we will try to prove convergence directly, which is motivated by [41,55] [see also [49,53,54] for recent developments]. Generally speaking, we will analyze the difference between two approximative solutions and directly find a space Y such that X → Y → Z and convergence (up to a subsequence) holds true in Y.…”

“…Motivated by [46,49], we introduce the concept on stability of exiting time in Sobolev spaces. Exiting time, as its name would suggest, is defined as the time when solution leaves a certain range.…”

“…Hypothesis H 2 ), the map u 0 → u is continuous, but one can not improve the stability of the exiting time and simultaneously the continuity of the map u 0 → u. Up to our knowledge, results of this type for SPDEs first appeared in [46,49]. We also refer to [3,43,55] for recent developments.…”