Asymptotically almost automorphic solutions to stochastic differential equations driven by a Lévy process

Abstract

In this paper, a new concept of Poisson asymptotically almost automorphy for stochastic processes is introduced. And then, some fundamental properties including composition theorems for the space of such processes are proved. Subsequently, this concept is applied to investigate the existence and uniqueness of asymptotically almost automorphic solutions in distribution to some linear and semilinear stochastic differential equations driven by a Lévy process under some suitable conditions. Finally, an example is given to illustrate the main results.

Keywords:

• Poisson asymptotically almost automorphy for stochastic processes
• stochastic differential equations
• Lévy process

Acknowledgements

The authors are very thankful to anonymous referees for their valuable comments to improve this manuscript.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by NSF of China [11361032], Program for Longyuan Youth Innovative Talents of Gansu Province of China [2014-4-80], and the Fundamental Research Funds for the Central Universities of China [JB160713].