A new result on averaging principle for Caputo-type fractional delay stochastic differential equations with Brownian motion

Abstract

In this paper, we mainly explore the averaging principle of Caputo-type fractional delay stochastic differential equations with Brownian motion. Firstly, the solutions of this considered system are derived with the aid of the Picard iteration technique along with the Laplace transformation and its inverse. Secondly, we obtain the unique result by using the contradiction method. In addition, the averaging principle is discussed by means of the Burkholder-Davis-Gundy inequality, Jensen inequality, Hölder inequality and Grönwall-Bellman inequality under some hypotheses. Finally, an example with numerical simulations is carried out to prove the relevant theories.

Keywords:

• Fractional stochastic differential equations
• delay
• existence
• uniqueness
• averaging principle

AMS Subject Classifications:

• 26A33
• 65C30

Acknowledgements

The authors are grateful to the editors and reviewers for their constructive comments, which have improved the quality of their article.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Natural Science Special Research Fund Project of Guizhou University, China [grant number 202002].