Sobolev-type stochastic differential equations driven by G-Brownian motion

Abstract

In this paper, we introduce a class of Sobolev-type stochastic differential equations driven by G-Brownian motion (G-SSDEs, in short). We prove the existence and uniqueness of the mild solution for G-SSDEs. By means of two integral inequalities, the attracting and quasi-invariant sets of the equations are obtained. As a byproduct, the exponentially stability of the solution in mean-square sense is derived. An example is given to illustrate the obtained theoretical results.

Keywords:

• Sobolev-type
• G-Brownian motion
• attracting set
• quasi-invariant set
• exponential stability

Acknowledgments

The authors wish to thank the editor and three anonymous referees for their valuable comments, correcting errors and improving written language.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [11871076].