Stochastic calculus for tempered fractional Brownian motion and stability for SDEs driven by TFBM

Abstract

The objective of this article is to introduce and study Itô type stochastic integrals with respect to tempered fractional Brownian motion (TFBM) of Hurst index $H\in (\frac{1}{2},1)$ and tempering parameter $\lambda >0$, by using the Wick product. The main tools are fractional calculus and Malliavin calculus. The Itô formula for this stochastic integral is established for the Itô type processes driven by TFBM. Based on this new Itô formula, we analyze the stability of stochastic differential equations driven by TFBM in the sense of $p$-th moment. A numerical example is given to illustrate our stability results.

Keywords:

• Tempered fractional Brownian motion
• Stochastic integral
• Malliavin calculus
• Wick product
• Itô formula
• stochastic differential equation
• general decay
• stochastic stability

2020 Mathematics Subject Classification:

• 60H05
• 60H10
• 60H30
• 60G22

Acknowledgments

The authors express their sincere thanks to the editor for his/her kind help, and the anonymous reviewers for their careful reading of the article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Data availability statement

All data generated or used during the study are available from the corresponding author (Yejuan Wang) by request.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China under Grant No. 41875084, the Innovative Groups of Basic Research in Gansu Province under Grant No. 22JR5RA391, and by an NSERC RGPIN-2018-05687.