Bismut type derivative formulae and gradient estimate for multiplicative SDEs with fractional noises

Abstract

In this article, we study multiplicative SDE with fractional noise in a suitable sense, which can be regarded as a fractional Gruschin type process. Using the transfer principle and fractional integral and derivative operators, two kinds of Bismut type derivative formulae are established in this non-Markovian context under incompatible conditions. As an application, an explicit gradient estimate is derived.

Keywords:

• Derivative formula
• gradient estimate
• fractional Brownian motion
• Malliavin calculus

AMS subject Classification:

• 60H10

Acknowledgments

The authors would like to thank the editors and the referee for their careful reading of this paper and for detailed comments and valuable suggestions which have greatly improve its presentation.

Disclosure statement

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

Additional information

Funding

This is supported in part by the Natural Science Foundation of Anhui Province [grant number 2008085MA10], [grant number 1908085MA07], the National Science Foundation of China [grant number 11501009], [grant number 11871076].