On mixed fractional SDEs with discontinuous drift coefficient

Abstract

We prove existence and uniqueness of the solution for a class of mixed fractional stochastic differential equations with discontinuous drift driven by both standard and fractional Brownian motion. Additionally, we establish a generalized Itô rule valid for functions with absolutely continuous derivative and applicable to solutions of mixed fractional stochastic differential equations with Lipschitz coefficients, which plays a key role in our proof of existence and uniqueness. The proof of such a formula is new and relies on showing the existence of a density of the law under mild assumptions on the diffusion coefficient.

Stochastic differential equations with irregular coefficients:~mind the gap!

“…Existence and uniqueness of solutions to SDEs with discontinuous drift is true under very general conditions and has been studied in various setups, see, e.g., [43,36,37,38,14,42,20,34,15,39,16,40,32,21,33,35].…”