On a SDE driven by a fractional Brownian motion and with monotone drift

Abstract

Let ${B_{t}^{H},t\in \lbrack 0,T]}$ be a fractional Brownian motion with Hurst parameter $H \gt \frac{1}{2}$. We prove the existence of a weak solution for a stochastic differential equation of the form $X_{t}=x+B_{t}^{H}+ \int_{0}^{t}\left( b_{1}(s,X_{s})+b_{2}(s,X_{s})\right) ds$, where $ b_{1}(s,x)$ is a Holder continuous function of order strictly larger than $1-\frac{1}{2H}$ in $x$ and than $H-\frac{1}{2}$ in time and $b_{2}$ is a real bounded nondecreasing and left (or right) continuous function.