Time Reversal for Drifted Fractional Brownian Motion with Hurst Index $H>1/2$

Abstract

Let $X$ be a drifted fractional Brownian motion with Hurst index $H > 1/2$. We prove that there exists a fractional backward representation of $X$, i.e. the time reversed process is a drifted fractional Brownian motion, which continuously extends the one obtained in the theory of time reversal of Brownian diffusions when $H=1/2$. We then apply our result to stochastic differential equations driven by a fractional Brownian motion.