Reflected stochastic differential equations driven by standard and fractional Brownian motion

Abstract

The reflection problem on the positive half-line with reflection at zero for a time-dependent stochastic differential equations driven by standard and fractional Brownian motion with Hurst parameter $H>\frac{1}{2}$ is considered. We prove the existence of weak solutions by using Euler scheme. Moreover, we show that pathwise uniqueness holds and a strong solution exists in the case of additive fractional noise and also up to a stopping time $\tau$ for the multiplicative case, but remains an open question beyond $\tau$.