Extended Lévy’s theorem for a two-sided reflection

We aim to set forth an extension of the result found in paper [6], which finds an explicit realisation of a reflecting Brownian motion with drift −μ, started at x, reflecting above zero, and its local time at zero. In this paper we find a corresponding realisation for a reflecting Brownian motion with drift −μ, started at x, reflected both above zero and below one, along with a corresponding expression in terms of associated local times, namely as the difference between the local time at zero and the local time at one.