On Reflecting Brownian Motion with Drift

Abstract

Let B ＝ (B_t)_t≥0 be a standard Brownian motion started at zero, and let μ ∈ R be a given and fixed constant. Set B^μ_t ＝ B_t +μt and S^μ_t ＝ max_0≤s≤t B^μ_s for t ≥ 0. Then the process:

(x ∨ S^μ) － B^μ ＝ ((x ∨ S^μ_t) － B^μ_t)_t≥0

realizes an explicit construction of the reflecting Brownian motion with drift －μ started at x in R₊. Moreover, if the latter process is denoted by Z^X ＝ (Z^x_t)_t≥0, then the classic Lévy's theorem extends as follows:

((x ∨ S^μ) － B^μ, (x ∨ S^μ) － x ) ＝^law (Z^x, l⁰(Z^x))

where l⁰(Z^x) is the local time of Z^x at O. The Markovian argument for (x ∨ S^μ) － B^μ remains valid for any other process with stationary independent increments in place of B^μ. This naturally leads to a class of Markov processes which are referred to as reflecting Lévy processes. A point of view which both unifies and complements various approaches to these processes is provided by the extended Skorohod lemma.