An explicit formula for the Skorokhod map on [0, a]

Abstract

The Skorokhod map is a convenient tool for constructing solutions to stochastic differential equations with reflecting boundary conditions. In this work, an explicit formula for the Skorokhod map Γ0,a on [0, a] for any a > 0 is derived. Specifically, it is shown that on the space D[O, ∞) of right-continuous functions with left limits taking values in ℝ, Γ0,a = Λa o Γ0, where Aa : D[0, ∞) → D[0, ∞) is defined by Λa(φ)(t)=φ(t)-supsε[0,t][(φ(s)-a) + ∧ infuε[s,t] φ(u)] and Γ0 : D[0, ∞) → D[0, ∞] is the Skorokhod map on [0, ∞], which is given explicitly by Γ0(ψ)(t)=ψ(t)+ aup sε[0,t][-ψ(s)+. In addition, properties of Λa are developed and comparison properties of Γ0,a are established. © Institute of Mathematical Statistics, 2007.