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

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[0,∞) of right-continuous functions with left limits taking values in ℝ, Γ0, a=Λa○Γ0, where Λₐ: D[0,∞)[0,∞ ) is defined by Λₐ(φ)(t)=φ(t)-s∈[0,t][(φ(s)-a)⁺infu∈[s,t]φ(u)] and Γ₀: D[0,∞)[0,∞) is the Skorokhod map on [0, ∞), which is given explicitly by Γ₀(ψ)(t)=ψ(t)+s∈[0,t][-ψ(s)]⁺. In addition, properties of Λa are developed and comparison properties of Γ0, a are established.