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 Γ 0,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, Γ 0,a = Λ a o Γ 0 , where Λ a : D[0, ∞) → D[0, ∞) is defined by Λ a (Φ) (t) = Φ(t)- sups∈[0,t] [(Φ(s)-a) + Λ infu∈[s,t] Φ(u)] and Γ 0 :D[0, ∞) → D[0, ∞) is the Skorokhod map on [0, ∞), which is given explicitly by Γ 0 (ψ)(t) = ψ(t) + sup s∈[0,t] [-ψ(s)] + . In addition, properties of Λ a are developed and comparison properties of Γ 0,a are established.