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

By Łukasz Kruk, John P. Lehoczky, Kavita Ramanan, Steven E. Shreve. 2007

172 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/2736ba14ae58e066c1163cace92d7c129c301624

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Citations in the corpus

Heavy traffic analysis for EDF queues with reneging (2011, background): “4 from [25], ΛH(0)(φ) is the function in D[0,∞) obtained by double reflection of φ at 0 and H(0).”
Occupation densities in solving exit problems for Markov additive processes and their reflections (2011, background): “ with R∗(0) = R∗(0) = 0, such that their points of increase are contained in {t≥ 0 : H(t) = −a} and {t≥ 0 : H(t) = 0} respectively. It is known that such a triplet (H,R∗,R∗) exists and is unique, see [25]. Observe that the process R∗ is continuous, because Xhas no positive jumps. It can be seen as t”