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An explicit formula for the Skorokhod map on \([0,a]\). (English) Zbl 1139.60017

Let \({\mathcal D}[0, \infty)\) be the space of right-continuous functions with left limits mapping \([0, \infty)\) into \(\mathbb R\), and let \({\mathcal C}[0, \infty)\), \({\ell} [0, \infty)\) and \(\mathcal{BV}[0, \infty)\) be the subspaces of continuous, nondecreasing and bounded variation functions, respectively. The Skorokhod map \(\Gamma_{0}\) on \([0, \infty)\) has the following explicit representation \[ \Gamma_{0}(\psi)(t) = \psi(t) + \sup_{s \in [0, t]} \big[-\psi(t)]\big]^+, \;\;\;\psi \in {\mathcal C}[0, \infty), \] which is of importance in both theory and various applications. Let \(a > 0\) be a given constant. For \(\psi \in {\mathcal D}[0, \infty)\) there exists a unique pair of functions \((\overline{\phi}, \overline{\eta}) \in {\mathcal D}[0, \infty) \times \mathcal{BV}[0, \infty)\) satisfying the following properties:
[1.] For every \(t \in [0, \infty)\), \(\overline{\phi}(t) = \psi(t) + \overline{\eta}(t) \in [0, a]\);
[2.] \(\overline{\eta}(0-) = 0\) and \(\overline{\eta}\) has the decomposition \(\overline{\eta} = \overline{\eta}_\ell - \overline{\eta}_u\) as the difference of functions \(\overline{\eta}_\ell, \overline{\eta}_u \in \ell[0, \infty)\) satisfying \[ \int_0^\infty {\mathbf 1}_{\{\overline{\phi}(s) > 0\}} \, d\overline{\eta}_\ell(s) = 0 \;\;\text{ and }\;\;\int_0^\infty {\mathbf 1}_{\{\overline{\phi}(s) < a\}} \, d\overline{\eta}_u(s) = 0. \] The mapping \(\Gamma_{0, a}: {\mathcal D}[0, \infty) \to {\mathcal D}[0, \infty)\) that maps \(\psi\) to \(\overline{\phi}\) is called the Skorokhod map (or the two-sided reflection map) on \([0, a]\). The pair \((\overline{\phi}, \overline{\eta})\) is said to solve the Skorokhod problem on \([0, a]\) for \(\psi\). In this paper under review, the authors provide an explicit formula for the Skorokhod map \(\Gamma_{0, a}\) on \([0, a]\) in terms of the Skorokhod map \(\Gamma_{0}\) and the mapping \(\Lambda_a: {\mathcal D}[0, \infty) \to {\mathcal D}[0, \infty)\) defined by \[ \Lambda_a(\phi)(t) {=:}\, \phi(t) - \sup_{s \in [0, t]} \bigg[\big(\phi(s) - a\big)^+ \wedge \inf_{u \in [s, t]}\phi(u)\bigg]. \] Namely, they prove that \(\Gamma_{0, a} = \Lambda_a \circ \Gamma_{0}.\)

MSC:

60G15 Gaussian processes
60G17 Sample path properties
60J60 Diffusion processes
90B05 Inventory, storage, reservoirs
90B22 Queues and service in operations research
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References:

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