Journal of Theoretical Probability

, Volume 27, Issue 4, pp 1292–1315 | Cite as

Occupation Times of Refracted Lévy Processes

Article

Abstract

A refracted Lévy process is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More precisely, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation
$$\begin{aligned} {\mathrm{d}}U_t=-\delta \mathbf 1 _{\{U_t>b\}}{\mathrm{d}}t +{\mathrm{d}}X_t,\quad t\ge 0 \end{aligned}$$
where \(X=(X_t, t\ge 0)\) is a Lévy process with law \(\mathbb{P }\) and \(b,\delta \in \mathbb{R }\) such that the resulting process \(U\) may visit the half line \((b,\infty )\) with positive probability. In this paper, we consider the case that \(X\) is spectrally negative and establish a number of identities for the following functionals
$$\begin{aligned} \int \limits _0^\infty \mathbf 1 _{\{U_t<b\}}{\mathrm{d}}t, \quad \int \limits _0^{\kappa _c^+}\mathbf 1 _{\{U_t<b\}}{\mathrm{d}}t, \quad \int \limits _0^{\kappa ^-_a}\mathbf 1 _{\{U_t<b\}}{\mathrm{d}}t, \quad \int \limits _0^{\kappa _c^+\wedge \kappa ^-_a}\mathbf 1 _{\{U_t<b\}}{\mathrm{d}}t, \end{aligned}$$
where \(\kappa ^+_c=\inf \{t\ge 0: U_t> c\}\) and \(\kappa ^-_a=\inf \{t\ge 0: U_t< a\}\) for \(a<b<c\). Our identities extend recent results of Landriault et al. (Stoch Process Appl 121:2629–2641, 2011) and bear relevance to Parisian-type financial instruments and insurance scenarios.

Keywords

Occupation times Fluctuation theory Refracted Lévy processes 

Mathematics Subject Classification (2010)

60G51 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK
  2. 2.Centro de Investigación en MatemáticasGuanajuatoMexico
  3. 3.Department of StatisticsITAMMexicoMexico

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