Occupation Times of Refracted Lévy Processes

A. E. Kyprianou (), J. C. Pardo () and J. L. Pérez ()
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A. E. Kyprianou: University of Bath
J. C. Pardo: Centro de Investigación en Matemáticas
J. L. Pérez: ITAM

Journal of Theoretical Probability, 2014, vol. 27, issue 4, 1292-1315

Abstract: 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 $$\begin{aligned} {\mathrm{d}}U_t=-\delta \mathbf 1 _{\{U_t>b\}}{\mathrm{d}}t +{\mathrm{d}}X_t,\quad t\ge 0 \end{aligned}$$ d U t = − δ 1 { U t > b } d t + d X t , t ≥ 0 where $$X=(X_t, t\ge 0)$$ X = ( X t , t ≥ 0 ) is a Lévy process with law $$\mathbb{P }$$ P and $$b,\delta \in \mathbb{R }$$ b , δ ∈ R such that the resulting process $$U$$ U may visit the half line $$(b,\infty )$$ ( b , ∞ ) with positive probability. In this paper, we consider the case that $$X$$ X is spectrally negative and establish a number of identities for the following functionals $$\begin{aligned} \int \limits _0^\infty \mathbf 1 _{\{U_t c\}$$ κ c + = inf { t ≥ 0 : U t > c } and $$\kappa ^-_a=\inf \{t\ge 0: U_t

Keywords: Occupation times; Fluctuation theory; Refracted Lévy processes; 60G51 (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s10959-013-0501-4

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