# Occupation times of refracted Lévy processes

A. E. Kyprianou, J. C. Pardo, J. L. Pérez

Research output: Contribution to journalArticlepeer-review

25 Citations (SciVal)
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.