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Occupation times of refracted Lévy processes

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 \[ \ud U_t=-δ\mathbf{1}_{\{U_t>b\}}\ud t +\ud X_t, \] where $X=(X_t, t\ge 0)$ is a Lévy process with law $\p$ and $b,δ\in \R$ 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 \[ \int_0^\infty\mathbf{1}_{\{U_t<b\}}\ud t, \quad\int_0^{ρ_a^+}\mathbf{1}_{\{U_t<b\}}\ud t, \quad\int_0^{ρ^-_c}\mathbf{1}_{\{U_t<b\}}\ud t, \quad\int_0^{ρ_a^+\landρ^-_c}\mathbf{1}_{\{U_t<b\}}\ud t, \] where $ρ^+_a=\inf\{t\ge 0: U_t> a\}$ and $ρ^-_c=\inf\{t\ge 0: U_t< c\}$ for $c<b<a$. Our identities extend recent results of Landriault et al. \cite{LRZ} and bear relevance to Parisian-type financial instruments and insurance scenarios.