Occupation times of refracted L\'evy processes

A refracted L\'evy process is a L\'evy 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 L\'evy process is described by the unique strong solution to the stochastic differential equation $$\ud U_t=-\delta\mathbf{1}_{\{U_t>b\}}\ud t +\ud X_t,$$ where $X=(X_t, t\ge 0)$ is a L\'evy process with law $\p$ and $b,\delta\in \R$ 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 $$\int_0^\infty\mathbf{1}_{\{U_t<b\}}\ud t, \quad\int_0^{\rho_a^+}\mathbf{1}_{\{U_t<b\}}\ud t, \quad\int_0^{\rho^-_c}\mathbf{1}_{\{U_t<b\}}\ud t, \quad\int_0^{\rho_a^+\land\rho^-_c}\mathbf{1}_{\{U_t<b\}}\ud t,$$ where $\rho^+_a=\inf\{t\ge 0: U_t> a\}$ and $\rho^-_c=\inf\{t\ge 0: U_t< c\}$ 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.