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 $$\begin{aligned} {\mathrm{d}}U_t=-\delta \mathbf 1 _{\{U_t>b\}}{\mathrm{d}}t +{\mathrm{d}}X_t,\quad t\ge 0 \end{aligned}$$ 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