Refracted Levy processes

Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted L'evy processes. The latter 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 formally, whenever it exists, a refracted L'evy process is described by the unique strong solution to the stochastic differential equation $$\text{D}{U}_t=-\delta {\mathbf{1}}_{\{U_t>b\}}\text{D}t+\text{D}{X}_t$$ where $X={X_t :t\geq 0}$isaL\acute{e}vyprocesswithlaw$$\mathbb{P}$$and$b, \delta\in $\mathbb{R}$$suchthattheresultingprocess$U$mayvisitthehalfline$(b,\infty)$withpositiveprobability\mathrm{.}Weconsiderinparticularthecasethat$X$ is spectrally negative and establish a suite of identities for the case of one and two sided exit problems. All identities can be written in terms of the $q$-scale function of the driving L'evy process and its perturbed version describing motion above the level $b$. We remark on a number of applications of the obtained identities to (controlled) insurance risk processes.