Refracted Levy processes

Motivated by classical considerations from risk theory, we investigate boundary crossing problems for refracted Lévy processes. The latter 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 formally, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation ¶ dUt=−δ1{Ut>b} dt+dXt, ¶ where X={Xt : t≥0} is a Lévy process with law ℙ and b, δ∈ℝ such that the resulting process U may visit the half line (b, ∞) with positive probability. We consider in particular the case that 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 Lévy 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.