Refracted Levy processes

TL;DR

This paper studies boundary crossing problems for refracted Lévy processes, deriving identities related to exit problems using scale functions, with applications to insurance risk modeling.

Contribution

It introduces new identities for exit problems of refracted Lévy processes, especially for spectrally negative cases, linking them to scale functions and applications in risk theory.

Findings

Derived identities for one and two-sided exit problems.

Expressed identities in terms of scale functions of Lévy processes.

Discussed applications to insurance risk processes.

Abstract

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 \[ \D U_t = - \delta \mathbf{1}_{\{U_t >b\}}\D t + \D X_t \] where $X=\{X_t :t\geq 0\}$ is a L\'evy process with law $\mathbb{P}$ and $b, \delta\in \mathbb{R}$ such that the resulting process $U$ may visit the half line $(b,\infty)$ 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 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.