Fluctuation theory for level-dependent L\'evy risk processes

TL;DR

This paper develops a fluctuation theory for level-dependent Le9vy risk processes, introducing scale functions and integral equations to analyze their behavior and solve related stochastic differential equations.

Contribution

It introduces a comprehensive framework for analyzing level-dependent Le9vy processes using scale functions and integral equations, extending classical fluctuation theory.

Findings

Derived fluctuation identities using scale functions

Established that derivatives of scale functions satisfy Volterra integral equations

Provided solutions to stochastic differential equations for level-dependent Le9vy processes

Abstract

A level-dependent L\'evy process solves the stochastic differential equation $dU(t) = dX(t)-{\phi}(U(t)) dt$, where $X$ is a spectrally negative L\'evy process. A special case is a multi-refracted L\'evy process with $\phi_k(x)=\sum_{j=1}^k\delta_j1_{\{x\geq b_j\}}$. A general rate function $\phi$ that is non-decreasing and continuously differentiable is also considered. We discuss solutions of the above stochastic differential equation and investigate the so-called scale functions, which are counterparts of the scale functions from the theory of L\'evy processes. We show how fluctuation identities for $U$ can be expressed via these scale functions. We demonstrate that the derivatives of the scale functions are solutions of Volterra integral equations.