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

A level-dependent L'evy process solves the stochastic differential equation $dU(t)=dX(t)-\varphi (U(t))dt$, where $X$ is a spectrally negative L'evy process. A special case is a multi-refracted L'evy process with ${\varphi }_k(x)={\sum }_{j=1}^k{\delta }_j{1}_{\{x\ge b_j\}}$. A general rate function $\varphi$ 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.