Fluctuation theory for level-dependent Lévy risk processes
Abstract
A level-dependent Lévy process solves the stochastic differential equation $dU(t) = dX(t)-{\phi}(U(t)) dt$, where $X$ is a spectrally negative Lévy process. A special case is a multi-refracted Lévy 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 Lévy 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.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2017
- DOI:
- 10.48550/arXiv.1712.00050
- arXiv:
- arXiv:1712.00050
- Bibcode:
- 2017arXiv171200050C
- Keywords:
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- Mathematics - Probability