Lévy processes under level-dependent Poissonian switching

TL;DR

This work analyzes a surplus process that switches between two Lévy dynamics at Poisson observation times depending on its level relative to a barrier $b$, producing a hybrid SDE for $U$. It develops a pathwise construction and proves the strong Markov property for the augmented process, then introduces generalized scale functions to express fluctuation identities, including two-sided and one-sided exit problems and potential measures. The main contributions are the fluctuation identities for the hybrid model in terms of $\mathcal{U}^{(q,\lambda)}_{b,a}$, $\mathcal{V}^{(q,\lambda)}_{b,a}$ and their limits, and an application to ruin theory with delayed dividends by choosing $Y=X-\delta t$. This framework extends Lévy fluctuation theory to level-dependent switching processes and provides tools for actuarial analysis of hybrid risk models with time delays. The results offer explicit formulas and limiting behaviors that connect to classical results when the two driving processes coincide.

Abstract

In this paper, we derive identities for the upward and downward exit problems and resolvents for a process whose motion changes between two Lévy processes if it is above (or below) a barrier $b$ and coincides with a Poissonian arrival time. This can be expressed in the form of a (hybrid) stochastic differential equation, for which the existence of its solution is also discussed. All identities are given in terms of new generalisations of scale functions (counterparts of the scale functions from the theory of Lévy processes). To illustrate the applicability of our results, the probability of ruin is obtained for a risk process with delays in the dividend payments.

Theorems & Definitions (28)

• Theorem 1
• Remark 2
• Lemma 3
• Lemma 4
• proof
• Corollary 5
• proof
• Theorem 6
• proof
• Theorem 7