Hybrid Risk Processes: A Versatile Framework for Modern Ruin Problems

TL;DR

This paper introduces the hybrid risk process framework, unifying various modern ruin models with features like Markov-modulation and reserve dependence, and provides computational methods for ruin analysis.

Contribution

It presents the hybrid risk process as a versatile framework that unifies multiple ruin concepts and offers computational techniques for practical analysis.

Findings

Unified framework for modern ruin models.

Efficient matrix-analytic computational methods.

Generalized Omega ruin model synthesis.

Abstract

We introduce the hybrid risk process, constructed via a time-change transformation applied to the solution of a hybrid stochastic differential equation. The framework covers several modern ruin settings, incorporating features like Markov-modulation and reserve-dependent parameters through an interdependent structure where the surplus level influences the dynamics of the background environment. The approach lets us define and analyze the Generalized Omega ruin model, a novel definition of insolvency that synthesizes concepts like Erlangian, cumulative Parisian and Omega ruin into a unified competing-risks framework. Finally, we show that the models are computationally tractable. By adapting recent matrix-analytic techniques, we provide an efficient way to compute a wide range of ruin-related quantities.