Solving an Integral Equation Arising from the Ruin Probability of Long-term Bonus-Malus Systems

TL;DR

This paper analyzes an integral equation related to ruin probabilities in Bonus-Malus insurance systems, providing solutions, conditions for meromorphic Laplace transforms, and extending results to stochastic cases using complex analysis.

Contribution

It offers new closed-form solutions and characterizations for ruin probabilities, along with conditions for their Laplace transforms and extensions to stochastic models.

Findings

Closed-form solutions for ruin probabilities in specific cases

Conditions for Laplace transform extension to meromorphic functions

Extension of results to doubly stochastic Bonus-Malus systems

Abstract

This article studies in detail the solution of an integral equation due to Rongming et al. [13]. The methods involve complex analysis. As an application, we find the ruin probability of a given Bonus-Malus system in a steady state. We obtain closed form solutions for the ruin probability in certain cases, and we characterize these cases. We give conditions for the Laplace transform of a ruin probability to extend to a meromorphic function in the complex plane, we prove a very general and almost sharp inequality of Lundberg type, and we extend our results to a doubly stochastic situation.