A characterization of equivalent martingale probability measures in a mixed renewal risk model with applications in Risk Theory

TL;DR

This paper characterizes all probability measures that transform a compound mixed renewal process into a compound mixed Poisson process, with implications for risk theory, ruin probabilities, and premium calculations in insurance markets.

Contribution

It extends existing results by providing a comprehensive characterization of equivalent martingale measures in mixed renewal models, applicable to risk theory and insurance mathematics.

Findings

Characterization of all measures making the process a compound mixed Poisson

Implications for ruin probability analysis in insurance

Applications to premium calculation principles

Abstract

If a given aggregate process $S$ is a compound mixed renewal process under a probability measure $P$, we provide a characterization of all probability measures $Q$ on the domain of $P$ such that $Q$ and $P$ are progressively equivalent and $S$ is converted into a compound mixed Poisson process under $Q$. This result extends earlier works of Delbaen & Haezendonck [2], Embrechts & Meister [5], Lyberopoulos & Macheras [11], and of the authors [14]. Implications to the ruin problem and to the computation of premium calculation principles in an insurance market possessing the property of no free lunch with vanishing risk are also discussed.