A characterization of equivalent martingale measures in a renewal risk model with applications to premium calculation principles

TL;DR

This paper characterizes all equivalent martingale measures in a renewal risk model, showing how to transform any compound renewal process into a compound Poisson process and linking this to premium calculation methods.

Contribution

It extends previous work by providing a comprehensive characterization of equivalent measures in renewal models and demonstrates measure changes that simplify processes for premium calculations.

Findings

Any renewal process can be converted into a Poisson process via measure change.

The characterization links measure changes to premium calculation principles.

The approach generalizes earlier results for compound renewal processes.

Abstract

Generalizing earlier work of Delbaen and Haezendonck for given compound renewal process $S$ under a probability measure $P$ we characterize all probability measures $Q$ on the domain of $P$ such that $Q$ and $P$ are progressively equivalent and $S$ remains a compound renewal process under $Q$. As a consequence, we prove that any compound renewal process can be converted into a compound Poisson process through a change of measures and we show how this approach is related to premium calculation principles.