A characterization of progressively equivalent probability measures preserving the structure of a compound mixed renewal process

TL;DR

This paper characterizes all probability measures that are progressively equivalent to a given measure and preserve the compound mixed renewal process structure, enabling transformation into a compound mixed Poisson process for applications in insurance risk modeling.

Contribution

It provides a comprehensive characterization of measure changes that maintain the process structure, extending previous work and enabling process transformation.

Findings

All such measures are characterized explicitly.

Any compound mixed renewal process can be converted into a compound mixed Poisson process.

Applications include ruin probability analysis and premium calculation in arbitrage-free insurance markets.

Abstract

Generalizing earlier works of Delbaen & Haezendonck [5] as well as of [18] and [16] for given compound mixed renewal process S under a probability measure P, we characterize all those probability measures Q on the domain of P such that Q and P are progressively equivalent and S remains a compound mixed renewal process under Q with improved properties. As a consequence, we prove that any compound mixed renewal process can be converted into a compound mixed Poisson process through a change of measures. Applications related to the ruin problem and to the computation of premium calculation principles in an insurance market without arbitrage opportunities are discussed in [26] and [27], respectively.