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

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.

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

“…For applications of Theorem 4.5 to a characterization of equivalent martingale measures for CMRPs, and to the pricing of actuarial risks (premium calculation principles) in an insurance market possessing the property of no free lunch with vanishing risk we refer to Tzaninis and Macheras (2020).…”

“…Appendix A collects some long proofs. For applications of Theorem 4.5 to a characterization of equivalent martingale measures for CMRPs, and to the pricing of actuarial risks (premium calculation principles) in an insurance market possessing the property of no free lunch with vanishing risk we refer to Tzaninis and Macheras (2020), while for applications of Theorem 4.5 to the ruin problem for CMRPs we refer to Tzaninis (2022) and Tzaninis and Macheras (2020).…”