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

Spyridon M. Tzaninis, Nikolaos D. Macheras

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

Abstract

If a given aggregate process is a compound mixed renewal process under a probability measure , we provide a characterization of all probability measures on the domain of such that and are progressively equivalent and is converted into a compound mixed Poisson process under . 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.

Paper Structure

This paper contains 6 sections, 10 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Characterization of Progressively Equivalent Martingale Measures for Compound Mixed Renewal Processes
  4. An Application to the Ruin Problem
  5. Mixed Premium Calculation Principles and Change of Measures
  6. Examples

Key Result

Proposition 2.6

(See mt2, Corollary 4.8 and Remark 4.9(c)). For $P\inmsl$ with $P(\{\mathbb E_P[W_1\mid\varTheta]<\infty\})=1$ the following hold true:

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.5
  • Proposition 2.6
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • Lemma 4.1
  • ...and 27 more