Non-homogeneous time convolutions, renewal processes and age dependent mean number of motorcar accidents

TL;DR

This paper introduces non-homogeneous time convolutions and renewal processes, extending renewal theory to better model real-world actuarial phenomena like motorcar accidents with age-dependent factors.

Contribution

It defines non-homogeneous time convolutions, formalizes non-homogeneous renewal processes, and applies these concepts to real motorcar insurance data.

Findings

Established the mathematical framework for non-homogeneous renewal processes.

Demonstrated the application to motorcar accident data.

Provided numerical methods for analyzing these processes.

Abstract

Non-homogeneous renewal processes are not yet well established. One of the tools necessary for studying these processes is the non-homogeneous time convolution. Renewal theory has great relevance in general in economics and in particular in actuarial science, however most actuarial problems are connected with the age of the insured person. The introduction of non-homogeneity in the renewal processes brings actuarial applications closer to the real world. This paper will define the non-homogeneous time convolutions and try to give order to the non-homogeneous renewal processes. The numerical aspects of these processes are dealt with and, finally, a real data application to an aspect of motorcar insurance is proposed.