Integro-differential equations linked to compound birth processes with infinitely divisible addends

TL;DR

This paper develops integro-differential equations linked to compound birth processes with infinitely divisible addends, generalizing classical models to include more complex, accelerating damage processes and their fractional and subordinator-based extensions.

Contribution

It introduces a novel integro-differential framework for modeling damage accumulation using a Yule process and infinitely divisible distributions, extending classical renewal models.

Findings

Derived PDEs for transition densities with exponential addends

Generalized equations using convolution-type space derivatives

Analyzed special cases like stable, tempered stable, gamma, and Poisson distributions

Abstract

Stochastic modelling of fatigue (and other material's deterioration), as well as of cumulative damage in risk theory, are often based on compound sums of independent random variables, where the number of addends is represented by an independent counting process. We consider here a cumulative model where, instead of a renewal process (as in the Poisson case), a linear birth (or Yule) process is used. This corresponds to the assumption that the frequency of \textquotedblleft damage" increments accelerates according to the increasing number of \textquotedblleft damages". We start from the partial differential equation satisfied by its transition density, in the case of exponentially distributed addends, and then we generalize it by introducing a space-derivative of convolution type (i.e. defined in terms of the Laplace exponent of a subordinator). Then we are concerned with the solution of integro-differential equations, which, in particular cases, reduce to fractional ones. Correspondingly, we analyze the related cumulative jump processes under a general infinitely divisible distribution of the (positive) jumps. Some special cases (such as the stable, tempered stable, gamma and Poisson) are presented.