Stochastic applications of Caputo-type convolution operators with non-singular kernels

TL;DR

This paper explores Caputo-type convolution operators with non-singular kernels, linking solutions of related integro-differential equations to Lévy subordinators, and extends the analysis to kernels with randomly distributed parameters.

Contribution

It establishes a connection between solutions of certain integro-differential equations and Lévy subordinators, and generalizes the results to kernels with random parameters.

Findings

Solutions coincide with transition densities of Lévy subordinators.

Extension to kernels with randomly distributed parameters.

Greater flexibility in kernel and jump density selection.

Abstract

We consider here convolution operators, in the Caputo sense, with non-singular kernels. We prove that the solutions to some integro-differential equations with such operators (acting on the space variable) coincide with the transition densities of a particular class of L\'evy subordinators (i.e. compound Poisson processes with non-negative jumps). We then extend these results to the case where the kernels of the operators have random parameters, with given distribution. This assumption allows greater flexibility in the choice of the kernel's parameters and, consequently, of the jumps' density function.