Renewal-scaled solutions of the Kolmogorov forward equation for residual times

TL;DR

This paper derives a measure-valued solution formula for the residual time density in renewal processes, demonstrating continuous evolution in measure space under the Kolmogorov forward equation for various holding time distributions.

Contribution

It introduces a novel measure-valued solution formula for residual times in renewal processes, extending understanding of their evolution under the Kolmogorov forward equation.

Findings

Solution formula for residual time density derived

Solutions evolve continuously in measure space

Applicable to a wide range of holding time distributions

Abstract

Let $N(\tau)$ be a renewal process for independent holding times $\{X_i\}_{k \ge 0}$ ,where $\{X_k\}_{k\ge 1}$ are identically distributed with density $p(x)$. If the associated residual time $R(\tau)$ has a density $u(x,\tau)$, its Kolmogorov forward equation is given by \begin{equation*} \partial_\tau u(x,\tau)-\partial_x u(x,\tau) = p(x)u(0,\tau), \quad x,\tau \in [0, \infty), \end{equation*} with an initial holding time density $u(x,0)=u_0(x)$. We derive a measure-valued solution formula for the density of residual times after an expected number of renewals occur. Solutions under this time scale are then shown to evolve continuously in the space of measures with the weak topology for a wide variety of holding times.