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

Abstract

Let N (τ ) be a renewal process for holding times {Xi} k≥0 ,where {X k } k≥1 are iid with density p(x). If the associated residual time R(τ ) has a density u(x, t), its Kolmogorov forward equation is given bywith an initial holding time density u(x, 0) = u0(x). We derive a measurevalued 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.