Using differential equations to obtain joint moments of first-passage times of increasing Levy processes

Let {D(s), s >= 0} be a Levy subordinator, that is, a non-decreasing process with stationary and independent increments and suppose that D(0) = 0. We study the first-hitting time of the process D, namely, the process E(t) = inf{s : D(s) > t}, t >= 0. The process E is, in general, non-Markovian with non-stationary and non-independent increments. We derive a partial differential equation for the Laplace transform of the n-time tail distribution function P[E(t(1)) > s(1), ... , E(t(n)) > s(n)]. This PDE can be used to derive all n-time moments of the process E. As an application, we give a recursive formula for multiple-time moments of the local time of a Markov process in terms of its transition density. (C) 2010 Elsevier B.V. All rights reserved.