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

Let {D(s),s≥0}{D(s),s≥0} be a Lévy subordinator, that is, a non-decreasing process with stationary and independent increments and suppose that D(0)=0D(0)=0. We study the first-hitting time of the process DD, namely, the process E(t)=inf{s:D(s)>t}E(t)=inf{s:D(s)>t}, t≥0t≥0. The process EE is, in general, non-Markovian with non-stationary and non-independent increments. We derive a partial differential equation for the Laplace transform of the nn-time tail distribution function P[E(t1)>s1,…,E(tn)>sn]P[E(t1)>s1,…,E(tn)>sn]. This PDE can be used to derive all nn-time moments of the process EE. 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.