Using differential equations to obtain joint moments of first-passage times of increasing Lévy processes
Taqqu, Murad S.
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Citation (published version)Mark Veillette, Murad S Taqqu. 2010. "Using differential equations to obtain joint moments of first-passage times of increasing Lévy processes." Statistics & Probability Letters, Volume 80, pp. 697 - 705. https://doi.org/10.1016/j.spl.2010.01.002
Let D(s),s≥0 be a Lévy 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)=infs:D(s)\textgreatert, 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(t1)\textgreaters1,…,E(tn)\textgreatersn]. 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.