Using Differential Equations to Obtain Joint Moments of First-Passage Times of Increasing Levy Processes

Details Using Differential Equations to Obtain Joint Moments of First-Passage Times of Increasing Levy Processes Using Differential Equations to Obtain Joint Moments of First-Passage Times of Increasing Levy Processes

2009-06-27

arxiv.org/abs/0906.5083v1

Mathematics

Probability

13 pages, one figure

Scientific paper

Let $\{D(s), s \geq 0 \}$ be a L\'evy 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 $E(t) = \inf \{s: D(s) > t \}$, $t \geq 0$. 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 $P[E(t_1) > s_1,...,E(t_n) > s_n]$, and show that this PDE has a unique solution given natural boundary conditions. This PDE can be used to derive all $n$-time moments of the process $E$.

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