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On an integral equation for first-passage-time probability densities. (English) Zbl 0551.60081

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by C. Park and S. R. Paranjape [Pac. J. Math. 53, 579–583 (1974; Zbl 0292.60117)] but also for all boundaries of the type \(S(t)=a+bt^{1/p}\) (p\(\geq 2\), a,b\(\in {\mathbb{R}})\) for which no explicit analytical results have previously been available.

MSC:

60J60 Diffusion processes
60J65 Brownian motion

Citations:

Zbl 0292.60117
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