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Smoothness of First Passage Time Distributions and a new Integral Equation for the First Passage Time Density of Countinuous Markov Processes

by Lehmann, A..

Series: 2002-18, Preprints

60G40 Stopping times; optimal stopping problems; gambling theory
60J25 Continuous-time Markov processes on general state spaces
60J65 Brownian motion

Let $X$ be a one-dimensional strong Markov process with
continuous sample paths. Using Volterra-Stieljes integral
equation techniques we investigate Hölder continuity and
differentiability of first passage time distributions of
$X$ with respect to continuous lower and upper moving
boundaries. Under mild assumptions on the transition
function of $X$ we prove the existence of a continuous
first passage time density to one-sided differentiable
moving boundaries and derive a new integral equation for
this density. We apply our results to Brownian motion
and its nonrandom Markovian transforms, in particular, to
the Ornstein-Uhlenbeck process.

First passage time density, Moving boundaries, Continuous Markov processes, Brownian motion, Ornstein-Uhlenbeck process, Volterra Integral equation