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Hölder Continuity and Differentiability of First Passage Time Distributions for Continuous Markov Processes

by Lehmann, A..

Series: 1996-16, Preprints

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

t X T be a one-dimensional Markov process with
continuous sample paths. We investigate continuity and differ-
entiability properties of first passage time (FPT) distributions of
X T with respect to continuous upper and lower moving bound-
aries. Using Volterra-Stieltjes integral equation techniques we
give sufficient conditions for Hölder continuity of the FPT distri-
bution function and the existence of a FPT density. We discuss
our results for Brownian motion and its nonrandom Markovian
transforms, in particular, for the Ornstein-Uhlenbeck process.

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