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Smoothness of first passage time distributions and a new integral equation for the first passage time density of continuous Markov processes

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Axel Lehmann
Axel Lehmann*
Affiliation:
Otto-von-Guericke-Universität Magdeburg
*
∗ Postal address: Faculty of Mathematics, Institute for Mathematical Stochastics, Otto-von-Guericke-Universität Magdeburg, PSF 4120, 39016 Magdeburg, Germany. Email address: axel.lehmann@mathematik.uni-magdeburg.de
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Abstract

Let X be a one-dimensional strong Markov process with continuous sample paths. Using Volterra-Stieltjes 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.

Keywords

First passage time densitymoving boundariescontinuous Markov processesBrownian motionOrnstein-Uhlenbeck processVolterra integral equation

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J65: Brownian motion
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 4 , December 2002 , pp. 869 - 887
Copyright
Copyright © Applied Probability Trust 2002 

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