by Lehmann, A..

**Series:** 2002-18, Preprints

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

**Abstract:**

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.

**Keywords:**

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