by Lehmann, A..

**Series:** 1996-16, Preprints

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

**Abstract:**

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.

**Keywords:**

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