02-18
Smoothness of First Passage Time Distributions and a new Integral Equation for the First Passage Time Density of Countinuous Markov Processes
by Lehmann, A.
Preprint series: 02-18, Preprints
- MSC:
- 60G40 Stopping times; optimal stopping problems; gambling theory, See also {62L15, 90D60}
- 60J25 Markov processes with continuous parameter
- 60J65 Brownian motion, See also {58G32}
Abstract: Let $X$ be a one-dimensional strong Markov process with continuous sample paths. Using Volterra-Stieljes integral equation techniques we investigate H lder 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
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