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Some results involving the maximum of Brownian motion

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

R. A. Doney
R. A. Doney*
Affiliation:
University of Manchester
*
∗ Postal address: Statistical Laboratory, Department of Mathematics, The University, Manchester M13 9PL, UK.
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Abstract

If X is a Brownian motion with drift and γ = inf{t > 0: Mt = t} we derive the joint density of the triple {U, γ, Δ}, where and Δ= γ —Xγ. In the case δ = 0 it follows easily from this that Δ has an Exp(2) distribution and this in turn implies the rather surprising result that if τ= inf{t > 0: Xt = Mt = t}, then Pr{τ = 0} = 0 and . We also derive various other distributional results involving the pair (X, M), including for example the distribution of ; in particular we show that, in case δ. = 1, when Pr{0 < τ < ∞} = 1, the ratio τ+/τ has the arc-sine distribution.

Keywords

DENSITY FACTORIZATIONHITTING TIMESEXPONENTIAL BARRIERS

MSC classification

Primary: 60G17: Sample path properties
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 4 , December 1993 , pp. 805 - 818
Copyright
Copyright © Applied Probability Trust 1993 

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