1 results
Some results involving the maximum of Brownian motion
- Part of
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- Journal:
- Journal of Applied Probability / Volume 30 / Issue 4 / December 1993
- Published online by Cambridge University Press:
- 14 July 2016, pp. 805-818
- Print publication:
- December 1993
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- Article
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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.
and 
and Δ= 
. We also derive various other distributional results involving the pair (
; in particular we show that, in case 