6533b82afe1ef96bd128cc40

RESEARCH PRODUCT

On first exit times and their means for Brownian bridges

Antti LuotoPaavo SalminenChristel Geiss

subject

Statistics and ProbabilityBessel processGeneral Mathematics010102 general mathematicsMathematical analysisProbability (math.PR)Brownian bridge01 natural sciencesBridge (interpersonal)010104 statistics & probabilitysymbols.namesakeDistribution (mathematics)Diffusion processMathematics::ProbabilitysymbolsFOS: MathematicsBinomial options pricing model0101 mathematicsStatistics Probability and UncertaintyMathematics - ProbabilityBessel functionBrownian motionMathematics

description

For a Brownian bridge from $0$ to $y$ we prove that the mean of the first exit time from interval $(-h,h), \,\, h>0,$ behaves as $O(h^2)$ when $h \downarrow 0.$ Similar behavior is seen to hold also for the 3-dimensional Bessel bridge. For Brownian bridge and 3-dimensional Bessel bridge this mean of the first exit time has a puzzling representation in terms of the Kolmogorov distribution. The result regarding the Brownian bridge is applied to prove in detail an estimate needed by Walsh to determine the convergence of the binomial tree scheme for European options.

yearjournalcountryeditionlanguage
2017-11-16
https://dx.doi.org/10.48550/arxiv.1711.06107