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Some Explicit Distributions Related to the First Exit Time from a Bounded Interval for Certain Functionals of Brownian Motion

Authors:	Aimé Lachal

Institution:	1. Institut National des Sciences Appliquées de Lyon, P?le de Mathématiques, Batiment Léonard de Vinci, 20 avenue Albert Einstein, 69621, Villeurbanne Cedex, France

Abstract:	Let (B _t)_{t≥ 0} be standard Brownian motion starting at y and set X _t = ${x+\int_{0}^{t} V(B_{s}) ds}$ for $x\in (a, b)$ , with V(y) = y ^γ if y≥ 0, V(y) = −K(−y)^γ if y≤ 0, where γ and K are some given positive constants. Set ${\tau_{ab} = inf\{t > 0: X_{t} \notin (a, b)\}}$ . In this paper, we provide some formulas for the probability distribution of the random variable $B_{\tau_{ab}}$ as well as for the probability ${\mathbb{P}\{X_{\tau_{ab}}=a}$ (or b)}. The formulas corresponding to the particular cases x = a or b are explicitly expressed by means of hypergeometric functions.

Keywords:	First exit time Laplace transform Kummer and hypergeometric functions
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