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First Exit Time from a Bounded Interval for a Certain Class of Additive Functionals of Brownian Motion

Authors:	Aimé Lachal

Institution:	(1) Laboratoire de Mathématiques Appliquées de Lyon, Institut National des Sciences Appliquées de Lyon, 69621 Villeurbanne Cedex, France

Abstract:	Let (B _t)_t0 be standard Brownian motion starting at y, X _t = x + ^t ₀ V(B _s) ds for x (a, b), with V(y) = y if y0, V(y)=–K(–y) if y0, where >0 and K is a given positive constant. Set _ab=inf{t>0: X _t(a, b)} and ₀=inf{t>0: B _t=0}. In this paper we give several informations about the random variable _ab. We namely evaluate the moments of the random variables $B_{\tau _{ab} } and B_{\tau _{ab} \wedge \sigma _0 }$ , and also show how to calculate the expectations ${\mathbb{E}}\left( {\tau _{ab}^m B_{\tau _{ab} }^n } \right) and {\mathbb{E}}\left( {\left( {\tau _{ab} \wedge \sigma _0 } \right)^m B_{\tau _{ab} \wedge \sigma _0 }^n } \right)$ . Then, we explicitly determine the probability laws of the random variables $B_{{\tau }_{ab} } and B_{{\tau }_{ab} \wedge \sigma _0 }$ as well as the probability ${\mathbb{P}}\left\{ {X_{\tau _{ab} } = a\left( {or b} \right)} \right\}$ by means of special functions.

Keywords:	first exit time excursion process Abel's integral equation hypergeometric functions
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