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

Aimé Lachal ()
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Aimé Lachal: Bâtiment Léonard de Vinci

Journal of Theoretical Probability, 2006, vol. 19, issue 4, 757-771

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; primary 60J65; primary 60G40; secondary 60J25 (search for similar items in EconPapers)
Date: 2006
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DOI: 10.1007/s10959-006-0039-9

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