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

摘要

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 $xin (a, b)$ , 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.