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

Let (B t) t=> 0 be standard Brownian motion starting at y, X t = x + (integral) t 0 V(B s) ds for x (element of) (a, b), with V(y) = y (gamma)if y => 0, V(y)=– K(–y) (gamma)if y <= 0, where (gamma) >0 and K is a given positive constant. Set (sigma) ab=inf{t>0: X t (not element of) (a, b)} and (sigma) 0=inf{t>0: B t=0}. In this paper we give several informations about the random variable (tau)ab. We namely evaluate the moments of the random variables B(tau) ab and B(tau) ab ^ (sigma)0 , and also show how to calculate the expectations E (((tau)ab ^ (sigma) 0)m B (tau)ab ^ (sigma) 0). Then, we explicitly determine the probability laws of the random variables B(tau) ab and B(tau) ab ^ (sigma)0 as well as the probability P {X(tau)ab = a (or b)} by means of special functions.