First exit time from a bounded interval for a certain class of additive functionals of Brownian motion

Let (B-t)(t greater than or equal to 0) be standard Brownian motion starting at y, X-t = x + integral(0)(t) V(B-s) ds for x is an element of (a, b), with V(y) = y(gamma) if y greater than or equal to 0, V(y) = -K( -y)gamma if y less than or equal to 0, where gamma > 0 and K is a given positive constant. Set tau(ab) = inf{t > 0; X-t is not an element of (a, b)} and sigma(0) = inf{t > 0: B-t = 0}. In this paper we give several informations about the random variable z,b. 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)(m)B(tau ab)(n)) and E((tau(ab) (<^>) (sigma 0))(m) B-tau ab(n) (<^>) (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.