Brownian Motion Hitting Probabilities for General Two-Sided Square-Root Boundaries

Let B(t) be a Brownian motion, g(t) = a root t + c, f(t) = b root t + c, t >= 0, a < b, c > 0, T > 0, and tau be the first hitting time of B(t) either in f(t) or in g(t). We study the hitting probabilities v(t, x) = P(t,x) (tau <= T, B tau = f (tau)) for 0 < t < T and g(t) < x < f (t), where P(t,x) is a probability such that P(t,x)(B(t) = x) = 1. We give general description of v(t,x) and find explicit series expansion for it in case of some special boundaries. The case of more general diffusion processes is discussed as well.