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

Abstract

Let B _t be a Brownian motion, $g(t) = a\sqrt{t+c}$, $f(t) = b\sqrt{t+c}$, t ≥ 0, a < b, c > 0, T > 0, and τ be the first hitting time of B _t either in f(t) or in g(t). We study the hitting probabilities $v(t,x)=P_{t,x}\left(\tau\leq T,\phantom{1}B_{\tau}=f\left(\tau\right)\right)$ 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.