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

Let Bt be a Brownian motion, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g(t) = a\sqrt{t+c}$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(t) = b\sqrt{t+c}$\end{document}, t ≥ 0, a < b, c > 0, T > 0, and τ be the first hitting time of Bt either in f(t) or in g(t). We study the hitting probabilities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$v(t,x)=P_{t,x}\left(\tau\leq T,\phantom{1}B_{\tau}=f\left(\tau\right)\right)$\end{document} for 0 < t < T and g(t) < x < f(t), where Pt,x is a probability such that Pt,x(Bt = 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.