TITLE:
An Evaluation for the Probability Density of the First Hitting Time
AUTHORS:
Shih-Yu Shen, Yi-Long Hsiao
KEYWORDS:
Brownian Motion; First Hitting Time; Heat Equation; Boundary Value Problem
JOURNAL NAME:
Applied Mathematics,
Vol.4 No.5,
May
20,
2013
ABSTRACT: Let h(t) be a smooth function, Bt a standard Brownian motion and th=inf{t; Bt=h(t)} the first hitting time. In this paper, new formulations are derived to evaluate the probability density of the first hitting time. If u(x, t) denotes the density function of x=Bt for t th, then uxx=2ut and u(h(t),t)=0. Moreover, the hitting time density dh(t) is 1/2ux(h(t),t). Applying some partial differential equation techniques, we derive a simple integral equation for dh(t). Two examples are demonstrated in this article.