An Evaluation for the Probability Density of the First Hitting Time

• 作者: Shih-Yu Shen Yi-Long Hsiao
• 来源期刊: 应用数学(英文)
• 学科: 数学
• 关键词: BROWNIAN MOTION FIRST Hitting Time Heat EQUATION BOUNDARY VALUE Problem
• 年，卷（期）: yysxyw_2013,（5）
• 页码范围: 792-796
• 页数: 5页
• 分类号: O1

摘要

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.