An Evaluation for the Probability Density of the First Hitting Time

Shih-Yu Shen, Yi-Long Hsiao

Department of Finance, National Dong Hwa University, Hualien County, Chinese Taipei.
Department of Mathematics, National Cheng-Kung University, Tainan, Chinese Taipei.

DOI: 10.4236/am.2013.45108                

Abstract

Let h(t) be a smooth function, B_t a standard Brownian motion and t_h=inf{t; B_t=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=B_t for t < t_h, then u_xx=2u_t and u(h(t),t)=0. Moreover, the hitting time density d_h(t) is 1/2u_x(h(t),t). Applying some partial differential equation techniques, we derive a simple integral equation for d_h(t). Two examples are demonstrated in this article.

Keywords

Brownian Motion; First Hitting Time; Heat Equation; Boundary Value Problem

Conflicts of Interest

The authors declare no conflicts of interest.

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