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J. Cuzick, “Boundary Crossing Probabilities for Stationary Gaussian Processes and Brownian Motion,” Transactions of the American Mathematical Society, Vol. 263, No. 2, 1981, pp. 469-492. doi:10.1090/S0002-9947-1981-0594420-5

has been cited by the following article:

• 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.