AM  Vol.4 No.5 , May 2013
An Evaluation for the Probability Density of the First Hitting Time
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.


Cite this paper
S. Shen and Y. Hsiao, "An Evaluation for the Probability Density of the First Hitting Time," Applied Mathematics, Vol. 4 No. 5, 2013, pp. 792-796. doi: 10.4236/am.2013.45108.
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