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An Evaluation for the Probability Density of the First Hitting Time

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DOI: 10.4236/am.2013.45108    3,912 Downloads   5,794 Views  


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.

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The authors declare no conflicts of interest.

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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