An Evaluation for the Probability Density of the First Hitting Time


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.

Share and Cite:

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.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] F. Black and M. Scholes, “The Pricing of Options and Corporate Liabilities,” Journal of Political Economy, Vol. 81, No. 3, 1973, pp. 637-659. doi:10.1086/260062
[2] R. C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Economics and Management Science, Vol. 4, No. 1, 1973, pp. 141-183. doi:10.2307/3003143
[3] C. Profeta, B. Roynette and M. Yor, “Option Prices as Probabilities,” Springer, New York, 2010. doi:10.1007/978-3-642-10395-7
[4] B. Ferebee, “The Tangent Approximation to One-Sided Brownian Exit Densities,” Z. Wahrscheinlichkeitsth, Vol. 61, No. 3, 1982, pp. 309-326. doi:10.1007/BF00539832
[5] G. R. Grimmett and D. R. Stirzaker, “Probability and Random Processes,” Oxford University Press, New York, 1982.
[6] 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
[7] J. Durbin, “The First-Passage Density of a Continuous Gaussian Process to a General Boundary,” Journal of Applied Probability, Vol. 22, No. 1, 1985, pp. 99-122. doi:10.2307/3213751
[8] P. Salminen, “On the First Hitting Time and the Last Exit Time for a Brownian Motion to/from a Moving Boundary,” Advances in Applied Probability, Vol. 20, No. 2, 1988, pp. 411-426. doi:10.2307/1427397
[9] A. Martin-L?f, “The Final Size of a Nearly Critical Epidemic and the First Passage Time of a Wiener Process to a Parabolic Barrier,” Journal of Applied Probability, Vol. 35, No. 3, 1998, pp. 671-682. doi:10.1239/jap/1032265215
[10] L. Breiman, “Probability,” SIAM, Philadelphia, 1992. doi:10.1137/1.9781611971286
[11] J. Kevorkian, “Partial Differential Equations: Analytical Solution Techniques,” Wadsworth, Belmont, 1990.
[12] F. John, “Partial Differential Equations,” 4th Edition, Springer-Verlag, New York, 1982.
[13] S. M. Ross, “Stochastic Processes,” John Wiley & Sons, New York, 1983.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.