AM  Vol.4 No.5 , May 2013
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.

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.

[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.