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Solutions for some diffusion processes with two barriers

Published online by Cambridge University Press:  14 July 2016

A. L. Sweet  and
J. C. Hardin
A. L. Sweet
Affiliation:
Purdue University, Lafayette, Indiana
J. C. Hardin
Affiliation:
NASA Langley Research Center, Hampton, Virginia
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Extract

Use is often made of the Wiener and Ornstein-Uhlenbeck (O.U.) processes in various applications of stochastic processes to problems of engineering interest. These applications frequently involve the presence of barriers. Although mathematical methods for solving Kolmogorov's forward equation for the above processes have previously been discussed ([1], [2]), many solutions for problems with two barriers do not seem to be available in the literature. Instead, one finds solutions for unrestricted processes or simulation used in place of analytical solutions in various applications ([3], [4], [5]). In this paper, solutions of Kolmogorov's forward equations in the presence of constant absorbing and/or reflecting barriers are obtained by means of separation of variables. This enables one to obtain expressions for the probability density functions for first passage times when absorbing barriers are present. The solution for the O.U. process is used to obtain a result of Breiman's [6] concerning first passage times.

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 7 , Issue 2 , August 1970 , pp. 423 - 431
Copyright
Copyright © Applied Probability Trust 1970 

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References

[1] Bharucha-Reid, A. T. (1960) Elements of the Theory of Markov Processes and Their Applications. McGraw–Hill, New York. 139–142.Google Scholar
[2] Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. J. Wiley and Sons, New York. Chapter 5.Google Scholar
[3] Hinze, J. O. (1959) Turbulence. McGraw-Hill, New York. Chapter 5.Google Scholar
[4] Savas, E. L. (1967) Computers in urban air pollution control systems. Socio-Economic Planning Sciences 1, 157–183.Google Scholar
[5] Bugliarello, G. and Jackson, E. D. (1964) Random walk study of convective diffusion. J. Eng. Mech. Div., A.S.C.E. 90, 49–77.Google Scholar
[6] Breiman, L., (1967) First exit times from a square root boundary. Proc. Fifth Berk. Symp. on Math. Statist. and Prob. 2, 9–16.Google Scholar
[7] Chandrasekhar, S. (1943) Stochastic problems in physics and astronomy. Rev. Modern Phys. 15, 1–89.Google Scholar
[8] Gaver, D. P. (1968) Diffusion approximations and models for certain congestion problems. J. Appl. Prob. 5, 607–623.Google Scholar
[9] Hardin, J. C. (1969) A Stochastic Model of Turbulent Channel Flow. Ph. D. Thesis, Purdue University.Google Scholar
[10] Smith, M., Editor (1968) Recommended Guide for the Prediction of the Dispersion of Airborne Effluents. Amer. Soc. Mech. Eng. 47.Google Scholar
[11] Carslaw, H. S. and Jaegar, J. C. (1947) Conduction of Heat in Solids. Clarendon Press, Oxford.Google Scholar
[12] Coddington, E. A. and Levinson, N. (1955) Theory of Ordinary Differential Equations. McGraw-Hill, New York. Chapter 7.Google Scholar
[13] Morse, P. M. and Feshbach, H. (1953) Methods of Theoretical Physics. McGraw-Hill, New York. Chapter 6.Google Scholar
[14] Pierce, B. O. (1929) A Short Table of Integrals. Ginn and Co., New York. 64.Google Scholar
[15] Crank, J. (1956) The Mathematics of Diffusion. Clarendon Press, Oxford.Google Scholar
[16] Uhlenbeck, G. E. and Ornstein, L. S. (1930) On the theory of Brownian motion. Phys. Rev. 36, 823–841.Google Scholar
[17] Darling, D. A. and Siegert, A. J. E. (1953) The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639.Google Scholar
[18] Whittaker, E. T. and Watson, G. N. (1952) A Course of Modern Analysis. Cambridge University Press, 347.Google Scholar
[19] Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions. Nat. Bur. Stand. Appl. Math. Series No. 55, 504.Google Scholar