Skip to main content

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
Cancel
×
×
Home
  • Home
Article
  • Cited by 18
  • Cited by
    Crossref Citations
    Crossref logo
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.
    Chang, Liu Chen, Shouting and Zhu, Ailin 2015. Properties of the Cox–Ingersoll–Ross Interest Rate Processes with Two-sided Reflections. Communications in Statistics - Theory and Methods, Vol. 44, Issue. 4, p. 657.
    Yu Liu and Li, X. Rong 2013. Average run length function of CUSUM test with independent but non-stationary log-likelihood ratios. p. 2803.
    Giorno, V. Nobile, A. G. and Ricciardi, L. M. 2011. On the densities of certain bounded diffusion processes. Ricerche di Matematica, Vol. 60, Issue. 1, p. 89.
    Zhang, Xiaowei and Glynn, Peter W. 2011. On the dynamics of a finite buffer queue conditioned on the amount of loss. Queueing Systems, Vol. 67, Issue. 2, p. 91.
    Dominé, Marco 1996. First passage time distribution of a Wiener process with drift concerning two elastic barriers. Journal of Applied Probability, Vol. 33, Issue. 01, p. 164.
    Louchard, G. 1994. Large finite population queueing systems. The single-server model. Stochastic Processes and their Applications, Vol. 53, Issue. 1, p. 117.
    Kobayashi, H. and Ren, Q. 1993. A diffusion approximation analysis of an ATM statistical multiplexer with multiple types of traffic. - I. Equilibrium state solutions. Vol. 2, Issue. , p. 1047.
    Berger, Arthur W. and Whitt, Ward 1992. The Brownian approximation for rate-control throttles and the G/G/1/C queue. Discrete Event Dynamic Systems, Vol. 2, Issue. 1, p. 7.
    Sivazlian, B. D. and Wang, K. H. 1990. Diffusion approximation to theG/G/R machine repair problem with warm standby spares. Naval Research Logistics, Vol. 37, Issue. 5, p. 753.
    Pronios, N.B. and Polydoros, A. 1988. Approximate methods for analyzing slotted random-access spread-spectrum networks. p. 916.
    GOLDMAN, M. BARRY SOSIN, HOWARD B. and SHEPP, LAWRENCE A. 1979. On Contingent Claims that Insure Ex-post Optimal Stock Market Timing. The Journal of Finance, Vol. 34, Issue. 2, p. 401.
    Bagshaw, Michael and Johnson, Richard A. 1975. The Effect of Serial Correlation on the Performance of CUSUM Tests II. Technometrics, Vol. 17, Issue. 1, p. 73.
    Beekman, John A. 1975. Asymptotic distributions for the Ornstein-Uhlenbeck process. Journal of Applied Probability, Vol. 12, Issue. 01, p. 107.
    Bagshaw, Michael and Johnson, Richard A. 1975. Sequential detection of a drift change in a wiener process. Communications in Statistics, Vol. 4, Issue. 8, p. 787.
    Dirkse, John P. 1975. An absorption probability for the Ornstein-Uhlenbeck process. Journal of Applied Probability, Vol. 12, Issue. 03, p. 595.
    Lindenberg, K. Shuler, K. E. Freeman, J. and Lie, T. J. 1975. First passage time and extremum properties of Markov and independent processes. Journal of Statistical Physics, Vol. 12, Issue. 3, p. 217.
    Samuel-Cahn, Ester 1974. Repeated significance tests I and II. generalizations. Communications in Statistics, Vol. 3, Issue. 8, p. 735.
    Durbin, J. 1971. Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. Journal of Applied Probability, Vol. 8, Issue. 03, p. 431.
    ×
  • Get access
    Add to cart USD35.00
    Check if you have access via personal or institutional login
    Log in Register Recommend to librarian

Solutions for some diffusion processes with two barriers

  • A. L. Sweet (a1) and J. C. Hardin (a2)
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.

Copyright
COPYRIGHT: © Applied Probability Trust 1970 
References
Hide All
[1] Bharucha-Reid, A. T. (1960) Elements of the Theory of Markov Processes and Their Applications. McGraw–Hill, New York. 139–142.
[2] Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. J. Wiley and Sons, New York. Chapter 5.
[3] Hinze, J. O. (1959) Turbulence. McGraw-Hill, New York. Chapter 5.
[4] Savas, E. L. (1967) Computers in urban air pollution control systems. Socio-Economic Planning Sciences 1, 157–183.
[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.
[6] Breiman, L., (1967) First exit times from a square root boundary. Proc. Fifth Berk. Symp. on Math. Statist. and Prob. 2, 9–16.
[7] Chandrasekhar, S. (1943) Stochastic problems in physics and astronomy. Rev. Modern Phys. 15, 1–89.
[8] Gaver, D. P. (1968) Diffusion approximations and models for certain congestion problems. J. Appl. Prob. 5, 607–623.
[9] Hardin, J. C. (1969) A Stochastic Model of Turbulent Channel Flow. Ph. D. Thesis, Purdue University.
[10] Smith, M., Editor (1968) Recommended Guide for the Prediction of the Dispersion of Airborne Effluents. Amer. Soc. Mech. Eng. 47.
[11] Carslaw, H. S. and Jaegar, J. C. (1947) Conduction of Heat in Solids. Clarendon Press, Oxford.
[12] Coddington, E. A. and Levinson, N. (1955) Theory of Ordinary Differential Equations. McGraw-Hill, New York. Chapter 7.
[13] Morse, P. M. and Feshbach, H. (1953) Methods of Theoretical Physics. McGraw-Hill, New York. Chapter 6.
[14] Pierce, B. O. (1929) A Short Table of Integrals. Ginn and Co., New York. 64.
[15] Crank, J. (1956) The Mathematics of Diffusion. Clarendon Press, Oxford.
[16] Uhlenbeck, G. E. and Ornstein, L. S. (1930) On the theory of Brownian motion. Phys. Rev. 36, 823–841.
[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.
[18] Whittaker, E. T. and Watson, G. N. (1952) A Course of Modern Analysis. Cambridge University Press, 347.
[19] Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions. Nat. Bur. Stand. Appl. Math. Series No. 55, 504.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

CAPTCHA *

Skip to the audio challenge
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 8 *
Loading metrics...

Abstract views

Total abstract views: 121 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 12th June 2018. This data will be updated every 24 hours.

Cancel
Confirm
×