Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-29T21:05:27.383Z Has data issue: false hasContentIssue false
  • English
  • Français

First passage time distribution of a Wiener process with drift concerning two elastic barriers

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Marco Dominé
Marco Dominé*
Affiliation:
University of Magdeburg
*
∗Postal address: Department of Mathematical Stochastics, Otto-von-Guericke-University of Magdeburg, PSF 4120, 39016 Magdeburg, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

We solve the Fokker-Planck equation for the Wiener process with drift in the presence of elastic boundaries and a fixed start point. An explicit expression is obtained for the first passage density. The cases with pure absorbing and/or reflecting barriers arise for a special choice of a parameter constellation. These special cases are compared with results in Darling and Siegert [5] and Sweet and Hardin [15].

Keywords

FIRST PASSAGE TIMEFIRST EXIT TIMEELASTIC BOUNDARIESREFLECTIONABSORPTIONWIENER PROCESS

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60J65: Brownian motion 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 1 , March 1996 , pp. 164 - 175
Copyright
Copyright © Applied Probability Trust 1996 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bharucha-Reid, A. T. (1960) Elements of the Theory of Markov Processes and their Applications. McGraw-Hill, New York.Google Scholar
[2] Bhattacharya, R. N. and Waymire, E. C. (1990) Stochastic Processes with Applications. Wiley, New York.Google Scholar
[3] Blake, I. F. and Lindsey, W. C. (1973) Level crossing problem for random processes. IEEE Trans. Inf. Theory 19, 295–315.CrossRefGoogle Scholar
[4] Cox, D. R. and Miller, H. D. (1968) The Theory of Stochastic Processes. Methuen, London.Google Scholar
[5] Darling, D. A. and Siegert, A. J. F. (1953) The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639.CrossRefGoogle Scholar
[6] Domine, M. (1995) Moments of the first passage time of a Wiener process with drift between two elastic barriers. J. Appl. Prob. 32, 1007–1014.Google Scholar
[7] Domine, M. and Pieper, V. (1993) First passage time distribution of a two dimensional Wiener process with drift. Prob. Eng. Inf. Sci. 7, 545–555.Google Scholar
[8] Fienberg, S. E. (1974) Stochastic models for single neuron firing trains: a survey. Biometrics 30, 399–427.Google Scholar
[9] Feller, W. (1954) Diffusion processes in one dimension. Trans. Amer. Math. Soc. 77, 1–31.CrossRefGoogle Scholar
[10] Feller, W. (1952) The parabolic differential equations and the associated semigroups of transformations. Ann. Math. 55, 468–519.Google Scholar
[11] Gardiner, C. W. (1990) Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. 2nd edn. Springer, Berlin.Google Scholar
[12] Gerstein, G. and Mandelbrot, B. (1964) Random walk models for the spike activity of a single neuron. Biophys. J. 4, 41–68.Google Scholar
[13] Iyengar, S. (1985) Hitting lines with two-dimensional Brownian motion. SIAM J. Appl. Math. 45, 983–989.Google Scholar
[14] Ricciardi, L. M. (1977) Diffusion Processes and Related Topics in Biology. (Lect. Notes Biomath. 14), Springer, Berlin.Google Scholar
[15] Sweet, A. L. and Hardin, J. C. (1970) Solutions for some diffusion processes with two barriers. J. Appl. Prob. 7, 423–431.CrossRefGoogle Scholar