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On the first-exit time problem for temporally homogeneous Markov processes

Published online by Cambridge University Press:  14 July 2016

Henry C. Tuckwell
Henry C. Tuckwell*
Affiliation:
University of Chicagocor1corresp
*
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Abstract

Using an integral equation of Darling and Siegert in conjunction with the backward Kolmogorov equation for the transition probability density function, recurrence relations are derived for the moments of the time of first exit of a temporally homogeneous Markov process from a set in the phase space. The results, which are similar to those for diffusion processes, are used to find the expectation of the time between impulses of a Stein model neuron.

Keywords

FIRST-EXIT TIMESMARKOV PROCESSESRECURRENCE RELATIONSNEURON
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 1 , March 1976 , pp. 39 - 48
Copyright
Copyright © Applied Probability Trust 1976 

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References

Borovkov, A. A. (1965) On the first passage time for one class of processes with independent increments. Theor. Prob. Appl. 10, 331–334.Google Scholar
Chuang, K. (1970) A study of first passage times and their control. Int. J. Control 12, 849–856.Google Scholar
Darling, D. A. and Siegert, A. J. F. (1953) The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639.Google Scholar
Darling, D. A. and Siegert, A. J. F. (1957) A systematic approach to a class of problems in the theory of noise and other random phenomena — part I. IRE Trans. Inf. Theory 3, 32–37.Google Scholar
Eccles, J. C. (1964) The Physiology of Synapses. Academic Press, New York.Google Scholar
Gihman, I. I. and Skorohod, A. V. (1972) Stochastic Differential Equations. Springer-Verlag, New York, Heidelberg, Berlin.Google Scholar
Gusak, D. V. and Koralyuk, V. S. (1968) On the first passage time across a given level for processes with independent increments. Theor. Prob. Appl. 13, 448–456.Google Scholar
Keilson, J. (1963) The first passage time density for homogeneous skip-free walks on the continuum. Ann. Math. Statist. 34, 1003–1011.Google Scholar
Redman, S. J., Lampard, D. G. and Annal, P. (1968) Monosynaptic stochastic stimulation of cat spinal motoneurons II. Frequency transfer characteristics of tonically discharging motoneurons. J. Neurophysiol. 31, 499–508.Google Scholar
Rogozin, B. A. (1965) On some classes of processes with independent increments. Theor. Prob. Appl. 10, 479–483.Google Scholar
Shtatland, E. S. (1965) On the distribution of the maximum of a process with independent increments. Theor. Prob. Appl. 10, 483–487.Google Scholar
Siegert, A. J. F. (1951) On the first passage time probability problem. Phys. Rev. 81, 617–623.Google Scholar
Stein, R. B. (1965) A theoretical analysis of neuronal variability. Biophys. J. 5, 173–194.Google Scholar
Tuckwell, H. C. (1975) Determination of the inter-spike times of neurons receiving randomly arriving post-synaptic potentials. Biol. Cybernetics 17, 225–237.Google Scholar
Wang, M. C. and Uhlenbeck, G. E. (1945) On the theory of Brownian motion II. Rev. Mod. Phys. 17, 323–342.Google Scholar
Zolotarev, V. M. (1964) The first passage time of a level and the behaviour at infinity for a class of processes with independent increments. Theor. Prob. Appl. 9, 653–662.Google Scholar