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Comparison results for diffusions conditioned on positivity

Published online by Cambridge University Press:  14 July 2016

Martin V. Day
Martin V. Day*
Affiliation:
Virginia Polytechnic Institute and State University
*
∗ Postal address: Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A.
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Abstract

We consider a diffusion process on the reals subject to the conditional probability that the process is positive from t = 0 to the present. We establish comparison results between the conditioned diffusion and a second unconditioned Markov diffusion. One result allows the initial process to be non-Markov before conditioning. A stronger comparison theorem is shown to hold in the Markov case.

Keywords

DIFFUSION PROCESSCONDITIONAL PROBABILITYCOMPARISON THEOREM
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 4 , December 1983 , pp. 766 - 777
Copyright
Copyright © Applied Probability Trust 1983 

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References

[1] Darden, T. and Kurtz, T. (1983), To appear.Google Scholar
[2] Day, M. (1980) On a stochastic control problem with exit constraints. Appl. Math. Optimization 6, 181–188.Google Scholar
[3] Day, M. (1983) On the exponential exit law in the small parameter exit problem. Stochastics 8, 297–323.Google Scholar
[4] Doob, J. L. (1959) A Markov chain theorem. In Probability and Statistics, The Harald Cramér Volume, ed. Grenander, U.. Wiley, New York.Google Scholar
[5] Dynkin, E. B. (1965) Markov Processes, Vol. 1. Springer-Verlag, Berlin.Google Scholar
[6] Ewens, W. J. (1973) Conditional diffusion processes in population genetics. Theoret. Popn Biol. 4, 21–30.Google Scholar
[7] Fleming, W. H. (1978) Exit probabilities and optimal stochastic control. Appl. Math. Optimization 4, 329–346.Google Scholar
[8] Girsanov, I. B. (1960) On transforming a certain class of stochastic processes by absolutely continuous substitution of measures. Theory Prob. Appl. 5, 285–301.Google Scholar
[9] Ikeda, N. and Watanabe, S. (1977) A comparison theorem for solutions of stochastic differential equations and its applications. Osaka J. Math. 14, 619–633.Google Scholar
[10] Jamison, B. (1975) The Markov processes of Schrödinger. Z. Wahrscheinlichkeitsth. 32, 323–331.Google Scholar
[11] Schuss, Z. (1980) Singular perturbation methods in stochastic differential equations of mathematical physics. SIAM Rev. 22, 119–155.Google Scholar
[12] Ventcel, A. D. and Freidlin, M. I. (1970) On small random perturbations of dynamical systems. Uspehi Mat. Nauk 24, 1–56. (English translation: Russian Math. Surveys 25, 1–55.) Google Scholar