Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-15T01:32:04.175Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

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] 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