Mean stochastic comparison of diffusions

  • Bruce Hajek


Stochastic bounds are derived for one dimensional diffusions (and somewhat more general random processes) by dominating one process pathwise by a convex combination of other processes. The method permits comparison of diffusions with different diffusion coefficients. One interpretation of the bounds is that an optimal control is identified for certain diffusions with controlled drift and diffusion coefficients, when the reward function is convex. An example is given to show how the bounds and the Liapunov function technique can be applied to yield bounds for multidimensional diffusions.


Diffusion Coefficient Stochastic Process Probability Theory Random Process Mathematical Biology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dellacherie, C., Meyer, P.A.: Probabilities and Potential B. New York: North-Holland 1982Google Scholar
  2. 2.
    Doob, J.L.: Stochastic Processes. New York: Wiley 1953Google Scholar
  3. 3.
    Ikeda, N., Watanabe, S.: A comparison theorem for solutions of stochastic differential equations and its applications. Osaka J. Math. 14, 619–633 (1977)Google Scholar
  4. 4.
    Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. New York: North-Holland 1981Google Scholar
  5. 5.
    Le Gall, J.F.: Application du temps local aux équations différentielles stochastiques unidimensionelles. Séminaire de Probabilités XVII, Lecture Notes in Mathematics 986, 15–31. Berlin Heidelberg New York: Springer 1982Google Scholar
  6. 6.
    Malliavin, P.: Géometrie Differentielle Stochastique. Montréal: Les Presse de l'Université de Montréal 1978Google Scholar
  7. 7.
    Meyer, P.A.: Un cours sur les intégrales stochastiques. Séminaire de Probabilités X, Lecture Notes in Mathematics 511, 245–400. Berlin Heidelberg New York: Springer 1976Google Scholar
  8. 8.
    Perkins, E.: Local time and pathwise uniqueness of solutions of stochastic differential equations. Seminaire de Probabilités XVI, Lecture Notes in Mathematics 920, 201–208. Berlin Heidelberg New York: Springer 1982Google Scholar
  9. 9.
    Skorohod, A.V.: Stochastic equations for a diffusion process in a bounded region. Theory of Probability and its Applications 6, 264–274 (1961)Google Scholar
  10. 10.
    Skorokhod, A.V.: Studies in the Theory of Random Processes. Reading, Massachusetts: Addison Wesley 1965Google Scholar
  11. 11.
    Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes, Berlin Heidelberg New York: Springer 1979Google Scholar
  12. 12.
    Wonham, W.M.: Liapunov criteria for weak stochastic stability. J. of Differential Equations 2, 195–207 (1966)Google Scholar
  13. 13.
    Yamada, T.: On the uniqueness of solution of stochastic differential equations with reflecting barrier conditions. Seminaire de Probabilités X, Lecture Notes in Mathematics 511, 240–244. Berlin Heidelberg New York: Springer 1976Google Scholar
  14. 14.
    Yor, M.: Sur la continuité des temps locaux associes á certaines semimartingales. Astérisque 52–53, 23–35 (1978)Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Bruce Hajek
    • 1
  1. 1.Department of Electrical and Computer Engineering and the Coordinated Science LaboratoryUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations