Summary
In this paper we study the asymptotic behaviour of the solution of the stochastic differential equation dX t=g(X t)dt+σ(X t)dW t, where σ and g are positive functions and W tis a Wiener process. We clarify, under which conditions X tmay be approximated on {X t→∞} by means of a deterministic function. Further the question is treated, whether X tconverges in distribution on {X t→∞. We deal with the Ito-solution as well as the Stratonovitch-solution and compare both.
Article PDF
Similar content being viewed by others
References
Arnold, L.: Stochastic differential equations: Theory and applications. New York: Wiley 1974
Feller, W.: An introduction to probability theory and its applications. Vol. II. New York: Wiley 1971
Fleming, H.W., Rishel, R.W.: Deterministic and stochastic optimal control. Berlin, Heidelberg, New York: Springer 1975
Fristedt, B., Orey, S.: The tail σ-field of one-dimensional diffusion. Stochastic analysis, Friedman, A., Pinsky, M. (eds.). New York: Academic Press 1978
Gihman, I.I., Skorohod, A.V.: Stochastic differential equations. Berlin, Heidelberg, New York: Springer 1972
Ibragimov, I.: On the composition of unimodal distributions. Theor. Probability Appl. 1, 255–260 (1956)
Karlin, S., Taylor, H.M.: A second course in stochastic processes. New York: Academic Press 1981
Mandl, P.: Analytical treatment of one-dimensional Markov processes. Berlin, Heidelberg, New York: Springer 1968
Rösler, U.: The tail σ-field of time-homogeneous one-dimensional diffusion-processes. Ann. Probability 7, 847–857 (1979)
Rösler, U.: Unimodality of passage times for one-dimensional strong Markov processes. Ann. Probability 8, 853–859 (1980)
Author information
Authors and Affiliations
Additional information
Partially supported by the SFB 123 “Stochastische Mathematische Modelle”, Heidelberg
Rights and permissions
About this article
Cite this article
Keller, G., Kersting, G. & Rösler, U. On the asymptotic behaviour of solutions of stochastic differential equations. Z. Wahrscheinlichkeitstheorie verw Gebiete 68, 163–189 (1984). https://doi.org/10.1007/BF00531776
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00531776