Abstract.
We study a class of stochastic differential equations with non-Lipschitz coefficients. A unique strong solution is obtained and the non confluence of the solutions of stochastic differential equations is proved. The dependence with respect to the initial values is investigated. To obtain a continuous version of solutions, the modulus of continuity of coefficients is assumed to be less than |x-y| log Finally a large deviation principle of Freidlin-Wentzell type is also established in the paper.
Article PDF
Similar content being viewed by others
References
Azencott, R.: Grande deviations et applications. Lect. Notes in Math. Vol 774, Springer-Verlag
Deuschel, J.-D., Stroock, D.W.: Large Deviations. Academic Press, Boston, San Diego, New York, (1989)
Dembo, A., Zeitouni, A.: Large Deviations Techniques and Applications. Springer-Verlag, Berlin Heidelberg, 1998
Emery, M.: Non confluence des solutions d’une equation stochastique lipschitzienne. Seminaire Proba. XV. Lecture Notes in Mathematics, vol. 850 (587–589) Pringer, Berlin Heidelberg New York 1981
Fang, S.: Canonical Brownian motion on the diffeomorphism group of the circle. J. Funct. Anal. 196, 162–179 (2002)
Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems. Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984
Fang, S., Zhang, T.S.: Stochastic differential equations with non-Lipschitz coefficients: pathwise uniqueness and no explosion. C.R. Acad. Sci. Paris Ser. I. 337, 737–740 (2003)
Fang, S., Zhang, T.S.: On the small time behavior of Ornstein-Uhlenbeck processes with unbounded linear drifts. Probability Theory and Related Fields 114 (4), 487–504 (1999)
Fang, S., Zhang, T.S.: Large deviations for the Brownian motion on loop groups. Journal of Theoretical Probability 14 (2), 463–483 (2001)
Ikeda, I., Watanabe, S.: Stochastic differential equations and Diffusion processes. North-Holland, Amsterdam, 1981
Kunita, H. : Stochastic flows and stochastic differential equations. Cambridge University Press 1990
LaSalle, J.: Uniqueness theorems and successive approximations. Annals of Mathematics 50 (3), 722–730 (1949)
Le Jan, Y., Raimond, O.: Integration of Brownian vector fields. Annals of Prob. 30 (2), 826–873 (2002)
Le Jan, Y., and Raimond, O.: Flows, coalescence and noise. Annals of Prob. 2003
Malliavin, P.: The Canonical diffusion above the diffeomorphism group of the circle. C.R. Acad. Sci. Paris, Série I 329, 325–329 (1999)
Mao, X.: Exponential stability of stochastic differential equations. Marcel Dekker, Inc. 1994
Protter, P. : Stochastic integration and differential equations. Berlin Heidelberg New York: Springer-Verlag 1990
Revuz, D., Yor, M.: Continuous martingales and Brownian motion. Grund. der Math. Wissenschaften 293, 1991, Springer-Verlag
Stroock, D.W.: An Introduction to the Theory of Large Deviations. Springer-Verlag, Berlin, 1984
Stroock, D.W., Varadhan S.R.S.: Multidimensional diffusion processes. Springer-Verlag 1979
Yamada, T., Ogura, Y.: On the strong comparison theorems for solutions of stochastic differential equations. Z. Wahrscheinlichkeitstheorie verw. Gebiete 56, 3–19 (1981)
Zhang, T.S.: On the small time asymptotics of diffusions on Hilbert spaces. Annals of Probability 28 (2), 537–557 (2000)
Zhang, T.S.: A large deviation principle of diffusions on configuration spaces. Stochastic Processes and Applications 91, 239–254 (2001)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fang, S., Zhang, T. A study of a class of stochastic differential equations with non-Lipschitzian coefficients. Probab. Theory Relat. Fields 132, 356–390 (2005). https://doi.org/10.1007/s00440-004-0398-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-004-0398-z
Keywords
- Gronwall lemma
- Non-Lipschitz conditions
- Pathwise uniqueness
- Non-explosion
- Non confluence
- Large deviation principle
- Euler approximation