Electronic Communications in Probability

Strong Feller property and continuous dependence on initial data for one-dimensional stochastic differential equations with Hölder continuous coefficients

Hua Zhang

Full-text: Open access

Abstract

In this paper, under the assumption of Hölder continuous coefficients, we prove the strong Feller property and continuous dependence on initial data for the solution to one-dimensional stochastic differential equations whose proof are based on the technique of local time, coupling method and Girsanov’s transform.

Article information

Source
Electron. Commun. Probab., Volume 25 (2020), paper no. 3, 10 pp.

Dates
Received: 1 November 2018
Accepted: 4 January 2020
First available in Project Euclid: 7 January 2020

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1578387617

Digital Object Identifier
doi:10.1214/20-ECP284

Zentralblatt MATH identifier
07149366

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
strong Feller property continuous dependence of initial data stochastic differential equations Hölder continuous coefficients local time coupling method Girsanov’s transform

Rights
Creative Commons Attribution 4.0 International License.

Citation

Zhang, Hua. Strong Feller property and continuous dependence on initial data for one-dimensional stochastic differential equations with Hölder continuous coefficients. Electron. Commun. Probab. 25 (2020), paper no. 3, 10 pp. doi:10.1214/20-ECP284. https://projecteuclid.org/euclid.ecp/1578387617


Export citation

References

  • [1] Elworthy K. D. and Li X. M.: Formulae for the derivatives of heat semigroups. J. Func. Anal., 125, (1994), 252–286.
  • [2] Ikeda N. and Watanabe S.: Stochastic Differential Equations and Diffusion Processes. North-Holland, Amsterdam, 1981.
  • [3] Karatzas I. and Shreve S.: Brownian Motion and Stochastic Calculus. Springer, New York, 1988.
  • [4] Li H. Q., Luo D. J. and Wang J.: Harnack inequalities for SDEs with multiplicative noise and non-regular drift. Stoch. Dynam., 15, (2015), 1550015 (18 pages).
  • [5] Wang F. Y.: Harnack inequalities for stochastic partial differential equations. Springer, New York, 2013.
  • [6] Wang Z.: Euler scheme and measurable flows for stochastic differential equations with non-Lipschitz coefficients. Acta Mathematica Scientia 2018, 38B, (2018), 157–168.
  • [7] Yan L. Q.: The Euler scheme with irregular coefficients. Ann. Probab., 30, (2002), 1172–1194.
  • [8] Zhang X. C.: Exponential ergodicity of non-Lipschitz stochastic differential equations. Proc. Amer. Math. Soc., 137, (2009), 329–337.
  • [9] Zhang X. C.: Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients. Electron. J. Probab., 16, (2011), 1096–1116.