The Annals of Probability

On approximate continuity and the support of reflected stochastic differential equations

Jiagang Ren and Jing Wu

Full-text: Open access

Abstract

In this paper we prove an approximate continuity result for stochastic differential equations with normal reflections in domains satisfying Saisho’s conditions, which together with the Wong–Zakai approximation result completes the support theorem for such diffusions in the uniform convergence topology. Also by adapting Millet and Sanz-Solé’s idea, we characterize in Hölder norm the support of diffusions reflected in domains satisfying the Lions–Sznitman conditions by proving limit theorems of adapted interpolations. Finally we apply the support theorem to establish a boundary-interior maximum principle for subharmonic functions.

Article information

Source
Ann. Probab., Volume 44, Number 3 (2016), 2064-2116.

Dates
Received: October 2014
Revised: March 2015
First available in Project Euclid: 16 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1463410039

Digital Object Identifier
doi:10.1214/15-AOP1018

Mathematical Reviews number (MathSciNet)
MR3502601

Zentralblatt MATH identifier
1347.60072

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H99: None of the above, but in this section
Secondary: 60F99: None of the above, but in this section

Keywords
Reflected stochastic differential equation approximate continuity support limit theorem maximum principle

Citation

Ren, Jiagang; Wu, Jing. On approximate continuity and the support of reflected stochastic differential equations. Ann. Probab. 44 (2016), no. 3, 2064--2116. doi:10.1214/15-AOP1018. https://projecteuclid.org/euclid.aop/1463410039


Export citation

References

  • [1] Aida, S. and Sasaki, K. (2013). Wong–Zakai approximation of solutions to reflecting stochastic differential equations on domains in Euclidean spaces. Stochastic Process. Appl. 123 3800–3827.
  • [2] Anderson, R. F. and Orey, S. (1976). Small random perturbation of dynamical systems with reflecting boundary. Nagoya Math. J. 60 189–216.
  • [3] Doss, H. and Priouret, P. (1982). Support d’un processus de réflexion. Z. Wahrsch. Verw. Gebiete 61 327–345.
  • [4] Evans, L. C. and Stroock, D. W. (2011). An approximation scheme for reflected stochastic differential equations. Stochastic Process. Appl. 121 1464–1491.
  • [5] Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland Mathematical Library 24. North-Holland, Amsterdam.
  • [6] Lions, P.-L. and Sznitman, A.-S. (1984). Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 511–537.
  • [7] Millet, A. and Sanz-Solé, M. (1994). The support of the solution to a hyperbolic SPDE. Probab. Theory Related Fields 98 361–387.
  • [8] Millet, A. and Sanz-Solé, M. (1994). A simple proof of the support theorem for diffusion processes. In Séminaire de Probabilités, XXVIII. Lecture Notes in Math. 1583 36–48. Springer, Berlin.
  • [9] Pettersson, R. (1999). Wong–Zakai approximations for reflecting stochastic differential equations. Stoch. Anal. Appl. 17 609–617.
  • [10] Ren, J. and Xu, S. (2010). Support theorem for stochastic variational inequalities. Bull. Sci. Math. 134 826–856.
  • [11] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin.
  • [12] Saisho, Y. (1987). Stochastic differential equations for multidimensional domain with reflecting boundary. Probab. Theory Related Fields 74 455–477.
  • [13] Stroock, D. W. and Varadhan, S. R. S. (1972). On the support of diffusion processes with applications to the strong maximum principle. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability Theory 333–359. Univ. California Press, Berkeley, CA.
  • [14] Tanaka, H. (1979). Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 163–177.
  • [15] Wong, E. and Zakai, M. (1965). On the relation between ordinary and stochastic differential equations. Internat. J. Engrg. Sci. 3 213–229.
  • [16] Zhang, T. (2014). Strong convergence of Wong–Zakai approximations of reflected SDEs in a multidimensional general domain. Potential Anal. 41 783–815.