Electronic Communications in Probability

On mean numbers of passage times in small balls of discretized Itô processes

Frédéric Bernardin, Mireille Bossy, Miguel Martinez, and Denis Talay

Full-text: Open access

Abstract

The aim of this note is to prove estimates on mean values of the number of times that Itô processes observed at discrete times visit small balls in $\mathbb{R}^d$. Our technique, in the innite horizon case, is inspired by Krylov's arguments in [2, Chap.2]. In the finite horizon case, motivated by an application in stochastic numerics, we discount the number of visits by a locally exploding coeffcient, and our proof involves accurate properties of last passage times at 0 of one dimensional semimartingales.

Article information

Source
Electron. Commun. Probab., Volume 14 (2009), paper no. 30, 302-316.

Dates
Accepted: 25 July 2009
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465234739

Digital Object Identifier
doi:10.1214/ECP.v14-1479

Mathematical Reviews number (MathSciNet)
MR2524981

Zentralblatt MATH identifier
1189.60108

Subjects
Primary: 60G99: None of the above, but in this section

Keywords
Diffusion processes sojourn times estimates discrete times

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Bernardin, Frédéric; Bossy, Mireille; Martinez, Miguel; Talay, Denis. On mean numbers of passage times in small balls of discretized Itô processes. Electron. Commun. Probab. 14 (2009), paper no. 30, 302--316. doi:10.1214/ECP.v14-1479. https://projecteuclid.org/euclid.ecp/1465234739


Export citation

References

  • Bernardin, F.; Bossy, M.; Talay, D.. Viscosity solutions of parabolic differential inclusions and weak convergence rate of discretizations of multi-valued stochastic differential equations.In preparation.
  • Krylov, N. V. Controlled diffusion processes. Translated from the Russian by A. B. Aries. Applications of Mathematics, 14. Springer-Verlag, New York-Berlin, 1980. xii+308 pp. ISBN: 0-387-90461-1
  • Krylov, N. V.; Liptser, R. On diffusion approximation with discontinuous coefficients. Stochastic Process. Appl. 102 (2002), no. 2, 235–264.
  • Martinez, M.; Talay, D.. Time discretization of one dimensional Markov processes with generators under divergence form with discontinuous coefficients.Submitted for publication, 2008.
  • Protter, P. E. Stochastic integration and differential equations. Second edition. Version 2.1. Corrected third printing. Stochastic Modelling and Applied Probability, 21. Springer-Verlag, Berlin, 2005. xiv+419 pp. ISBN: 3-540-00313-4
  • Revuz, D.; Yor, M.. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7 357
  • Talay, Denis. Probabilistic numerical methods for partial differential equations: elements of analysis. Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995), 148–196, Lecture Notes in Math., 1627, Springer, Berlin, 1996.