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

Frédéric Bernardin (CETE de Lyon)
Mireille Bossy (INRIA Sophia Antipolis Méditerrannée)
Miguel Martinez (Université Paris-Est Marne-la-Vallée, LAMA, UMR 8050 CNRS)
Denis Talay (INRIA Sophia Antipolis Méditerrannée)

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.

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Pages: 302-316

Publication Date: July 25, 2009

DOI: 10.1214/ECP.v14-1479

References

  1. 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.
  2. 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 MR0601776 (82a:60062)
  3. Krylov, N. V.; Liptser, R. On diffusion approximation with discontinuous coefficients. Stochastic Process. Appl. 102 (2002), no. 2, 235--264. MR1935126 (2003i:60056)
  4. Martinez, M.; Talay, D.. Time discretization of one dimensional Markov processes with generators under divergence form with discontinuous coefficients.Submitted for publication, 2008.
  5. 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 MR2273672 (2008e:60001)
  6. 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 MR1725 357
  7. 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. MR1431302 (98j:60092)
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