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

Frédéric Bernardin,Mireille Bossy,Denis Talay,Miguel Martinez

doi:10.1214/ecp.v14-1479

Abstract

The aim of this note is to prove estimates on mean values of the number of times that Ito 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.

Highlights

The present work is motivated by two convergence rate analyses concerning discretization schemes for diffusion processes whose generators have non smooth coefficients: Bernardin et al [1] study discretization schemes for stochastic differential equations with multivalued drift coefficients; Martinez and Talay [4] study discretization schemes for diffusion processes whose infinitesimal generators are of divergence form with discontinuous coefficients
To have reasonable accuracies, a scheme needs to mimic the local behaviour of the exact solution in small neighborhoods of the discontinuities of the coefficients, in the sense that the expectations of the total times spent by the scheme and the exact solution in these neighborhoods need to be close to each other
The objective of this paper is to provide two estimates for the expectations of such total times spent by fairly general Itô processes observed at discrete times; these estimates concern in particular the discretization schemes studied in the above references

Summary

Introduction

The present work is motivated by two convergence rate analyses concerning discretization schemes for diffusion processes whose generators have non smooth coefficients: Bernardin et al [1] study discretization schemes for stochastic differential equations with multivalued drift coefficients; Martinez and Talay [4] study discretization schemes for diffusion processes whose infinitesimal generators are of divergence form with discontinuous coefficients. The objective of this paper is to provide two estimates for the expectations of such total times spent by fairly general Itô processes observed at discrete times; these estimates concern in particular the discretization schemes studied in the above references (see section 7 for more detailed explanations). To this end, we carefully modify the technique developed by Krylov [2] to prove inequalities of the type. It suffices to prove Theorems 1.1 and 1.2 for all one dimensional Itô process t

An estimate for a localizing stopping time

Estimate for the localized problem in the infinite horizon setting