Quasi sure quadratic variation of local times of smooth semimartingales.
type: Item
instance of: scholarly article
title: Quasi sure quadratic variation of local times of smooth semimartingales.
zbMATH Open document ID: 1055.60054
author: Kai He
author: Jiagang Ren
published in: Bulletin des Sciences Mathématiques
publication date: 2004-03-14T00:00:00Z
review text: Let \(L^x_t\) denote the local time of a continuous semimartingale \((M_t)\). Let \(\Delta_n= (a_{n,i}, a_{n,i+1})\) be any sequence of partitions of \([a,b]\) with mesh size converging to zero. It was shown by \textit{N. Bouleau} and \textit{M. Yor} [C. R. Acad. Sci., Paris, Sér. I 292, 491--494 (1981; Zbl 0476.60046)] that \[ \lim_{n\to\infty} \sum_{\Delta_n} (L^{a_{n,i+1}}_t- L^{a_{n,i}}_t)^2= 4\int^b_a L^x t \,dx\;\text{ in probability}. \tag{+} \] In the present paper it is shown for continuous semimartingales \((M_t)\) which are smooth in the sense of \textit{P. Malliavin} and \textit{D. Nualart} [J. Funct. Anal. 112, No. 2, 287--317 (1993; Zbl 0783.60006)] that the convergence in \((+)\) even holds almost surely for partitions \(\Delta_n= (a_{n,i}, a_{n,i+1})\) such that \(a_{n,i}= i(b- a)/2^n\), \(i= 0,1,\dots, 2^n\).
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 31C15
Mathematics Subject Classification ID: 31C25
zbMATH DE Number: 2055854
reviewed by: Klaus Schürger
MaRDI profile type: Publication
full work available at URL: j.bulsci.2003.11.001
OpenAlex ID: W1965044052
cites work: Smoothness of local times of semimartingales
cites work: Q3933721
cites work: Infinite dimensional quasi continuity, path continuity and ray continuity of functions with fractional regularity
cites work: Q3336457
cites work: Quasi sure analysis of stochastic flows and Banach space valued smooth functionals on the Wiener space
cites work: Smoothness of Brownian local times and related functionals
cites work: Local time is a semi-martingale
cites work: Q3488968
cites work: Quasi sure quadratic variation of smooth martingales
cites work: Q3997782
cites work: On a quasi everywhere existence of the local time of the 1-dimensional Brownian motion
cites work: Q4169289
cites work: Fractional order Sobolev spaces on Wiener space
DOI: 10.1016/J.BULSCI.2003.11.001
description: scientific article; zbMATH DE number 2055854