Regularities of local times of two-parameter martingales
type: Item
instance of: scholarly article
title: Regularities of local times of two-parameter martingales
zbMATH Open document ID: 1090.60039
author: Jicheng Liu
author: Jiagang Ren
published in: Journal of Mathematics of Kyoto University
publication date: 2006-03-09T00:00:00Z
review text: Let \(X\) be the space of all continuous \(\omega :[0,1]^{2}\rightarrow \)\textbf{R}, vanishing on the axes, \(H\) be the set of all absolutely continuous \(h\in X\), with finite \(\| h\|_{H}^{2}=\int h_{s t}dsdt\) and \(\mu\) be the 2-parameter Wiener measure on \(X\). Let \(L\) be the Ornstein Uhlenbeck operator, \(D_{r}^{p}\) be the Sobolev space with the norm \(\| F\|_{p,2 r}=\) \(\| (1-L)^{r}F\|_{p}\), \(w_{z}(\omega )=\omega (z)\) for \(z\in [0,1]^{2}\) and \(M\) be an \(L^{2}\)-bounded martingale with respect to the filtration of \((w_{z})\), vanishing on the axes, represented (Wong-Zakai) as \[ M_{z}=\int_{[0,z]}\varphi (\xi )dw(\xi )+\iint_{[0,z]\times [0,z]}\psi (\xi ,\eta )dw(\xi )dw(\eta ), \] with adapted \(\varphi ,\psi\). \(M_{z}\) can also be represented as \(\int_{[0,z]}q_{1}(z,\eta )dw_{\eta }=\int_{[0,z]}q_{2}(z,\xi )dw_{\xi }\) and let \(M_{z}^{\sim }\) denote \(\iint_{[0,z]\times [0,z]}q_{1}(\xi \vee \eta ,\eta )q_{2}(\xi \vee \eta ,\xi )\,dw_{\xi }\,dw_{\eta}\). If, for \(\omega \in X\), \(\tau_{1}(\cdot ,\omega )\), \(\tau_{2}(\cdot ,\omega )\) denote the measures on \([0,1]^{2}\) defined by \(M^{1}= M^{\sim }\), \(M^{2}= M\), respectively, then \(L_{i}(z,a,\omega )\), \(z\in [0,1]^{2}\), \(a\in {\mathbb R}\), \(i=1,2\), are known to exist such that \(\int F(a)L_{i}(z,a,\omega )da=\int_{[0,z]}F(M_{\xi }^{i})(\omega )\tau_{i}(d\xi ,\omega )\). The brackets \(\langle M\rangle\), \(\langle M_{\cdot t}\rangle\) can be represented as \(\langle M\rangle _{z}=\int_{[0,z]}g(u,v)\,dudv\) and \(\langle M_{\cdot t}\rangle _{s}=\int_{[0,s]}f(v,t)dv\). If \(L^{1}(s,t,a)\) is the \(M_{\cdot t}\)-local time, let \[ L^{1,(n)}(u,v,a)=L^{1}(u,v,a)-L^{1}(1/n,v,a), \] \[ L_{2,n}(z,a)=\int_{[(1/n,1/n),z]}g(u,v)f(u,v)^{-1}L^{1,(n)}(du,v,a)dv. \] The authors prove the following theorems: 3.1. If \(\varphi (\xi ),\psi (\xi ,\eta )\in \bigcap_{1< p<\infty }D_{1}^{p}\) and \(\int_{_{_{[0,1]}2}}\| \varphi (\xi )\|_{p,1}^{p}d\xi +\int \int_{[0,1]^2\times [0,1]^2}\| \psi …
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 5010261
zbMATH Keywords: Wong-Zakai representation of two parameter martingales
zbMATH Keywords: Sobolev space
reviewed by: Ion Cuculescu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250283081
OpenAlex ID: W1529151717
DOI: 10.1215/KJM/1250283081
description: scientific article; zbMATH DE number 5010261