On Itô stochastic integral with respect to vector stable random measures
type: Item
instance of: scholarly article
title: On Itô stochastic integral with respect to vector stable random measures
zbMATH Open document ID: 0885.60041
author: Dang Hung Thang
published in: Acta Mathematica Vietnamica
publication date: 1998-04-20T00:00:00Z
review text: Fix \(q\in [1,2]\) and a \(q\)-smoothable Banach space \(X\), that is to say satisfying: \(\mathbb{E} [|M_n|^q] \leq C\sum^n_{j=1} \mathbb{E} [|M_j -M_{j-1} |^q]\) for any \(X\)-valued martingale \(M\) and any \(n\in \mathbb{N}\). Fix then \(p<q\), and an \(X\)-valued \(p\)-stable random measure \(Z_p\), with characteristic measure \(Q_p\). The author shows that it exists a stochastic integral on \(L^p(|Q_p |\otimes \mathbb{P})\)-adapted real processes \(u\) on \([0, 1]\): \(u\mapsto \int^1_0 u dZ_p\), extending the obvious integral on simple processes \(u\), and continuous from \(L^p(|Q_p |\otimes \mathbb{P})\) into \(L^0_X (\Omega)\). It is also continuous from \(L^2(|Q_2 |\otimes \mathbb{P})\) into \(L^2_X (\Omega)\), for \(p=q=2\). In this last case, the covariance operator of \(\int^1_0udZ_2\) is \(\int^1_0 \mathbb{E} (|u_t |^2)dQ_2\).
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 46G10
Mathematics Subject Classification ID: 60G57
zbMATH DE Number: 1083728
zbMATH Keywords: stochastic integral
zbMATH Keywords: stable random measure
zbMATH Keywords: smoothable Banach space
zbMATH Keywords: characteristic measure
zbMATH Keywords: covariance operator
reviewed by: Jacques Franchi
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1083728