Vague convergence of locally integrable martingale measures
type: Item
instance of: scholarly article
title: Vague convergence of locally integrable martingale measures
zbMATH Open document ID: 0808.60009
DOI: 10.1016/0304-4149(94)90025-6
author: Yingchao Xie
published in: Stochastic Processes and their Applications
publication date: 1995-03-12T00:00:00Z
review text: The author introduces a concept of vague convergence of locally integrable (orthogonal) martingale measures as follows. Let \(M^ n\), \(n \in \mathbb{N}\), and \(M\) be \(F_ t\)-martingale measures on a locally compact Hausdorff space \(\mathbf E\), given a filtered probability space \((\Omega, F, F_ t, P)\). \(M^ n\) vaguely converges to \(M\) in distribution \((M^ n @>v{\mathcal L}>> M)\) if, for all functions \(f\in C_ K( \mathbb{R}_ + \times {\mathbf E})\) (continuous, with a compact support), \[ \int^ \bullet _ 0 \int_ E f(s,x) M^ n (ds, dx) @>{\mathcal L}>> \int^ \bullet_ 0 \int_ E f(s,x) M(ds,dx),\quad n \to \infty. \] (\(@>{\mathcal L}>>\) denotes the convergence in distribution.) Conditions for vague convergence are provided in terms of convergence of characteristics of martingale measures. The convergence of stochastic integrals \(f^ n \cdot M^ n @>v{\mathcal L}>> f \cdot M\) and \[ \int^ \bullet _ 0 \int_ E f^ n (s,x) M^ n (ds,dx) @>{\mathcal L}>> \int^ \bullet_ 0 \int_ E f(s, x) M(ds, dx), \quad n \to \infty, \] is also studied when \(f^ n \to f\) in a certain sense.
Mathematics Subject Classification ID: 60B10
Mathematics Subject Classification ID: 60G44
zbMATH DE Number: 691005
zbMATH Keywords: weak convergence
zbMATH Keywords: vague convergence of locally integrable (orthogonal) martingale measures
zbMATH Keywords: convergence of characteristics of martingale measures
zbMATH Keywords: stochastic integrals
reviewed by: Vigirdas Mackevičius
MaRDI profile type: Publication
full work available at URL: 0304-4149(94)90025-6
OpenAlex ID: W2072671615
cites work: Q5560061
cites work: Martingale measures and stochastic calculus
cites work: Q3774629
cites work: Q1099492
cites work: Q4776689
cites work: Limit theorems for point processes and their functionals
cites work: Q3979063
cites work: On the convergence of vector random measures
cites work: Q3747436
description: scientific article; zbMATH DE number 691005