A stability result for solutions of stochastic equations driven by point processes
type: Item
instance of: scholarly article
title: A stability result for solutions of stochastic equations driven by point processes
zbMATH Open document ID: 0827.60045
DOI: 10.1007/BF00970944
author: Pio Andrea Zanzotto
published in: Lithuanian Mathematical Journal
publication date: 1995-09-18T00:00:00Z
review text: We consider a sequence \((X^n)_n\) of processes which satisfy stochastic differential equations of the following form: \[ dX^n_t = \int_{\{|u |\leq 1\}} f^n \bigl( X^n_\bullet (\omega), s,u \bigr) q^n (ds,du) + \int_{\{|u |> 1\}} f^n \bigl( X^n_\bullet (\omega), s,u \bigr) p^n (ds,du), \] where \(p^n\) is a stochastic point process, i.e., an integer-valued random measure \(p^n (\omega, ds,du)\) (defined on a given stochastic basis) and \(q^n\) stands for the martingale measure \(p^n - \lambda^n\), \(\lambda^n\) denoting the compensator of \(p^n\); the coefficients \(f^n (X^n_\bullet (\omega), t,u)\), for every couple \((t,u)\), may depend on the whole past of the solution-process before time \(t\). For the processes \((X^n)_n\) we prove a stability theorem in the sense of weak convergence of their laws \((\Lambda^n)_n\) on the Skorokhod space. In fact, we state sufficient conditions under which, as \(n\) tends to infinity, \(\Lambda^n\) converges narrowly (or weakly) to the law of the solution \(X\) of the following equation: \[ dX_t = \int_{\{|u |\leq 1\}} f \bigl( X_\bullet (\omega), s,u \bigr) q(ds, du) + \int_{\{|u |> 1\}} f \bigl( X_\bullet (\omega), s,u \bigr) p(ds,du), \] where \(p\) is a Poisson random measure with a preassigned Lévy measure \(\alpha\) and \(q = p - ds \otimes \alpha (du)\).
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 93E15
zbMATH DE Number: 794495
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: stochastic point process
zbMATH Keywords: martingale measure
zbMATH Keywords: compensator
zbMATH Keywords: stability theorem
zbMATH Keywords: weak convergence
zbMATH Keywords: Poisson random measure
zbMATH Keywords: Lévy measure
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00970944
OpenAlex ID: W2043680337
description: scientific article; zbMATH DE number 794495