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Stability of Doob-Meyer decomposition under extended convergence. (English) Zbl 1052.60022

Summary: We consider the relation between Aldous’ extended convergence and weak convergence of filtrations. We prove that, for a sequence \((X^n)\) of \({\mathcal F}_t^n\)-special semimartingales, with canonical decomposition \(X^n=M^n+A^n\), if the extended convergence \((X^n,{\mathcal F}^n_.) \to(X,{\mathcal F}_.)\) holds with a quasi-left continnous \(({\mathcal F}_t)\)-special semimartingale \(X=M+A\), then, under an additional assumption of uniform integrability, we get the convergence in probability under the Skorokhod topology: \(M^n @>P>> M\) and \(A^n @>P>> A\).

MSC:

60F17 Functional limit theorems; invariance principles
60G07 General theory of stochastic processes
60G48 Generalizations of martingales
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