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On weak convergence of filtrations. (English) Zbl 0987.60009

Azéma, Jacques (ed.) et al., Séminaire de Probabilités XXXV. Berlin: Springer. Lect. Notes Math. 1755, 306-328 (2001).
Let \((\Omega,{\mathcal G}, P)\) be a probability space. Denote by \({\mathbb D}\) the space of càdlàg (right-continuous with left limits) real functions on \({\mathbb R}^{+}\). Fix a positive integer \(T\). A sequence of filtrations \({\mathcal F}^{n}=({\mathcal F}_{t}^{n})_{0\leq t\leq T}\) in \(\mathcal G\) is said to converge weakly to a filtration \({\mathcal F}=({\mathcal F}_{t})_{0\leq t\leq T}\) in \(\mathcal G\) if, for all \(B\in{\mathcal F}_{T}\), the sequence of càdlàg martingales \((E[1_{B}\mid {\mathcal F}_{t}^{n}])_{0\leq t\leq T}\) converges in probability under the Skorokhod topology on \(\mathbb D\) to the martingale \((E[1_{B}\mid {\mathcal F}_{t}])_{0\leq t\leq T}\). The authors give some examples of this kind of convergence of filtrations; then they study, under weak convergence of filtrations, the convergence in probability of processes \((E[X_{t}\mid {\mathcal F}_{t}^{n}])_{0\leq t\leq T}\) to \(X=(X_{t})_{0\leq t\leq T}\) in \(\mathbb D\) where \(X\) is an \({\mathcal F}_{t}\)-adapted semimartingale.
For the entire collection see [Zbl 0960.00020].

MSC:

60B10 Convergence of probability measures
60F05 Central limit and other weak theorems
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