On Weak Convergence of Filtrations

Coquet, François; Mémin, Jean; Słomiński, Leszek

doi:10.1007/978-3-540-44671-2_21

Cited by 32 publications

(74 citation statements)

References 11 publications

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“…On the other hand, we know (this is Proposition II.2 of [3]) that if a sequence of processes with independent increments converges in probability, then the convergence of associated filtrations also holds.…”

Section: P(b) Lc" ~ E(1aobit~') = P(a U B)i(t~l+l/qmentioning

confidence: 91%

“…How can one approximate 5r-martingales by martingales adapted to the filtrations generated by Y"? This question was studied in several directions in [2] with an application in the framework of backward stochastic differential equations; the connection with the convergence of filtrations was established in [3], where the relations between the convergence of processes and the convergence of the corresponding generated filtrations were studied in detail. In this paper, we are interested in the weak convergence of sequences of a-algebras and filtrations themselves.…”

Section: Introduction and Remindersmentioning

confidence: 99%

“…We are interested in the notions of convergence of cr-algebras and of filtrations that were introduced by Hoover [6] and generalized in [3] in the framework of the Skorokhod topology Jt as follows: if and only if, for all B 6 5rr, the sequence of the processes (E[1815r.n]),,~>l converges to the process E(1815r.) in probability for the Skorokhod topology Jl.…”

Section: Introduction and Remindersmentioning

confidence: 99%

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Some examples and counterexamples of convergence of σ-algebras and filtrations

2000

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Abstract. We study some properties of the weak convergence of filtrations, in particular, its behavior under elementary set operations. We also derive relations between the convergence of filtrations generated by point processes with a single jump and the convergence of their compensators or distributions of their jump moments. Finally, we apply a lemma on the intersection of cr-algebras to filtrations generated by different discretizations of a single process.

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Section: P(b) Lc" ~ E(1aobit~') = P(a U B)i(t~l+l/qmentioning

confidence: 91%

Section: Introduction and Remindersmentioning

confidence: 99%

Section: Introduction and Remindersmentioning

confidence: 99%

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Some examples and counterexamples of convergence of σ-algebras and filtrations

2000

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“…Let us assume that X n , (F n t ) → X, (F t ) then using Remark 1 and Lemma 1 of [4], we get an assertion which looks stronger: For every m, for every integrable F ∞ -measurable random variables U 1 , · · · , U m , the following convergence in probability under J 1 topology holds:…”

Section: Remarkmentioning

confidence: 92%

Stability of Doob-Meyer Decomposition Under Extended Convergence

Mémin

2003

Acta Mathematicae Applicatae Sinica, English Series

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In what follows, we consider the relation between Aldous's extended convergence and weak convergence of filtrations. We prove that, for a sequence (X n ) of F n t )-special semimartingales, with canonical decomposition X n = M n + A n , if the extended convergence (X n , F n . ) → (X, F.) holds with a quasi-left continuous (Ft)-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.

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“…In [8], the filtrations are indexed by a finite interval time [0, T ]. We generalise it to the case of filtrations indexed by R + .…”

Section: Let (Y Z) Be the Solution Of The Bsdementioning

confidence: 99%

Stability of solutions of BSDEs with random terminal time

Toldo¹

2006

ESAIM: PS

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Abstract. In this paper, we study the stability of the solutions of Backward Stochastic Differential Equations (BSDE for short) with an almost surely finite random terminal time. More precisely, we are going to show that if (W n ) is a sequence of scaled random walks or a sequence of martingales that converges to a Brownian motion W and if (τ n ) is a sequence of stopping times that converges to a stopping time τ , then the solution of the BSDE driven by W n with random terminal time τ n converges to the solution of the BSDE driven by W with random terminal time τ .Mathematics Subject Classification. 60H10, 60Fxx, 60G40.

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On Weak Convergence of Filtrations

Cited by 32 publications

References 11 publications

Some examples and counterexamples of convergence of σ-algebras and filtrations

Some examples and counterexamples of convergence of σ-algebras and filtrations

Stability of Doob-Meyer Decomposition Under Extended Convergence

Stability of solutions of BSDEs with random terminal time

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