2003
DOI: 10.1007/s10255-003-0094-2
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Stability of Doob-Meyer Decomposition Under Extended Convergence

Abstract: 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… Show more

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Cited by 14 publications
(24 citation statements)
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“…, q. The sequence (M k,i ) k∈N satisfies the conditions of [52,Corollary 12]. In the middle of the proof of the aforementioned corollary, we can find the required convergences.…”
Section: Appendix a Auxiliary Resultsmentioning
confidence: 91%
“…, q. The sequence (M k,i ) k∈N satisfies the conditions of [52,Corollary 12]. In the middle of the proof of the aforementioned corollary, we can find the required convergences.…”
Section: Appendix a Auxiliary Resultsmentioning
confidence: 91%
“…For the semimartingale case, we just need to apply Th. 11 in [37] to conclude M Y,k → M uniformly in probability as k → +∞ and this implies M Y,k → M weakly in B 2 (F) as k → +∞ so that Y is stable. Th 2 in [13] allows us to conclude stability for the Dirichlet case up to the fact that our partition is random, but one can easily check that all arguments in the proof of Th 2 in [13] apply to our case as well.…”
Section: Pathwise Dynamics Of the Skeleton Let Us Definementioning
confidence: 92%
“…In this case, because of [X k , A k,j ] = [M Y,k , A k,j ], Lemma 4.3 and (4.5), D Y X may not even exists or it will not coincide with H. This type of phenomena is well-known in time-deterministic discretizations of filtrations. See e.g [37,12] and other references therein. This is the reason why we restrict the computation of the weak derivative DX to stable structures and this is the best one might expect.…”
Section: Thenmentioning
confidence: 99%
“…In order to establish the convergence of BSDEs we also need to deploy the notions of extended weak convergence and weak convergence of filtrations, the definitions of which, we recall from Coquet et al (2004) and Mémin (2003), are given as follows: Remark 2.3. It is clear that the notion of extended weak convergence in general is stronger than the notion of weak convergence of filtration (see for instance Coquet et al (2004) and Mémin (2003) for a discussion). However, in the notation of the previous definition, if F n i converges to F i in L 1 for i = 1, .…”
Section: 2mentioning
confidence: 99%
“…in probability in the Skorokhod J 1 -topology. In particularly, if G n converges to G weakly, L n is a G n -martingale and L is a G-martingale then We recall (from Proposition 2 in Mémin (2003)) that (W (π) , X (π) ) converges to the Lévy process (W, X) in the sense of extended convergence, due to the independence of the increments of the two-coordinate processes W (π) and X (π) , in conjunction with the fact that the filtration F (π) is generated by the process (W (π) , X (π) ):…”
Section: 2mentioning
confidence: 99%