On certain almost Brownian filtrations

Émery, Michel

doi:10.1016/j.anihpb.2005.01.001

Cited by 10 publications

(8 citation statements)

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“…In the succinct language of filtrations (cf. Beghdadi-Sakrani and Emery [2], Émery [12,13]), the natural filtrations of the two processes must both be Coupling, local times, immersions 3 immersed in a common filtration (that is, the martingales of the natural filtrations must remain martingales in the larger common filtration). We therefore propose and adopt the new terminology of immersed couplings to replace the nomenclature of co-adapted or Markovian couplings: Definition 1.…”

Section: Introductionmentioning

confidence: 99%

“…In the stochastic differential framework (1), synchronous coupling corresponds to K = 0 and J = I, while rotation coupling corresponds to K = 0 and J equal to a d-dimensional rotation. (It is interesting to compare this direction of research with the work of Émery [12], Theorem 1; this characterizes Brownian filtrations using the notion of "self-coupling" -jointly immersed Brownian filtrations for which a prescribed scalar functional is approximately coupled. )…”

Section: Introductionmentioning

confidence: 99%

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Coupling, local times, immersions

Kendall¹

2015

Bernoulli

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This paper answers a question of Émery [In Séminaire de Probabilités XLII (2009) [383][384][385][386][387][388][389][390][391][392][393][394][395][396] by constructing an explicit coupling of two copies of the Beneš et al. [In Applied Stochastic Analysis (1991) 121-156 Gordon & Breach] diffusion (BKR diffusion), neither of which starts at the origin, and whose natural filtrations agree. The paper commences by surveying probabilistic coupling, introducing the formal definition of an immersed coupling (the natural filtration of each component is immersed in a common underlying filtration; such couplings have been described as co-adapted or Markovian in older terminologies) and of an equi-filtration coupling (the natural filtration of each component is immersed in the filtration of the other; consequently the underlying filtration is simultaneously the natural filtration for each of the two coupled processes). This survey is followed by a detailed case-study of the simpler but potentially thematic problem of coupling Brownian motion together with its local time at 0. This problem possesses its own intrinsic interest as well as being closely related to the BKR coupling construction. Attention focusses on a simple immersed (co-adapted) coupling, namely the reflection/synchronized coupling. It is shown that this coupling is optimal amongst all immersed couplings of Brownian motion together with its local time at 0, in the sense of maximizing the coupling probability at all possible times, at least when not started at pairs of initial points lying in a certain singular set. However numerical evidence indicates that the coupling is not a maximal coupling, and is a simple but non-trivial instance for which this distinction occurs. It is shown how the reflection/synchronized coupling can be converted into a successful equi-filtration coupling, by modifying the coupling using a deterministic time-delay and then by concatenating an infinite sequence of such modified couplings. The construction of an explicit equi-filtration coupling of two copies of the BKR diffusion follows by a direct generalization, although the proof of success for the BKR coupling requires somewhat more analysis than in the local time case.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Coupling, local times, immersions

Kendall¹

2015

Bernoulli

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“…Lemma below is a duplicate of Lemma 2 in [13], which could be proved identically in spite of this difference between the two notions of substantialness.…”

Section: ⊓ ⊔mentioning

confidence: 99%

“…The notion of substantial family of σ-fields appears in [13] in a slightly different form. Lemma below is a duplicate of Lemma 2 in [13], which could be proved identically in spite of this difference between the two notions of substantialness.…”

Section: ⊓ ⊔mentioning

confidence: 99%

On Standardness and I-cosiness

Laurent

2010

Séminaire De Probabilités XLIII

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Summary. The object of study of this work is the invariant characteristics of filtrations in discrete, negative time, pioneered by Vershik. We prove the equivalence between I-cosiness and standardness without using Vershik's standardness criterion. The equivalence between I-cosiness and productness for homogeneous filtrations is further investigated by showing that the I-cosiness criterion is equivalent to Vershik's first level criterion separately for each random variable. We also aim to derive the elementary properties of both these criteria, and to give a survey and some complements on the published and unpublished literature.Acknowledgments. Financial support from the IAP research network (grant nr. P6/03 of the Belgian government, Belgian Science Policy) is gratefully acknowledged. I am also indebted to M. Émery for helpful and encouraging comments and suggestions on earlier drafts of this paper.

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“…In Section 2 we introduce the basic notations and results needed in the sequel. For further use in the analysis of co-adapted couplings of spherical Brownian motions, in Lemma 2.1 we derive a characterization of all such couplings, similar to the one obtained in [11] or [15] in the case of Euclidean Brownian motions, and intimately related to Stroock's representation of spherical Brownian motion. Section 3 contains the analysis of Brownian couplings on R n (Theorem 3.1), and in Section 4 we present the analogous result for spherical Brownian motions (Theorem 4.1).…”

Section: Introductionmentioning

confidence: 94%

Couplings of Brownian Motions of Deterministic Distance in Model Spaces of Constant Curvature

Pascu

Popescu

2017

J Theor Probab

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We consider the model space M n K of constant curvature K and dimension n ≥ 1 (Euclidean space for K = 0, sphere for K > 0 and hyperbolic space for K < 0), and we show that given a functionfor every t ≥ 0 if and only if ρ is continuous and satisfies for almost every t ≥ 0 the differential inequality −(n − 1). In other words, we characterize all co-adapted couplings of Brownian motions on the model space M n K for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of ρ satisfying the above hypotheses.2010 Mathematics Subject Classification. Primary 60J65. Secondary 60G99, 58J65.

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On certain almost Brownian filtrations

Cited by 10 publications

References 10 publications

Coupling, local times, immersions

Coupling, local times, immersions

On Standardness and I-cosiness

Couplings of Brownian Motions of Deterministic Distance in Model Spaces of Constant Curvature

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