Modulus of Continuity for Stochastic Flows

Baldi, P.; Sanz–Solé, Marta

doi:10.1007/978-3-0348-8555-3_1

Cited by 5 publications

(13 citation statements)

References 7 publications

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“…All this is known (with an equality) for hypoelliptic diffusions on Nilpotent group in [1] and elliptic diffusions in [2], using a natural metric associated to the diffusion.…”

Section: Remark 17 Asmentioning

confidence: 99%

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Approximations of the Brownian rough path with applications to stochastic analysis

Friz¹,

Victoir²

2005

Annales de l'Institut Henri Poincare (B) Probability and Statis

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A geometric p-rough path can be seen to be a genuine path of finite p-variation with values in a Lie group equipped with a natural distance. The group and its distance lift (R d , +, 0) and its Euclidean distance.This approach allows us to easily get a precise modulus of continuity for the Enhanced Brownian Motion (the Brownian Motion and its Levy Area).As a first application, extending an idea due to Millet & Sanz-Sole, we characterize the support of the Enhanced Brownian Motion (without relying on correlation inequalities). Secondly, we prove Schilder's theorem for this Enhanced Brownian Motion. As all results apply in Hölder (and stronger) topologies, this extends recent work by Ledoux, Qian, Zhang [24]. Lyons' fine estimates in terms of control functions [22] allow us to show that the Itô map is still continuous in the topologies we introduced. This provides new and simplified proofs of the Stroock-Varadhan support theorem and the Freidlin-Wentzell theory. It also provides a short proof of modulus of continuity for diffusion processes along old results by Baldi. AbstractUn p-rough path est un chemin de p-variation finieà valeurs dans un groupe de Lie muni d'une distance sous-riemannienne. Le

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“…All this is known (with an equality) for hypoelliptic diffusions on Nilpotent group in [1] and elliptic diffusions in [2], using a natural metric associated to the diffusion.…”

Section: Remark 17 Asmentioning

confidence: 99%

“…For the rest of the paper, p denotes a fixed real in (2,3). The contributions of this paper may be summarized as follows:…”

Section: Introductionmentioning

confidence: 99%

Approximations of the Brownian rough path with applications to stochastic analysis

Friz¹,

Victoir²

2005

Annales de l'Institut Henri Poincare (B) Probability and Statis

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“…existence, uniqueness) of the uncontrolled versions of the flows. Unlike in [18,3] and [5] (which consider only finite dimensional flows), the proof of the LDP does not require any exponential probability estimates or discretization/approximation of the original model.…”

Section: For Details) Letting A(x Y T) =mentioning

confidence: 99%

“…For ε ∈ (0, ∞), when f i is replaced by εf i in (1.1), we write the corresponding flow as φ ε . Large deviations for φ ε inŴ k , as ε → 0, have been studied for the case k = 0 in [18,3] and for general k in [5].…”

Section: Introductionmentioning

confidence: 99%

“…One such example is given in Section 5. Thus, following Kunita's [15] notation for stochastic integration with respect to semi-martingales with a spatial parameter, the study of general Brownian flows of C k -diffeomorphisms leads to SDEs of the form [18,3,5], which consider stochastic flows driven by only finitely many real Brownian motions. The main technical difficulty in establishing the LDP inŴ k × W k is the proof of a result analogous to Proposition 4.10, which establishes tightness of certain controlled processes, when k − 1 is replaced by k.…”

Section: Introductionmentioning

confidence: 99%

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Large Deviations for Stochastic Flows of Diffeomorphisms

Budhiraja¹,

Dupuis²,

Maroulas³

2007

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Flow Decomposition and Large Deviations

Arous

Castell

1996

Journal of Functional Analysis

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We study large deviations properties related to the behavior, as = goes to 0, of diffusion processes generated by = 2 L 1 +L 2 , where L 1 and L 2 are two second-order differential operators, extending recent results of Doss and Stroock and Rabeherimanana. The main tool is the decomposition theorem for flows of stochastic differential equations proved by Bismut and Kunita. We give another application of flow decomposition in a nonlinear filtering problem.

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Modulus of Continuity for Stochastic Flows

Cited by 5 publications

References 7 publications

Approximations of the Brownian rough path with applications to stochastic analysis

Approximations of the Brownian rough path with applications to stochastic analysis

Large Deviations for Stochastic Flows of Diffeomorphisms

Flow Decomposition and Large Deviations

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