Strong existence and higher order Fréchet differentiability of stochastic flows of fractional Brownian motion driven SDE's with singular drift

Baños, David R.; Nilssen, Torstein; Proske, Frank

doi:10.48550/arxiv.1511.02717

Cited by 8 publications

(36 citation statements)

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“…In the case of SDEs driven by Lévy processes we mention [23]. Other results can be found in [6], [1] with respect to SDEs driven by fractional Brownian motion and related noise. See also [7] in the case of "skew fractional Brownian motion", [5] with respect to singular delay equations and [8] in the case of Brownian motion driven mean-field equations.…”

Section: Introductionmentioning

confidence: 78%

“…Applying Theorem B.1 we obtain the following crucial estimate (compare [1], [2], [6], and [7]): Proposition B.2 Let the functions f and κ be defined as in (42) and (43), respectively. Further, let 0 ≤ θ ′ < θ < t ≤ T and for some m ≥ 1…”

Section: Examplementioning

confidence: 99%

“…In this section we derive an integration by parts formula similar to [6] which is used in the proof of Theorem 4.10 to verify the conditions of the compactness criterion Theorem A.3. Before stating the integration by parts formula, we start by giving some definitions and notations frequently used during the course of this section.…”

Section: Appendix a Compactness Criterionmentioning

confidence: 99%

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Restoration of Well-Posedness of Infinite-dimensional Singular ODE's via Noise

Baños¹,

Bauer²,

Meyer‐Brandis³

et al. 2019

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In this paper we aim at generalizing the results of A. K. Zvonkin [41] and A. Y. Veretennikov [39] on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the Euclidean space to the case of infinite-dimensional state spaces. The regularizing driving noise in our equation is chosen to be a locally non-Hölder continuous Hilbert space valued process of fractal nature, which does not allow for the use of classical construction techniques for strong solutions from PDE or semimartingale theory. Our approach, which does not resort to the Yamada-Watanabe principle for the verification of pathwise uniqueness of solutions, is based on Malliavin calculus.

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Section: Introductionmentioning

confidence: 78%

Section: Examplementioning

confidence: 99%

Section: Appendix a Compactness Criterionmentioning

confidence: 99%

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Restoration of Well-Posedness of Infinite-dimensional Singular ODE's via Noise

Baños¹,

Bauer²,

Meyer‐Brandis³

et al. 2019

Preprint

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“…We hereby give a slight generalization of the above lemma, whose proof can be also found in [7]. This lemma will be used in Section 5.…”

Section: Frameworkmentioning

confidence: 91%

“…In showing this we employ a compactness criterion for sets in L 2 (Ω) based on Malliavin calculus combined with a "local time variational calculus" argument. See [7] for the existence of strong solutions of SDE's driven by B H • , H < 1 2 , when e.g. the drift coefficients b belong to [42] in the Wiener case.…”

Section: Introductionmentioning

confidence: 99%

Strong solutions of SDE's with generalized drift and multidimensional fractional Brownian initial noise

Baños¹,

Ortiz-Latorre²,

Proske³

2017

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In this paper we prove the existence of strong solutions to a SDE with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters H < 1 2 . Here the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a "local time variational calculus" argument.

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