Regularity Properties of the Stochastic Flow of a Skew Fractional Brownian Motion

Amine, Oussama; Baños, David R.; Proske, Frank

doi:10.48550/arxiv.1805.04889

Cited by 3 publications

(13 citation statements)

References 15 publications

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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: 80%

“…for all f ∈ I α a + (L p ), and with Hurst parameter H ∈ (0, 1 2 ) on a complete probability space (Ω, F , P) is defined as a centered Gaussian process with covariance function…”

Section: Fractional Calculusmentioning

confidence: 99%

“…Recall the following result (see [32,Proposition 5.1.3]) which gives the kernel of a fractional Brownian motion and an integral representation of R H (t, s) in the case of H < 1 2 .…”

Section: Fractional Calculusmentioning

confidence: 99%

“…More precisely, if H = R d , well-posedness of the ODE (1) can be restored via regularization by a Brownian (additive) noise, that is by a perturbation of the ODE (1) given by the SDE…”

Section: Introductionmentioning

confidence: 99%

“…Here {λ k } k≥1 ⊂ R, {e k } k≥1 is an orthonormal basis of H and {B H k • } k≥1 are independent one-dimensional fractional Brownian motions with Hurst parameters H k ∈ (0, 1 2 ), k ≥ 1, such that…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Baños¹,

Bauer²,

Meyer‐Brandis³

et al. 2019

Preprint

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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: 80%

“…for all f ∈ I α a + (L p ), and with Hurst parameter H ∈ (0, 1 2 ) on a complete probability space (Ω, F , P) is defined as a centered Gaussian process with covariance function…”

Section: Fractional Calculusmentioning

confidence: 99%

“…Recall the following result (see [32,Proposition 5.1.3]) which gives the kernel of a fractional Brownian motion and an integral representation of R H (t, s) in the case of H < 1 2 .…”

Section: Fractional Calculusmentioning

confidence: 99%

“…More precisely, if H = R d , well-posedness of the ODE (1) can be restored via regularization by a Brownian (additive) noise, that is by a perturbation of the ODE (1) given by the SDE…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Baños¹,

Bauer²,

Meyer‐Brandis³

et al. 2019

Preprint

Self Cite

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C-infinity-regularization by Noise of Singular ODE's

Amine,

Baños,

Proske

2017

Preprint

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In this paper we construct a new type of noise of fractional nature that has a strong regularizing effect on differential equations. We consider an equation with this noise with a highly irregular coefficient. We employ a new method to prove existence and uniqueness of global strong solutions, where classical methods fail because of the "roughness" and non-Markovianity of the driving process. In addition, we prove the rather remarkable property that such solutions are infinitely many times classically differentiable with respect to the initial condition in spite of the vector field being discontinuous. The technique used in this article corresponds to the Nash-Moser principle combined with a new concept of "higher order averaging operators along highly fractal stochastic curves". This approach may provide a general principle for the study of regularization by noise effects in connection with important classes of partial differential equations.

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A Bismut-Elworthy-Li Formula for Singular SDE's Driven by a Fractional Brownian Motion and Applications to Rough Volatility Modeling

Amine¹,

Coffie²,

Harang³

et al. 2018

Preprint

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In this paper we derive a Bismut-Elworthy-Li type formula with respect to strong solutions to singular stochastic differential equations (SDE's) with additive noise given by a multidimensional fractional Brownian motion with Hurst parameter H < 1/2. "Singular" here means that the drift vector field of such equations is allowed to be merely bounded and integrable. As an application we use this representation formula for the study of price sensitivities of financial claims based on a stock price model with stochastic volatility, whose dynamics is described by means of fractional Brownian motion driven SDE's.Our approach for obtaining these results is based on Malliavin calculus and arguments of a recently developed "local time variational calculus".

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Regularity Properties of the Stochastic Flow of a Skew Fractional Brownian Motion

Cited by 3 publications

References 15 publications

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

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

C-infinity-regularization by Noise of Singular ODE's

A Bismut-Elworthy-Li Formula for Singular SDE's Driven by a Fractional Brownian Motion and Applications to Rough Volatility Modeling

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