Regularity properties of the stochastic flow of a skew fractional Brownian motion

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

doi:10.1142/s0219025720500058

Cited by 8 publications

(7 citation statements)

References 12 publications

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“…is Malliavin differentiable with respect to Z = (Z (1) ,Z (2) ) * = (W,B H ) * with Malliavin derivative D = (D W ,D H ) * and we get…”

Section: Application: Stock Price Model With Stochastic Volatilitymentioning

confidence: 93%

“…Using techniques from Malliavin calculus and arguments of a "local time variational calculus" as recently developed in the series of works [5], [6], [2] in the case of fractional Brownian motion, we aim at obtaining in this paper an extension of the above mentioned results to the case of fractional Brownian motion driven singular SDE's. More precisely, we want to derive a BEL-formula of the type (1.6) with respect to strong solutions to SDE's of the form…”

Section: Introductionmentioning

confidence: 99%

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A Bismut–Elworthy–Li formula for singular SDEs driven by a fractional Brownian motion and applications to rough volatility modeling

Amine

Coffie

Harang

et al. 2020

Communications in Mathematical Sciences

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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 the δ price sensitivity 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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“…is Malliavin differentiable with respect to Z = (Z (1) ,Z (2) ) * = (W,B H ) * with Malliavin derivative D = (D W ,D H ) * and we get…”

Section: Application: Stock Price Model With Stochastic Volatilitymentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

A Bismut–Elworthy–Li formula for singular SDEs driven by a fractional Brownian motion and applications to rough volatility modeling

Amine

Coffie

Harang

et al. 2020

Communications in Mathematical Sciences

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“…assume that β ∈ (2H − 1, 0) . In view of Lemma 4.5 and the assumption H < 1 3 , we have that (ii) in Theorem 2.13 is fulfilled for p = q = ∞ as…”

Section: Existence Of a Path-by-path Solutionmentioning

confidence: 97%

“…We recall here a link between fractional Brownian motion and Brownian motion. For each H ∈ (0, 1 2 ), there exists an operator A and its inverse A −1 , where both can be given in terms of fractional integrals and derivatives (see (7.2) and [30,Th. 11]…”

Section: Definitions Of Solutionmentioning

confidence: 99%

“…This leads to a second class of interesting problems, namely solving Equation (1.1) when b is a distribution and B is a fractional Brownian motion with sufficiently small Hurst parameter H. A first attempt in this direction seems to be due to Nualart and Ouknine [29], who proved existence and uniqueness for some non-Lipschitz drifts. When b = aδ 0 with a ∈ R, the well-posedness of this equation was established for H < 1/4 by Catellier and Gubinelli [7] (who also treat drifts in negative Hölder spaces) and independently for H < 1/6 in [1,3] (with extensions to dimension d 1 in the three papers [1,3,7]). The solution is generally referred to as skew fractional Brownian motion.…”

Section: Introductionmentioning

confidence: 98%

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Regularisation by fractional noise for one-dimensional differential equations with nonnegative distributional drift

Anzeletti¹,

Richard²,

Tanré³

2021

Preprint

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We study existence and uniqueness of solutions to the equation dXt = b(Xt)dt + dBt, where b is a distribution in some Besov space and B is a fractional Brownian motion with Hurst parameter H 1/2. First, the equation is understood as a nonlinear Young integral equation. The integral is constructed in a p-variation space, which is well suited when b is a nonnegative (or nonpositive) distribution. Based on the Besov regularity of b, a condition on H is given so that solutions to the equation exist. The construction is deterministic, and B can be replaced by a deterministic path w which has a sufficiently smooth local time.We then consider probabilistic notions of weak and strong solution to this SDE. In particular, when b is a nonnegative finite measure, we show the existence of weak solutions for H < √ 2 − 1. We prove that solutions can also be understood as limits of strong solutions when b is regularised, which is used to establish pathwise uniqueness and existence of a strong solution when H 1/4. The proofs involve fine properties of the local time of the fractional Brownian motion, as well as new regularising properties of this process which are established using the stochastic sewing Lemma.

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$$C^{\infty }$$-Regularization by Noise of Singular ODE’s

Amine,

Baños,

Proske

2024

J Dyn Diff Equat

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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 driven by a highly irregular vector field and study the effect of this noise on such dynamical systems. 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, in a certain sense, to the Nash–Moser iterative scheme in combination 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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Regularity properties of the stochastic flow of a skew fractional Brownian motion

Cited by 8 publications

References 12 publications

A Bismut–Elworthy–Li formula for singular SDEs driven by a fractional Brownian motion and applications to rough volatility modeling

A Bismut–Elworthy–Li formula for singular SDEs driven by a fractional Brownian motion and applications to rough volatility modeling

Regularisation by fractional noise for one-dimensional differential equations with nonnegative distributional drift

$$C^{\infty }$$-Regularization by Noise of Singular ODE’s

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