Well-posedness of some non-linear stable driven SDEs

Frikha, Noufel; Konakov, Valentin; Menozzi, Stéphane

doi:10.3934/dcds.2020302

Cited by 7 publications

(5 citation statements)

References 34 publications

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“…Observe that when −ζ 0 ≥ 0, i.e. when the intrinsic Besov index is large enough, we could as well take Γ = −αζ 0 in (21). This choice would give a non exploding power of time, i.e.…”

Section: 1mentioning

confidence: 99%

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Weak well-posedness of multidimensional stable driven SDEs in the critical case

2020

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We establish weak well-posedness for critical symmetric stable driven SDEs in [Formula: see text] with additive noise [Formula: see text], [Formula: see text]. Namely, we study the case where the stable index of the driving process [Formula: see text] is [Formula: see text] which exactly corresponds to the order of the drift term having the coefficient [Formula: see text] which is continuous and bounded. In particular, we cover the cylindrical case when [Formula: see text] and [Formula: see text] are independent one-dimensional Cauchy processes. Our approach relies on [Formula: see text]-estimates for stable operators and uses perturbative arguments.

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Section: 1mentioning

confidence: 99%

“…This choice would give a non exploding power of time, i.e. the exponent − Γ α − ζ 0 = 0 and would also allow to take a supremum norm in time of the Besov norms in (21). Namely, sup s∈(t,S]…”

Section: 1mentioning

confidence: 99%

Weak well-posedness of multidimensional stable driven SDEs in the critical case

2020

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“…(i) In the regime α ∈ [2/3, 2], (2.3) coincides with the well-known condition (1.2), and thus Theorem 2.2 establishes strong convergence in the optimal range of β. We also remark that the threshold 2/3 appears in the theory of Lévy driven SDEs from time to time, e.g., in [CSZ18,FKM21]. We are unsure whether there is some connection between these appearances, or if this is just an instance of the "law of small numbers".…”

Section: Introductionmentioning

confidence: 94%

Strong rate of convergence of the Euler scheme for SDEs with irregular drift driven by Levy noise

Butkovsky¹,

Dareiotis²,

Gerencsér³

2022

Preprint

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We study the strong rate of convergence of the Euler-Maruyama scheme for a multidimensional stochastic differential equation (SDE)with irregular β-Hölder drift, β > 0, driven by a Lévy process with exponent α ∈ (0, 2]. For α ∈ [2/3, 2] we obtain strong L p and almost sure convergence rates in the whole range β > 1 − α/2, where the SDE is known to be strongly well-posed. This significantly improves the current state of the art both in terms of convergence rate and the range of α. In particular, the obtained convergence rate does not deteriorate for large p and is always at least n −1/2 ; this allowed us to show for the first time that the the Euler-Maruyama scheme for such SDEs converges almost surely and obtain explicit convergence rate. Furthermore, our results are new even in the case of smooth drifts. Our technique is based on a new extension of the stochastic sewing arguments.

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“…On the other hand, Lemma 8 in the Appendix (see also Lemma A.2 in [33]) ensures the existence of a family {P u } u≥0 of probability measures such that…”

Section: Proposition 3 (Decomposition) Let the Freezing Couplementioning

confidence: 99%

Weak well-posedness for degenerate SDEs driven by Lévy processes

Marino

Menozzi²

2021

Preprint

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In this article, we study the effects of the propagation of a non-degenerate Lévy noise through a chain of deterministic differential equations whose coefficients are Hölder continuous and satisfy a weak Hörmander-like condition. In particular, we assume some non-degeneracy with respect to the components which transmit the noise. Moreover, we characterize, for some specific dynamics, through suitable counter-examples, the almost sharp regularity exponents that ensure the weak well-posedness for the associated SDE. As a by-product of our approach, we also derive some Krylov-ype estimates for the density of the weak solutions of the considered SDE.

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Well-posedness of some non-linear stable driven SDEs

Cited by 7 publications

References 34 publications

Weak well-posedness of multidimensional stable driven SDEs in the critical case

Weak well-posedness of multidimensional stable driven SDEs in the critical case

Strong rate of convergence of the Euler scheme for SDEs with irregular drift driven by Levy noise

Weak well-posedness for degenerate SDEs driven by Lévy processes

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