Some Remarks on Davie’s Uniqueness Theorem

Шапошников, А. В.

doi:10.1017/s0013091515000589

Cited by 30 publications

(35 citation statements)

References 15 publications

(37 reference statements)

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“…This approach uses the flow property of strong solutions of SDEs driven by the Wiener process. Beside [8] our work has been inspired by Theorem 3.1 in [30] which deals with drifts b possibly unbounded in time and such that b(t, ·) is Hölder continuous. We mention that applications of Davie's uniqueness to Euler approximations for (1.1) are given in Section 4 of [8].…”

Section: Introductionmentioning

confidence: 99%

Davie’s type uniqueness for a class of SDEs with jumps

Priola¹

2018

Ann. Inst. H. Poincaré Probab. Statist.

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A result of A.M. Davie [Int. Math. Res. Not. 2007] states that a multidimensional stochastic equation dX t = b(t, X t ) dt + dW t , X 0 = x, driven by a Wiener process W = (W t ) with a coefficient b which is only bounded and measurable has a unique solution for almost all choices of the driving Wiener path. We consider a similar problem when W is replaced by a Lévy process L = (L t ) and b is β-Hölder continuous in the space variable, β ∈ (0, 1). We assume that L 1 has a finite moment of order θ, for some θ > 0. Using also a new càdlàg regularity result for strong solutions, we prove that strong existence and uniqueness for the SDE together with L p -Lipschitz continuity of the strong solution with respect to x imply a Davie's type uniqueness result for almost all choices of the Lévy path. We apply this result to a class of SDEs driven by non-degenerate α-stable Lévy processes, α ∈ (0, 2) and β > 1 − α/2.

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Section: Introductionmentioning

confidence: 99%

Davie’s type uniqueness for a class of SDEs with jumps

Priola¹

2018

Ann. Inst. H. Poincaré Probab. Statist.

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“…Since the fractional Brownian motion is not a semimartingale for H = 1 2 , we cannot pursue the same or similar proof strategy as, e.g., in [22] for the verification of strong uniqueness of solutions by using, e.g., the Itô-Tanaka formula. However, it is conceivable that our arguments combined with those in [4] which are based on results in [42] and a certain type of supremum concentration inequality in [44] will enable the construction of unique strong solutions to (1.1)-possibly even in the sense of Davie [15].…”

mentioning

confidence: 99%

Strong Solutions of Stochastic Differential Equations with Generalized Drift and Multidimensional Fractional Brownian Initial Noise

2021

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In this paper, we prove the existence of strong solutions to an stochastic differential equation with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters $$H<\frac{1}{2}.$$ 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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“…Proof. The result follows essentially by combining and extending [5], [16], and [18, Theorem 1.1] (the latter paper establishes weak differentiability of the solution map with respect to the spatial variable). We therefore simply give an outline of the proof.…”

Section: Examples Of (Local) Rds Generated By Sdesmentioning

confidence: 88%

The perfection of local semi-flows and local random dynamical systems with applications to SDEs

Ling¹,

Scheutzow²,

Vorkastner³

2021

Preprint

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We provide a rather general perfection result for crude local semi-flows taking values in a Polish space showing that a crude semi-flow has a modification which is a (perfect) local semi-flow which is invariant under a suitable metric dynamical system. Such a (local) semi-flow induces a (local) random dynamical system. Then we show that this result can be applied to several classes of stochastic differential equations driven by semimartingales with stationary increments such as equations with locally monotone coefficients and equations with singular drift. For these examples it was previously unknown whether they generate a (local) random dynamical system or not.

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Some Remarks on Davie’s Uniqueness Theorem

Abstract: We present a new approach to Davie's theorem on the uniqueness of solutions to the equation dX t = b(t, X t ) dt + dW t for almost all Brownian paths. A generalization of this result and a discussion of some close problems are given.

Cited by 30 publications

References 15 publications

Davie’s type uniqueness for a class of SDEs with jumps

Davie’s type uniqueness for a class of SDEs with jumps

Strong Solutions of Stochastic Differential Equations with Generalized Drift and Multidimensional Fractional Brownian Initial Noise

The perfection of local semi-flows and local random dynamical systems with applications to SDEs

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