Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven stochastic differential equations with smooth drift coefficient functions with at most polynomially growing derivatives

Arnulf Jentzen; Benno Kuckuck; Thomas Müller-Gronbach; Larisa Yaroslavtseva

doi:10.3934/dcdsb.2021203

Article Contents

2022, Volume 27, Issue 7: 3707-3724. Doi: 10.3934/dcdsb.2021203

This issue Previous Article Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations Next Article Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise

Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven stochastic differential equations with smooth drift coefficient functions with at most polynomially growing derivatives

1.
Seminar for Applied Mathematics, Department of Mathematics, ETH Zurich, Zurich, Switzerland
2.
Applied Mathematics: Institute for Analysis and Numerics, Faculty of Mathematics and Computer Science, University of Münster, Münster, Germany
3.
School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen, China
4.
Mathematical Institute, Faculty of Mathematics and Natural Sciences, University of Düsseldorf, Düsseldorf, Germany
5.
Faculty of Computer Science and Mathematics, University of Passau, Passau, Germany

^* Corresponding author
^* Corresponding author

Received: November 30, 2020

Revised: May 31, 2021

Published: July 2022

This work has been funded by DFG grant EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure

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Abstract

In the recent article [A. Jentzen, B. Kuckuck, T. Müller-Gronbach, and L. Yaroslavtseva, J. Math. Anal. Appl. 502, 2 (2021)] it has been proved that the solutions to every additive noise driven stochastic differential equation (SDE) which has a drift coefficient function with at most polynomially growing first order partial derivatives and which admits a Lyapunov-type condition (ensuring the existence of a unique solution to the SDE) depend in the strong sense in a logarithmically Hölder continuous way on their initial values. One might then wonder whether this result can be sharpened and whether in fact, SDEs from this class necessarily have solutions which depend in the strong sense locally Lipschitz continuously on their initial value. The key contribution of this article is to establish that this is not the case. More precisely, we supply a family of examples of additive noise driven SDEs, which have smooth drift coefficient functions with at most polynomially growing derivatives and whose solutions do not depend in the strong sense on their initial value in a locally Lipschitz continuous, nor even in a locally Hölder continuous way.

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Mathematics Subject Classification: Primary: 60H10.

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