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

Abstract

In the recent article [A. Jentzen, B. Kuckuck, T. Müller-Gronbach, and L. Yaroslavtseva, arXiv:1904.05963 (2019)] 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 the existence of a unique solution to the SDE) depend 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 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 whose solutions do not depend on their initial value in a locally Lipschitz continuous, nor even in a locally Hölder continuous way.

arXiv comment27 pages

References in corpus (10)

• Loss of regularity for Kolmogorov equations
• On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients
• Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations
• Global flows for stochastic differential equations without global Lipschitz conditions
• On stochastic differential equations with arbitrary slow convergence rates for strong approximation
• Lack of strong completeness for stochastic flows
• Strong completeness for a class of stochastic differential equations with irregular coefficients
• On the Itô-Alekseev-Gröbner formula for stochastic differential equations
• On the strong regularity of degenerate additive noise driven stochastic differential equations with respect to their initial values
• Strong convergence rates on the whole probability space for space-time discrete numerical approximation schemes for stochastic Burgers equations

Cited by in corpus (1)

• Regularity preservation in Kolmogorov equations for non-Lipschitz coefficients under Lyapunov conditions