论文标题
与本地Lipschitz和本地Hölder连续性相对于具有光滑漂移系数函数的添加噪声驱动的SDE的初始值，大多数多数值生长的衍生物
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
论文作者
论文摘要
在最近的文章中[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 （确保存在SDE的独特解决方案的存在）以对数hölder的连续方式取决于其初始值。然后，人们可能会怀疑该结果是否可以锐化，实际上，该类别的SDE是否必须具有不断依赖于Lipschitz的最初价值的解决方案。本文的关键贡献是确定事实并非如此。更确切地说，我们提供了一个具有添加噪声驱动的SDE的示例，它们具有光滑的漂移系数函数，其中最多是多项式生长的衍生物，这些衍生物的解决方案不取决于其在本地Lipschitz连续的初始值，甚至在本地Hölder的连续方式中也不取决于其最初的价值。
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.
