标题： 关于Euler-Maruyama格式在多维SDEs（带有不连续漂移系数）上的性能 显示英文标题

标题： On the performance of the Euler-Maruyama scheme for multidimensional SDEs with discontinuous drift coefficient

摘要： 我们研究了具有不连续漂移系数的$d$维随机微分方程（SDE）的强逼近问题。更精确地说，我们主要假设漂移系数是分段Lipschitz连续的，且例外集$\Theta\subset \mathbb{R}^d$是一个可定向的正达到度的$C^4$维超曲面；扩散系数假设为Lipschitz连续，并且在区域$\Theta$附近，这两个系数都受到有界约束，且扩散系数在正交于$\Theta$的部分是非退化的。近年来，文献中已经证明了许多关于此类SDE强逼近的结果，特别是Euler-Maruyama格式的表现得到了研究。 对于$d=1$和有限的$\Theta$，已证明欧拉-马略拉 scheme 在所有满足系数为Lipschitz连续的经典情况下的$p\geq 1$中，达到至少$1/2$的$L_p$-误差率。 对于 $d>1$，到目前为止，人们只知道如果附加地系数 $\mu$ 和 $\sigma$ 在全局范围内有界，Euler-Maruyama 方案能达到至少 $1/4-$ 的 $L_2$-误差率。 本文中，我们证明了在上述设定下，Euler-Maruyama格式实际上达到了至少 $1/2-$ 的 $L_{p}$-误差阶，对于所有 $d\in\mathbb{N}$ 和所有 $p\geq 1$ 均成立。 该结果的证明基于一种广为人知的方法：将此类随机微分方程（SDE）转化为具有全局Lipschitz连续系数的SDE，一种针对一类非全局 $C^2$ 函数的新Itô公式，以及对时间连续Euler-Maruyama格式的实际位置与其在底层网格上前一时刻位置相对于超曲面 $\Theta$ “不同侧”的预期总时间的详细分析。 显示英文摘要

摘要： We study strong approximation of $d$-dimensional stochastic differential equations (SDEs) with a discontinuous drift coefficient. More precisely, we essentially assume that the drift coefficient is piecewise Lipschitz continuous with an exceptional set $\Theta\subset \mathbb{R}^d$ that is an orientable $C^4$-hypersurface of positive reach, the diffusion coefficient is assumed to be Lipschitz continuous and, in a neighborhood of $\Theta$, both coefficients are bounded and the diffusion coefficient has a non-degenerate portion orthogonal to $\Theta$. In recent years, a number of results have been proven in the literature for strong approximation of such SDEs and, in particular, the performance of the Euler-Maruyama scheme was studied. For $d=1$ and finite $\Theta$ it was shown that the Euler-Maruyama scheme achieves an $L_p$-error rate of at least $1/2$ for all $p\geq 1$ as in the classical case of Lipschitz continuous coefficients. For $d>1$, it was only known so far, that the Euler-Maruyama scheme achieves an $L_2$-error rate of at least $1/4-$ if, additionally, the coefficients $\mu$ and $\sigma$ are globally bounded. In this article, we prove that in the above setting the Euler-Maruyama scheme in fact achieves an $L_{p}$-error rate of at least $1/2-$ for all $d\in\mathbb{N}$ and all $p\geq 1$. The proof of this result is based on the well-known approach of transforming such an SDE into an SDE with globally Lipschitz continuous coefficients, a new It\^{o} formula for a class of functions which are not globally $C^2$ and a detailed analysis of the expected total time that the actual position of the time-continuous Euler-Maruyama scheme and its position at the preceding time point on the underlying grid are on 'different sides' of the hypersurface $\Theta$.