论文标题
Euler-Maruyama方案的总变化中的收敛性，用于扩散过程，具有可测量的漂移系数和加性噪声
Convergence in total variation of the Euler-Maruyama scheme applied to diffusion processes with measurable drift coefficient and additive noise
论文作者
论文摘要
我们对以恒定扩散系数和有限的可测量漂移系数为$ d $的随机微分方程的Euler-Maruyama离散化感兴趣。在该方案中，使用时间变量的随机化来摆脱该变量中漂移的任何规律性假设。我们证明，总变化距离的订单$ 1/2 $差异很弱。当漂移在分布意义上具有空间分歧时，与$ρ$ th的功率相对于勒布斯格在空间中均匀地集成的电源，及时及时左右，对于一些$ρ\ ge d $，终端时的收敛顺序将提高到$ 1 $ $ 1 $。在尺寸$ d = 1 $中，当漂移的空间衍生物是空间中的一个量度时，将保留此结果。我们通过数值实验确认我们的理论分析。
We are interested in the Euler-Maruyama discretization of a stochastic differential equation in dimension $d$ with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get rid of any regularity assumption of the drift in this variable. We prove weak convergence with order $1/2$ in total variation distance. When the drift has a spatial divergence in the sense of distributions with $ρ$-th power integrable with respect to the Lebesgue measure in space uniformly in time for some $ρ\ge d$, the order of convergence at the terminal time improves to $1$ up to some logarithmic factor. In dimension $d=1$, this result is preserved when the spatial derivative of the drift is a measure in space with total mass bounded uniformly in time. We confirm our theoretical analysis by numerical experiments.
