关于具有任意慢收敛速度的强近似在二维空间中的随机微分方程
On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions
AI总结：
本文研究了二维空间中随机微分方程的强近似问题，证明了即使在二维和三维空间中，也存在收敛速度极慢的随机微分方程，这扩展了之前关于高维空间中此类现象的研究结果。
摘要：
在最近的文章[Jentzen, A., Müller-Gronbach, T., and Yaroslavtseva, L., Commun. Math. Sci., 14(6), 1477--1500, 2016]中，已经建立了对于任意慢的收敛速度和每一个自然数$d \in \{4,5,\ldots\}$，存在$d$维随机微分方程(SDEs)，其系数无限次可微且全局有界，使得基于有限次布朗运动观测的任何近似方法都无法以比给定收敛速度更快的速度在绝对均值下收敛到解。在本文中，我们通过证明这种慢收敛现象也出现在二维($d=2$)和三维($d=3$)空间中，加强了上述结果。
英文摘要：
In the recent article [Jentzen, A., Müller-Gronbach, T., and Yaroslavtseva, L., Commun. Math. Sci., 14(6), 1477--1500, 2016] it has been established that for every arbitrarily slow convergence speed and every natural number $d \in \{4,5,\ldots\}$ there exist $d$-dimensional stochastic differential equations (SDEs) with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence. In this paper we strengthen the above result by proving that this slow convergence phenomena also arises in two ($d=2$) and three ($d=3$) space dimensions.
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