In this paper we deal with global approximation of solutions of stochastic differential equations (SDEs) driven by countably dimensional Wiener process. Under certain regularity conditions imposed on the coefficients, we show lower bounds for exact asymptotic error behaviour. For that reason, we analyse separately two classes of admissible algorithms: based on equidistant, and possibly not equidistant meshes. Our results indicate that in both cases, decrease of any method error requires significant increase of the cost term, which is illustrated by the product of cost and error diverging to infinity. This is, however, not visible in the finite dimensional case. In addition, we propose an implementable, path-independent Euler algorithm with adaptive step-size control, which is asymptotically optimal among algorithms using specified truncation levels of the underlying Wiener process. Our theoretical findings are supported by numerical simulation in Python language.

翻译：本文研究由可数维维纳过程驱动的随机微分方程解全局逼近问题。在系数满足特定正则条件下，我们给出了精确渐近误差行为的下界。为此，我们分别分析了两类可容许算法：基于等距网格与可能非等距网格的算法。结果表明，两类算法中任何方法误差的降低都需要显著增加成本项，这通过成本与误差乘积趋于无穷大得以体现。然而，这一现象在有限维情形下并不明显。此外，我们提出了一种可实现的、与路径无关的自适应步长欧拉算法，该算法在指定维纳过程截断水平的算法中具有渐近最优性。理论结果通过Python语言数值模拟得到了验证。