In this paper, we study numerical approximations for stochastic differential equations (SDEs) that use adaptive step sizes. In particular, we consider a general setting where decisions to reduce step sizes are allowed to depend on the future trajectory of the underlying Brownian motion. Since these adaptive step sizes may not be previsible, the standard mean squared error analysis cannot be directly applied to show that the numerical method converges to the solution of the SDE. Building upon the pioneering work of Gaines and Lyons, we shall instead use rough path theory to establish convergence for a wide class of adaptive numerical methods on general Stratonovich SDEs (with sufficiently smooth vector fields). To the author's knowledge, this is the first error analysis applicable to standard solvers, such as the Milstein and Heun methods, with non-previsible step sizes. In our analysis, we require the sequence of adaptive step sizes to be nested and the SDE solver to have unbiased "L\'evy area" terms in its Taylor expansion. We conjecture that for adaptive SDE solvers more generally, convergence is still possible provided the method does not introduce "L\'evy area bias". We present a simple example where the step size control can skip over previously considered times, resulting in the numerical method converging to an incorrect limit (i.e. not the Stratonovich SDE). Finally, we conclude with a numerical experiment demonstrating a newly introduced adaptive scheme and showing the potential improvements in accuracy when step sizes are allowed to be non-previsible.

翻译：本文研究使用自适应步长的随机微分方程数值逼近方法。具体而言，我们考虑一般情形：允许基于底层布朗运动的未来轨迹来决策步长缩减。由于此类自适应步长可能不具有可预测性，标准均方误差分析无法直接证明数值方法收敛于随机微分方程的解。基于Gaines和Lyons的开创性工作，我们将转而利用粗糙路径理论，在一般的Stratonovich随机微分方程（具有足够光滑的向量场）上建立广泛自适应数值方法的收敛性。据作者所知，这是首个适用于Milstein和Heun方法等标准求解器在非可预测步长下的误差分析。在我们的分析中，要求自适应步长序列具有嵌套性，且随机微分方程求解器的泰勒展开中包含无偏的"Lévy面积"项。我们推测：对更一般的自适应随机微分方程求解器而言，只要方法不引入"Lévy面积偏差"，收敛性仍有可能成立。我们给出了一个简单算例：当步长控制可以跳过先前已考虑的时间节点时，数值方法将收敛于错误的极限（即非Stratonovich随机微分方程）。最后，我们通过数值实验展示了一种新提出的自适应方案，并证明允许非可预测步长时精度可能获得的提升。