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 our knowledge, this is the first convergence guarantee that applies 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 an experiment demonstrating the accuracy of Heun's method and a newly introduced Splitting Path-based Runge-Kutta scheme (SPaRK) when used with adaptive step sizes.

翻译：本文研究使用自适应步长的随机微分方程（SDE）数值逼近问题。特别地，我们考虑一种一般性设定，其中步长缩减决策可以依赖于底层布朗运动的未来轨迹。由于此类自适应步长可能不可预测，标准均方误差分析无法直接证明数值方法收敛于SDE解。基于Gaines与Lyons的开创性工作，我们将转而使用粗糙路径理论，在一般Stratonovich型SDE（具有足够光滑的向量场）上，为一类广泛的自适应数值方法建立收敛性。据我们所知，这是首个适用于标准求解器（如Milstein法和Heun法）且允许非预测性步长的收敛保证。分析中要求自适应步长序列具有嵌套性，且SDE求解器的泰勒展开中包含无偏的“Lévy面积”项。我们推测，对于更一般的自适应SDE求解器，只要方法不引入“Lévy面积偏差”，收敛性仍可能成立。通过简单算例展示：当步长控制跳过先前考虑的时间点时，数值方法将收敛到错误极限（即非Stratonovich型SDE）。最后，通过实验验证Heun法及新提出的基于分裂路径的龙格-库塔格式（SPaRK）在自适应步长下的计算精度。