We address the weak numerical solution of stochastic differential equations driven by independent Brownian motions (SDEs for short). This paper develops a new methodology to design adaptive strategies for determining automatically the step-sizes of the numerical schemes that compute the mean values of smooth functions of the solutions of SDEs. First, we introduce a general method for constructing variable step-size weak schemes for SDEs, which is based on controlling the match between the first conditional moments of the increments of the numerical integrator and the ones corresponding to an additional weak approximation. To this end, we use certain local discrepancy functions that do not involve sampling random variables. Precise directions for designing suitable discrepancy functions and for selecting starting step-sizes are given. Second, we introduce a variable step-size Euler scheme, together with a variable step-size second order weak scheme via extrapolation. Finally, numerical simulations are presented to show the potential of the introduced variable step-size strategy and the adaptive scheme to overcome known instability problems of the conventional fixed step-size schemes in the computation of diffusion functional expectations.

翻译：本文研究由独立布朗运动驱动的随机微分方程（简称SDEs）的弱数值求解问题。我们发展了一种新方法论，用于设计自适应策略，自动确定计算SDE解的光滑函数均值数值格式的步长。首先，我们引入了一种为SDE构造变步长弱格式的通用方法，该方法通过控制数值积分器增量的一阶条件矩与相应附加弱近似条件矩的匹配程度来实现。为此，我们采用了不涉及随机变量采样的局部差异函数。给出了设计合适差异函数和选择初始步长的精确指导准则。其次，我们引入了一种变步长欧拉格式，以及通过外推法得到的变步长二阶弱格式。最后，通过数值仿真展示了所引入的变步长策略和自适应格式在克服传统固定步长格式计算扩散泛函期望值时已知不稳定问题方面的潜力。