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:我们处理由独立的布朗动议驱动的随机差异方程式的微弱数字解决方案。本文件开发了一种新的方法,用于设计适应性战略,以自动确定计算SDE解决方案平滑功能平均值的数字办法的阶梯大小。首先,我们引入了一种通用方法,用于构建SDE的可变的阶梯大小弱方程式,其基础是控制数字集成器加量的第一个条件时刻与与额外微弱近似相匹配之间的匹配。为此,我们使用某些不包含抽样随机变量的本地差异功能。我们给出了设计适当差异函数和选择起步阶大小的精确方向。第二,我们引入了可变阶梯度 Euler 计划,同时通过外推法,引入了可变阶大小第二顺序弱方程式。最后,进行了数字模拟,以展示引入的可变梯度战略与在计算传播功能预期时的适应性办法的潜力,以克服已知的常规固定阶梯规模计划不稳定问题。