Asymptotically efficient Runge-Kutta methods for a class of ITOˆ and Stratonovich equations
Asymptotically efficient Runge-Kutta methods for a class of ITOˆ and Stratonovich equations
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一类 ITOˆ 和 Stratonovich 方程的渐近有效 Runge-Kutta 方法
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发表时间：
1991
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通讯作者：
Nigel J. Newton
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作者：
Nigel J. Newton
在随机微分方程的某些应用中，必须找到仅依赖于驾驶过程样本的近似解。据了解，这种近似的收敛顺序是有限的，并且有些是渐近有效的，在这个意义上说，他们最小化的领先系数在扩展的均方误差的幂级数的样本步长。本文发展了渐近有效的Runge-Kt tta方法，该方法涉及评价Ito方程的系数或相应的Stratonovich方程的系数。简单的近似，收敛的最大可能的顺序，但不是渐近有效的，也被定义。Ito方程的龙格-库塔方法不同于常微分方程的龙格-库塔方法，因为它们涉及样本步长的平方根。在五个算例上与经典的Euler方法一起对近似进行了沿着测试。模拟…
In certain applications of stochastic differential equations, approximate solutions must be found that depend only on samples of the driving process. It is known that the order of convergence of such approximations is limited, and that some are asymptotically efficient in the sense that they minimize the leading coefficient in the expansion of mean-square errors as power series in the sample step size. This article develops asymptotically efficient Runge-Kt tta methods that involve evaluations either of the coefficients of an Ito equation or of the coefficients of the corresponding Stratonovich equation. Simpler approximations, which converge with the maximum possible order but which are not asymptotically efficient, are also defined. The Runge-Kutta methods for Ito equations differ from those designed for ordinary differential equations in that they involve terms in the square root of the sample step size. The approximations are tested along with the classical Euler method on five examples. The simulatio...