In this paper we present a general procedure for designing higher strong order methods for It\^o stochastic differential equations on matrix Lie groups and illustrate this strategy with two novel schemes that have a strong convergence order of 1.5. Based on the Runge-Kutta--Munthe-Kaas (RKMK) method for ordinary differential equations on Lie groups, we present a stochastic version of this scheme and derive a condition such that the stochastic RKMK has the same strong convergence order as the underlying stochastic Runge-Kutta method. Further, we show how our higher order schemes can be applied in a mechanical engineering as well as in a financial mathematics setting.

翻译：在本文中,我们提出一个通用程序,用于设计Itçção schochatic 差异方程式在矩阵中的更强的顺序方法,并用两个具有1.5个强烈趋同顺序的新方案来说明这一战略。根据Lie组普通差别方程式的Runge-Kutta-Munthe-Kaas(RKMK)方法,我们提出这一方法的随机版,并得出一个条件,即Stochastic RKMK具有与基本随机龙格-Kutta方法相同的强烈组合顺序。此外,我们展示了如何在机械工程以及财务数学环境中应用我们的更高顺序方案。