Convenient, easy to implement stochastic integration methods are developed on the basis of abstract one-step deterministic order $p$ integration techniques. The abstraction as an arbitrary one step map allows the inspection of easy to implement stochastic exponential time differencing Runge-Kutta (SETDRK), stochastic integrating factor Runge-Kutta (SIFRK) and stochastic RK (SRK) schemes. Such schemes require minimal modifications to existing deterministic schemes and converging to the Stratonovich SDE, whilst inheriting many of their desirable properties. These schemes capture all symmetric terms in the Stratonovich-Taylor expansion, are order $p$ in the limit of vanishing noise, can attain at least strong order $p/2$ or $p/2-1/2$ (parity dependent) for drift commutative noise, strong order $1$ for commutative noise, and strong order $1/2$ for multidimensional non-commutative noise. Numerical convergence is demonstrated using different bases of noise for 2nd, 3rd and 4th order SETDRK, SIFRK and SRK schemes for a stochastic KdV equation.

翻译：基于抽象一步确定性阶$p$积分技术，本文发展了一类便捷且易于实现的随机积分方法。将方法抽象为任意一步映射，使得我们可以考察易于实现的随机指数时间差分龙格-库塔（SETDRK）、随机积分因子龙格-库塔（SIFRK）以及随机龙格-库塔（SRK）格式。此类格式仅需对现有确定性格式进行最小修改，即可收敛至Stratonovich随机微分方程，同时继承原格式的诸多优良特性。这些格式能够捕获Stratonovich-Taylor展开中的所有对称项，在噪声趋于零时具有阶$p$精度，对于漂移可交换噪声至少能达到强阶$p/2$或$p/2-1/2$（取决于奇偶性），对于可交换噪声可达强阶$1$，对于多维非交换噪声可达强阶$1/2$。本文通过采用不同噪声基，针对随机KdV方程对二阶、三阶和四阶SETDRK、SIFRK及SRK格式进行了数值收敛性验证。