In this paper, a family of arbitrary high order quadratic invariants and energy conservation parametric stochastic partitioned Runge-Kutta methods (SPRK) are constructed for stochastic canonical Hamiltonian systems where the parameters depend on some truncated random variables, step size and numerical solutions. We first apply the P-series and bi-coloured trees theory to analyze the mean-square and weak convergence order conditions of SPRK methods solving a class of single integrand stochastic differential equations. Then, a class of SPRK methods with parameters are obtained by means of W-transform and technique of truncated Wiener increments, and we prove that the methods are symplectic. Combining with order conditions, there exists a special parameter α* which enables convergence order in each iteration and can preserve the energy of the stochastic canonical Hamiltonian systems. Finally, the representative stochastic canonical Hamiltonian systems are selected to verify the good performance of the proposed parameter methods.

本文针对随机规范的哈密顿系统构造了一系列任意的高阶二次不变量和能量守恒的参数随机划分的Runge-Kutta方法（SPRK），其参数取决于一些截断的随机变量，步长和数值解。首先，我们应用P系列和双色树理论来分析SPRK方法的均方和弱收敛阶条件，该方法求解一类单整数整数随机微分方程。然后，通过W变换和截断维纳增量技术获得了一类带参数的SPRK方法，证明了该方法是辛的。结合订购条件，存在一个特殊的参数α*可以在每次迭代中实现收敛顺序，并可以保留随机规范哈密顿系统的能量。最后，选择代表性的典型规范哈密顿系统，以验证所提出参数方法的良好性能。