Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge–Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods.

中文翻译：

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随机离散哈密顿变分积分器

变分积分器用于随机哈密顿系统的结构保持模拟，其中几何力学中出现某种类型的乘法噪声。推导基于随机离散哈密顿量，它近似于哈密顿量系统的随机流的 II 型随机生成函数。生成函数是通过引入适当的随机作用函数及其相应的变分原理来获得的。我们的方法允许在一个统一的框架中重铸以前在文献中研究过的许多积分器，并提出了一种通用的方法来推导出新的结构保持数值方案。由此产生的积分器是辛的；它们保持与李群对称性相关的运动积分；它们包括作为特例的随机辛 Runge-Kutta 方法。提出了使用这种方法导出的几种新的均方阶数为 1.0 的低阶随机辛方法，并进行了数值测试，以证明与非辛方法相比，它们具有优越的长期数值稳定性和能量行为。