Stochastic differential equations are widely used in various fields; in particular, the usefulness of duality relations has been demonstrated in some models such as population models and Brownian momentum processes. In this study, a discussion based on combinatorics is made and applied to calculate the expectation values of functions in systems in which evolution is governed by stochastic differential equations. Starting with the duality theory of stochastic processes, some modifications to the interpretation and usage of time-ordering operators naturally lead to discussions on combinatorics. For demonstration, the first and second moments of the Ornstein-Uhlenbeck process are re-derived from the discussion on combinatorics. Furthermore, two numerical methods for practical applications are proposed. One method is based on a conventional exponential expansion and the Pade approximation. The other uses a resolvent of a time-evolution operator, along with the application of the Aitken series acceleration method. Both methods yield reasonable approximations. Particularly, the resolvent and Aitken acceleration show satisfactory results. These findings will provide a new way of calculating expectations numerically and directly without using time-discretization.

中文翻译：

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用于计算由随机微分方程控制的演化系统中函数期望值的组合学

随机微分方程广泛应用于各个领域；特别是，对偶关系的有用性已经在一些模型中得到了证明，例如人口模型和布朗动量过程。在这项研究中，基于组合学进行了讨论，并将其应用于计算进化受随机微分方程控制的系统中函数的期望值。从随机过程的对偶理论开始，对时间排序算子的解释和使用的一些修改自然会引发对组合学的讨论。为了演示，Ornstein-Uhlenbeck 过程的一阶矩和二阶矩是从组合数学的讨论中重新推导出来的。此外，还提出了两种实际应用的数值方法。一种方法是基于传统的指数展开和 Pade 近似。另一种使用时间演化算子的​​求解器，以及 Aitken 级数加速方法的应用。这两种方法都会产生合理的近似值。特别是溶剂和艾特肯加速表现出令人满意的结果。这些发现将提供一种在不使用时间离散化的情况下直接以数字方式计算期望的新方法。