B-Series and generalizations are a powerful tool for the analysis of numerical integrators. An extension named exotic aromatic B-Series was introduced to study the order conditions for sampling the invariant measure of ergodic SDEs. Introducing a new symmetry normalization coefficient, we analyze the algebraic structures related to exotic B-Series and S-Series. Precisely, we prove the relationship between the Grossman–Larson algebras over exotic and grafted forests and the corresponding duals to the Connes–Kreimer coalgebras and use it to study the natural composition laws on exotic S-Series. Applying this algebraic framework to the derivation of order conditions for a class of stochastic Runge–Kutta methods, we present a multiplicative property that ensures some order conditions to be satisfied automatically.

B 系列和概括是数值积分器分析的强大工具。引入了一种名为奇异芳香 B 系列的扩展来研究遍历 SDE 不变测度采样的阶次条件。引入新的对称归一化系数，我们分析了与奇异的 B 系列和 S 系列相关的代数结构。准确地说，我们证明了外来森林和嫁接森林上的 Grossman-Larson 代数与 Connes-Kreimer 代数的相应对偶之间的关系，并用它来研究外来 S 系上的自然组成定律。将该代数框架应用于一类随机龙格-库塔方法的阶次条件的推导，我们提出了一个乘法性质，确保自动满足某些阶次条件。