统一签名累积量与广义 Magnus 展开
Unified Signature Cumulants and Generalized Magnus Expansions
AI总结：
本文在一般半鞅框架下建立期望签名对数（签名累积量）的普适函数关系，借助 Magnus 展开推广 diamond 与累积量展开，并给出其非交换 Riccati 方程解释及多类示例。
摘要：
路径的签名可被描述为其完整的非交换指数。遵循 T. Lyons 的观点，我们将其期望（即期望签名）视为经典矩生成函数在路径空间中的类似物。在张量代数中取其对数，便定义了签名累积量。我们在一般半鞅背景下建立了一个普适的函数关系。我们的工作展示了 Magnus 展开在计算期望签名累积量这一算法问题中的重要性，并进一步对近期关于被称为 diamond 展开和累积量展开的特征指数的结果给出了影响深远的推广；其动机涵盖从金融数学到统计物理。从仿射过程的角度看，该函数关系可被解释为 Riccati 方程的无限维、非交换（“Hausdorff”）变体。文中给出了许多示例。
英文摘要：
The signature of a path can be described as its full non-commutative exponential. Following T. Lyons we regard its expectation, the expected signature, as path space analogue of the classical moment generating function. The logarithm thereof, taken in the tensor algebra, defines the signature cumulant. We establish a universal functional relation in a general semimartingale context. Our work exhibits the importance of Magnus expansions in the algorithmic problem of computing expected signature cumulants, and further offers a far-reaching generalization of recent results on characteristic exponents dubbed diamond and cumulant expansions; with motivation ranging from financial mathematics to statistical physics. From an affine process perspective, the functional relation may be interpreted as infinite-dimensional, non-commutative ("Hausdorff") variation of Riccati's equation. Many examples are given.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
