连续半鞅的指数李级数
The exponential Lie series for continuous semimartingales
摘要：
考虑由连续半鞅驱动的、由非交换向量场控制的随机微分系统。我们证明了流图的对数是一个指数李级数。这依赖于一个自然变化的基础，向量场的相关的二次协变过程，类似于Stratonovich校正。然后，流图可以展开为向量场的合成幂的级数，并且流图的对数可以因此在向量场的李代数中展开。此外，我们给出了一个直接的自包含证明相应的陈-李哈茨公式，提供了一个显式的李级数系数的公式。这样的指数李级数是重要的发展强李群积分计划，确保近似解本身位于任何齐次流形上的解决方案的发展。
英文摘要：
We consider stochastic differential systems driven by continuous semimartingales and governed by non-commuting vector fields. We prove that the logarithm of the flowmap is an exponential Lie series. This relies on a natural change of basis to vector fields for the associated quadratic covariation processes, analogous to Stratonovich corrections. The flowmap can then be expanded as a series in compositional powers of vector fields and the logaritm of the flowmap can thus be expanded in the Lie algebra of vector fields. Further, we give a direct self-contained proof of the corresponding Chen-Strichartz formula which provides an explicit formula for the Lie series coefficients. Such exponential Lie series are important in the development of strong Lie group integration schemes that ensure approximate solutions themselves lie in any homogeneous manifold on which the solution evolves.
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。
