The exponential Lie series for continuous semimartingales
The exponential Lie series for continuous semimartingales
复制标题
连续半鞅的指数李级数
DOI：
10.1098/rspa.2015.0429
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Anke Wiese
中科院分区：
文献类型：
--
作者：
K. Ebrahimi;S. Malham;F. Patras;Anke Wiese
考虑了由连续半鞅驱动的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上，向量场的相关二次协变过程，类似于Stratonovich校正。然后，流图可以扩展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扩展。进一步，我们给出了相应的Chen-Chenhartz公式的直接显式证明，该公式提供了李级数系数的显式公式.这样的指数李级数在强李群积分方案的发展中是重要的，该方案确保近似解本身位于解在其上演化的任何齐次流形中。
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 logarithm of the flowmap can thus be expanded in the Lie algebra of vector fields. Further, we give a direct explicit 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.