Abstract
We consider stochastic differential systems driven by continuous semimartingales and governed by noncommuting 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 ChenStrichartz 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.
Original language  English 

Number of pages  20 
Journal  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 
Volume  471 
Issue number  2184 
DOIs  
Publication status  Published  Dec 2015 
Keywords
 Ito stochastic flows
 quasishuffle product
 exponential Lie series
 ChenStrichartz formula
ASJC Scopus subject areas
 Mathematics(all)
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Profiles

Anke Wiese
 School of Mathematical & Computer Sciences  Associate Professor
 School of Mathematical & Computer Sciences, Actuarial Mathematics & Statistics  Associate Professor
Person: Academic (Research & Teaching)