Abstract
For general stochastic systems driven by continuous semimartingales an explicit formula for the logarithm of the Itô flow map is given. The computation relies on the lift to quasishuffle algebras of formulas involving products of Itô integrals of semimartingales. Whereas the Chen–Strichartz formula computing the logarithm of the Stratonovich flow map is classically expanded as a formal sum indexed by permutations, the analogous formula in Itô calculus is naturally indexed by surjections. This reflects the change of algebraic background involved in the transition between the two integration theories. Lastly, we extend our formula for the quasishuffle Chen–Strichartz series for the logarithm of the flow map to the noncommutative case. For linear matrixvalued SDEs driven by arbitrary semimartingales we obtain a similar formula.
Original language  English 

Article number  495202 
Number of pages  17 
Journal  Journal of Physics A: Mathematical and Theoretical 
Volume  48 
Issue number  49 
DOIs  
Publication status  Published  18 Nov 2015 
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Anke Wiese
 School of Mathematical & Computer Sciences  Associate Professor
 School of Mathematical & Computer Sciences, Actuarial Mathematics & Statistics  Associate Professor
Person: Academic (Research & Teaching)