Stratonovich, Ito and Numerical Analysis on Lie Groups

The description of stochastic mechanical systems naturally leads to Stratonovich stochastic differential equations on Lie groups. But calculating statistical properties is naturally done using Ito stochastic differential equations. All this is well known. But numerical implementations raise new issues. In particular, we show that the common practice of constructing numerical schemes for Stratonovich stochastic differential equations in Lie groups by following deterministic numerical schemes is in general flawed. Such numerical schemes do not in general converge to the correct limits. In a particular case, we give conditions showing when they do and when they don’t.