Stochastic Exponential in Lie Groups and its Applications

The aim of this article is to develop new simple proofs for the basic formulas of stochastic analysis in Lie groups, in particular the stochastic exponential and logarithm. We present applications to direct proofs of the (multiplicative) Doob­Meyer decomposition, Girsanov theorem for semimartingales in Lie groups and solution of stochastic Lax equations.