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Homogeneous space-valued semi-martingales. (Semi-martigales dans les espaces homogènes.) (French) Zbl 0779.60045
Consider a Lie group $$G$$ of Lie algebra $${\mathfrak G}$$ with neutral element $$e$$. Let $$Lg$$ $$(g\in G)$$ be the left application of $$G$$ on $$G:g'\to gg'$$, $$X\in{\mathfrak G}$$, $$M$$ be a semi-martingale on $${\mathfrak G}$$. Then the stochastic exponent $${\mathcal E}(M)$$ of $$M$$ is the solution of the stochastic differential equation of Stratonovich: $$\delta X=(L_ X)_ *\delta M$$, $$X_ 0=e$$. Then Lie groups and some homogeneous space-valued semi-martingales are considered (with the help of stochastic exponent) as developments of semi-martingales with values in the Lie algebra or a subspace of the Lie algebra. Homogeneous space-valued martingales and Brownian motion on noncompact symmetric spaces $$G/K$$ are characterized in that way. Also the Iwasawa and Cartan decomposition of Brownian motion obtained by M. P. and P. Malliavin are deduced.

##### MSC:
 60G48 Generalizations of martingales 53C30 Differential geometry of homogeneous manifolds 53C35 Differential geometry of symmetric spaces
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