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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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