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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.

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