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Local characteristics of semi-martingales and changes of probabilities. (Caractéristiques locales des semi-martingales et changements de probabilités.) (French) Zbl 0779.60043
Manifold-valued semi-martingales and the transformations of their local characteristics by change of probabilities are studied. A general Girsanov theorem is given: Let $$F({\mathcal D}\tilde X)={\mathcal B}(X)\alpha$$, where $$\alpha$$ is a previsible locally bounded process with values in $$T^*V$$ over $$X$$. Let $$N_ t=\int^ t_ 0\langle\alpha,d X\rangle$$, and $$Z={\mathcal E}(N)$$ is the stochastic exponent of $$N$$. Then $$N$$ is a local martingale and if $$Z$$ is a uniformly integrable martingale, $$X$$ is a $$Q$$-martingale with $$Q=Z\cdot P$$, $$Q$$ is equivalent to $$P$$. Then local characteristics of the product of two Lie group-valued semi-martingales $$X$$ and $$X'$$, and local characteristics of the projection of $$X$$ in a homogeneous space are computed. As an application Girsanov theorem in symmetric space of noncompact type is proved: conditions are given on a diffusion $$X$$, $$P$$-martingale in a symmetric space of noncompact type $$G/K$$, and a path $$g$$ in the Lie group $$G$$, for $$g\cdot X$$ to be a $$Q$$- martingale for some probability $$Q$$ equivalent to $$P$$.
##### MSC:
 60G44 Martingales with continuous parameter 58J65 Diffusion processes and stochastic analysis on manifolds 53C30 Differential geometry of homogeneous manifolds
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