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Semi-martingales à valeurs dans des espaces d’applications $$C^{\infty}$$ entre espaces vectoriels. I. (Semi-martingales with values in spaces of $$C^{\infty}$$-maps between vector spaces. I.). (French) Zbl 0617.60050
Let E, F, G be finite dimensional vector spaces, $$\Phi$$ a semi-martingale with values in $$C^{\infty}(E;F)$$, $$\Psi$$ a semi-martingale with values in $$C^{\infty}(F;G)$$, X a semi-martingale with values in E. We prove the following three theorems:
1. $$\Phi$$ (X) is a semi-martingale with values in F; 2. $$\Phi\circ \Phi$$ is a semi-martingale with values in $$C^{\infty}(E;G)$$; 3. If $$\Phi$$ is inversible, $$\Phi^{-1}$$ is a semi-martingale with values in $$C^{\infty}(F;E)$$. This theorem 3 will be proved in part II [see the following entry, Zbl 0617.60051].

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60G48 Generalizations of martingales 60J60 Diffusion processes
semi-martingale