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

Schwartz, Laurent

C. R. Acad. Sci., Paris, Sér. I 305, 31-35 (1987).

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

Keywords:

semi-martingale

Citations:

Zbl 0617.60051