XVI: 23, 257-267, LNM 920 (1982)
FLIESS, Michel;
NORMAND-CYROT, Dorothée
Algèbres de Lie nilpotentes, formule de Baker-Campbell-Hausdorff et intégrales itérées de K.T.~Chen (
Stochastic calculus)
Consider a s.d.e. in a manifold, $dX_t=\sum_i A_i(X)\,dM^i_t$ (Stratonovich differentials), driven by continuous real semimartingales $M^i_t$, and where the $A_i$ have the geometrical nature of vector fields. Such an equation has a counterpart in which the $M^i(t)$ are arbitrary deterministic piecewise smooth functions, and if this equation can be solved by some deterministic machinery, then the s.d.e. can be solved too, just by making the input random. Thus a bridge is drawn between s.d.e.'s and problems of deterministic control theory. From this point of view, the complexity of the problem reflects that of the Lie algebra generated by the vector fields $A_i$. Assuming these fields are complete (i.e., generate true one-parameter groups on the manifold) and generate a finite dimensional Lie algebra (which then is the Lie algebra of a matrix group), the problem can be linearized. If the Lie algebra is nilpotent, the solution then can be expressed explicitly as a function of a finite number of iterated integrals of the driving processes (Chen integrals), and this provides the required ``deterministic machine''. It thus appears that results like those of Yamato (
Zeit. für W-Theorie, 47, 1979) do not really belong to probability theory
Comment: See Kunita's paper
1432Keywords: Stochastic differential equations,
Lie algebras,
Chen's iterated integrals,
Campbell-Hausdorff formulaNature: Original
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