XIV: 32, 282-304, LNM 784 (1980)
KUNITA, Hiroshi
On the representation of solutions of stochastic differential equations (
Stochastic calculus)
This paper concerns stochastic differential equations in the standard form $dY_t=\sum_i X_i(Y_t)\,dB^i(t)+X_0(Y_t)\,dt$ where the $B^i$ are independent Brownian motions, the stochastic integrals are in the Stratonovich sense, and $X_i,X_0$ have the geometric nature of vector fields. The problem is to find a deterministic (and smooth) machinery which, given the paths $B^i(.)$ will produce the path $Y(.)$. The complexity of this machinery reflects that of the Lie algebra generated by the vector fields. After a study of the commutative case, a paper of Yamato settled the case of a nilpotent Lie algebra, and the present paper deals with the solvable case. This line of thought led to the important and popular theory of flows of diffeomorphisms associated with a stochastic differential equation (see for instance Kunita's paper in
Stochastic Integrals, Lecture Notes in M. 851)
Comment: On a closely related subject, see the paper of Fliess and Norman-Cyrot,
1623Keywords: Stochastic differential equations,
Lie algebras,
Campbell-Hausdorff formulaNature: Original
Retrieve article from Numdam
