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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, 1623

Keywords: Stochastic differential equations, Lie algebras, Campbell-Hausdorff formula

Nature: Original

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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 1432

Keywords: Stochastic differential equations, Lie algebras, Chen's iterated integrals, Campbell-Hausdorff formula

Nature: Original

Retrieve article from Numdam

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

Comment: On a closely related subject, see the paper of Fliess and Norman-Cyrot, 1623

Keywords: Stochastic differential equations, Lie algebras, Campbell-Hausdorff formula

Nature: Original

Retrieve article from Numdam

XVI: 23, 257-267, LNM 920 (1982)

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 (

Comment: See Kunita's paper 1432

Keywords: Stochastic differential equations, Lie algebras, Chen's iterated integrals, Campbell-Hausdorff formula

Nature: Original

Retrieve article from Numdam