Rational non-commutative formal power series and iterated integral representation of a class of Ito processes

In 1982 and 1983 two articles [M. Fliess and F. Lamnabhi-Lagarrigue, J. Math. Phys. 23 (1982), no. 4, 495--502; F. Lamnabhi-Lagarrigue and M. Lamnabhi, in Computer algebra 55--67, Lecture Notes in Comput. Sci., 162, Springer, Berlin, 1983] were published in which the previous study is used to analyze the solution of stochastic differential equations in Stratonovich form. A formal power series is associated to the Volterra series with analytic kernels, of the analytic causal functional solution of some SDE. The purpose is to solve the SDE `by series'. Then, the statistics of the solution are deduced by the formal series properties. In the present paper, starting from these ideas, we analyze this association for functionals for the Wiener process and we give a theorem of convergence, in suitable norm, for formal rational series. In Section 1 we recall some basic algebraic notions which are used in the sequel.