Statistical linearization has recently seen a particular surge of interest as
a numerically cheap method for robust control of stochastic differential
equations. Although it has already been successfully applied to control complex
stochastic systems, accessibility and controllability properties of statistical
linearization, which are key to make the robust control problem well-posed,
have not been investigated yet. In this paper, we bridge this gap by providing
sufficient conditions for the accessibility and controllability of statistical
linearization. Specifically, we establish simple sufficient algebraic
conditions for the accessibility and controllability of statistical
linearization, which involve the rank of the Lie algebra generated by the drift
only. In addition, we show these latter algebraic conditions are essentially
sharp, by means of a counterexample, and that they are generic with respect to
the drift and the initial condition.Comment: 23 page