Canonical forms for stochastic nonlinear affine systems with unknown parameters

This note proposes several necessary and sufficient conditions for the existence of diffeomorphisms which could transform stochastic nonlinear systems with unknown parameters to their canonical forms. Firstly, an equivalence is built between stochastic nonlinear systems and their deterministic partners under same transformation with Itô's differential formula. Secondly, several necessary and sufficient conditions are presented to transform the stochastic nonlinear systems to their strict-feedback forms, minimum-phase forms and observable normal forms with the equivalent relation.