It is known that knowledge of a symmetry of a scalar Ito stochastic differential equations leads, thanks to the Kozlov substitution, to its integration. In the present paper we provide a classification of scalar autonomous Ito stochastic differential equations with simple noise possessing symmetries; here "simple noise" means the noise coefficient is of the form $\s (x,t) = s x^k$, with $s$ and $k$ real constants. Such equations can be taken to a standard form via a well known transformation; for such standard forms we also provide the integration of the symmetric equations. Our work extends previous classifications in that it also consider recently introduced types of symmetries, in particular standard random symmetries, not considered in those.

Comment: published version; 49 pages in OCNMP format

• 34C14 - Symmetries, invariants of ordinary differential equations
• 34C20 - Transformation and reduction of ordinary differential equations and systems, normal forms
• 34F05 - Ordinary differential equations and systems with randomness
• 60K99 - Special processes