A Note on the Itô and Stratonovich Stochastic Relative Degree and Normal Form

In this note we compare two notions of stochastic relative degree. Specifically, we consider nonlinear stochastic systems defined by the same stochastic differential equation and interpreted in either Itô's or Stratonovich's sense. We then recall the Itô stochastic relative degree and we introduce the concept of Stratonovich stochastic relative degree and Stratonovich normal form. We show, by means of examples, that the stochastic relative degrees arising from the two different interpretations of the same stochastic differential equations are, in general, different. We finally point out that this discrepancy can be eliminated through conversion formulas between Itô and Stratonovich integrals.