Converting between Ito vs Stratonovich Calculus

We want to be able to convert between Stratonovich Calculus and Ito Calculus. We differentiate the two following our global Notation choices.

$$\begin{array}{rcl}\text{Ito}& \iff & \text{Stratonovich}\\ dX=Fdt+GdW& \Rightarrow & \left(F-\frac{1}{2}\frac{\partial G}{\partial x}G\right)dt+G\circ dW\\ dX=\left(F+\frac{1}{2}\frac{\partial G}{\partial x}G\right)dt+GdW& \Leftarrow & dX=Fdt+G\circ dW\end{array}$$