Differential equation to Ito equation

Question

I have a differential equation: $$\frac{dX}{dt}=f(X)+\epsilon X n(t)$$ where $f$ is a deterministic function, $\epsilon$ is a constant, $n(t)$ is a white Gaussian noise and $X$ is a random process.

I want to transfer it to an Ito form: $$dX=g(X)dt+\epsilon XdW_t$$ where $W_t$ is a Weiner process.

I know that the Stratonovich form is $$dX=f(X)dt+\epsilon XdW_t~.$$ How can I find $g(X)~$?

I found in a textbook that it equal to $$f(X)+\frac{1}{2}\epsilon^2 X~.$$

But I do not know if it is right, or any justification for that result.

Thanks a lot.

Answer 1

This follows immediately from Ito's formula and the fact that $$ \int X \circ dY= \int X dY+ \langle X,Y\rangle$$ for continuous semi-martingales $X,Y$. Here $\circ dY$ denotes the Stratonovich integral with respect to $Y$, and $dY$ without the $\circ$ denotes the Ito integral with respect to $Y$.