# Ito's lemma for a Forward

I'm trying to understand the derivation of Ito's process with respect to a Forward $$F$$ on a stock $$S$$ that pays a constant dividend yield, say $$y$$. Stock follows brownian motion $$\\$$ $$dS_{t} = S_{t}(\mu dt + \sigma dW_{t})$$ $$\\$$ and $$r$$ is interest rate.

Can someone verify that the Ito's process of $$F$$ is the following:$$\\$$

$$F_{t} = S_{t}\exp{(r-y)(T-t)}\\$$

$$dF_{t} = F_{t}((y-r+\mu)dt + \sigma dW_{t})$$

If the above is correct then how would this change under a risk neutral measure?