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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?

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Your equation looks ok. If interest rates are deterministic then forwards (being the same as futures) are driftless under the risk neutral measure. Otherwise, Forwards are driftless (i.e. martingales) under the corresponding forward measure while futures are martingales under the risk neutral measure.

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