Suppose that $V(S, t)$ satisfies the Black-Scholes PDE:
$$\frac{\partial V}{\partial t} + \frac{1}{2} \sigma ^2 S^2 \frac{\partial ^2V}{\partial S^2} + (r-q)S \frac{\partial V}{\partial S} - rV = 0, \space S > 0, \space t < T,$$ $$V(S, T) = P_{o}(S), \space S > 0.$$
I want to consider the forward price of $S$ over the time interval $[t, T]$, i.e. $F = S\exp((r-q)(T-t)).$ Let $t' = t, \hat{V}(F, t') = V(S,t)$. I want to show that $\hat{V}$ satisfies the PDE:
$$\frac{\partial \hat{V}}{\partial t'} + \frac{1}{2} \sigma ^2 F^2 \frac{\partial ^2 \hat{V}}{\partial F^2} - r \hat{V} = 0, \space F > 0, \space t' < T,$$ $$\hat{V}(F, T) = P_{o}(F), \space F > 0.$$ I think this should be just an application of the chain rule, but I keep getting terms like $\frac{\partial ^2 V}{\partial t \partial S}$ that I don't know how to get rid of. So I think I am confusing something, most likely not getting the right formulas for the chain rule...