EDIT: One thing I forgot to mention before is that this is all under the $\mathbb{Q}$ measure in case that changes anything
I was just wondering if someone could explain how to solve this problem. I have an equation $X(t, S_{t}) = s\partial_{s}u - u$ where we have $u(t,S_{t})$. Now, according to the paper I'm reading, if we evaluate $dX$ then this gives: $dX = S_{t}\partial_{ss}u dS_{t}$. Can someone tell me what happens inbetween? I'm not sure how to apply Ito's formula in such a way as to yield this result (Note that the paper kinda implies that $s$ and $S_{t}$ are the 'same' variable).
Here's what I think I have to do: By applying Ito's formula to $X(t,S_{t})$ we can look at the terms $s\partial_{s}u$ and $u$ separately. By definition Ito's formula states that $du(t, S_{t}) = \frac{\partial u}{\partial s}(S_{t}, t)dS_{t} + \frac{\partial u}{\partial t}(S_{t}, t)dt + \frac{1}{2}\sigma^{2}_{u}(t)\frac{\partial^{2} u}{\partial s^{2}}(S_{t}, t)dt$. From this we can write the following:
Applying Ito's formula to the term $s\partial_{s}u$ gives:
$s\partial_{s}u = s\frac{\partial^{2}u}{\partial s^{2}}dS_{t} + s\frac{\partial^{2}u}{\partial s\partial t}dt + s\frac{\sigma^{2}}{2}\frac{\partial^{3}u}{\partial s^{3}}dt$ (I think the last term gets cancelled out though)
And, applying Ito's formula to the term $u$ gives:
$\frac{\partial u}{\partial s}dS_{t} + \frac{\partial u}{\partial t}dt + \frac{1}{2}\sigma^{2}_{u}(t)\frac{\partial^{2} u}{\partial s^{2}}dt$
Therefore, putting the two terms together, this means that applying Ito's formula to $X = s\partial_{s}u - u$ gives:
$dX = s\frac{\partial^{2}u}{\partial s^{2}}dS_{t} + s\frac{\partial^{2}u}{\partial s\partial t}dt + s\frac{\sigma^{2}}{2}\frac{\partial^{3}u}{\partial s^{3}}dt - \large[\frac{\partial u}{\partial s}dS_{t} + \frac{\partial u}{\partial t}dt + \frac{1}{2}\sigma^{2}_{u}(t)\frac{\partial^{2} u}{\partial s^{2}}dt\large]$
Which is of course completely different to the what the paper says, since the paper states that $dX$ should only be equal to $dX = S_{t}\partial_{ss}u dS_{t}$ Naturally, I feel as if I've messed this up quite a bit and that quite a bit of simplification must be going on at the very least. The problem I have is that I have no idea how to get this result. Am I applying Ito's formula the wrong way?
Thanks in advance.