It has been a while since I studied stochastic calculus but I have encountered a problem (unrelated to options pricing) at work which requires me to brush up on it.
Looking at this pdf document, we see in section B that $G$ is a function of $S$ only. But $S$ follows a Wiener process, which means it is a function of $t$, so therefore $G$ must also be a function of $t$, no?
The process for $S$ typically looks like $$dS = \mu S dt+\sigma S dz$$ so why (in the link above) is $\frac{\partial G}{\partial t}=0$? Should it not be $\frac{\partial G}{\partial t}=\frac{\partial G}{\partial S}\frac{\partial S}{\partial t}$?
The referenced example is confusing: "$G$ is only a function of $S$". $G$ is a new process with nowhere differentiable trajectories. Formally, $G_t=f(t,S_t)$, where $f(t,x)=\ln(x)$. Itô's lemma is applied to $f$ for which $\partial f/\partial t=0$.