Let $g$ be a smooth (as smooth as necessary) function from $\mathbb{R}\to \mathbb{R}$, and consider the process: $$ X_t = \int_0^t g(W_s)ds $$ where here $W_s$ is the standard Wiener process.
My question: How is one to compute/interpret the quantity $dX_t$ ? Intuitively it does not seem to be the case that $dX_t = g(W_t) dt$ and that we would need to apply some sort of chain rule, however the fact that $W_t$ is not differentiable is throwing me off a bit. How does one interpret a quantity like this? Is it through integration by parts?
There's nothing special here. This is a Riemann integral of an (almost surely) continuous function. The standard fundamental theorem of calculus says that $X_t$ is literally differentiable.