When can we integrate Wiener process?

Question

I get confused with the Itô integral. For example, if I look at geometric Brownian motion then we say $\int \alpha X_t dW_t$ is an Itô integral, one cannot integrate it, so one uses Itô formula. Then $\int \alpha dW_t$ comes out and you can suddenly integrate it into $\alpha W_t$. Why can't you integrate $\int \alpha X_tdW_t$ but $\int \alpha dW_t$?

Answer 1

You can integrate $\int \alpha X_t dW_t$ (assuming $X$ is reasonably well-behaved), it just usually doesn't have a closed form solution. There are very few cases where you do get a closed form solution. The case where $X_t = 1$ for all $t$ is one of those cases.

It's no different from saying we can integrate $\int x dx$ to get $\frac 12 x^2$, but can't integrate $\int e^{-\frac 12 x^2} dx$: We can integrate the second one, it just doesn't have a closed form solution.