This is an exercise in Bjork's book, (not an assignment or homework, I am trying to solve all the exercises to improve my fundamentals as a beginner)
$$ Z_t = \int_0^t g_s \, dW_s $$
The question asks to find the stochastic differential where g is an adapted stochastic process.
I'm not sure if this problem requires some form of substitution but I am getting thrown off by the fact that g(s) is a stochastic adapted process. (I assume that means it is a stochastic process with some form of diffusion from W.)
Naively applying Ito's lemma gives us a time derivative (Which I have no idea what to do with), a derivative with respect to W, which I assume cancels out the integral and gives us the integrand, and then a second derivative with respect to the stochastic process (assumed to be deterministic before plugging in the stochastic variable)
$$ dZ_t = \frac{dZ}{dt}.dt + \frac{dZ}{dx}.dx + \frac{1}{2}\frac{d^2Z}{dx^2}.dx^2 $$
Where $$ dX_t = 1.dW_t$$
I can intuitively see what happens to the second term but the rest of the two terms are something I struggle to make sense of. I imagine this is a pretty basic problem since I cannot find any thread with exactly this (where g is non-deterministic) so I must be missing something basic.
Thanks for your help
It's simply $dZ_t = g_t\,dW_t$. The differential notion is just a shorthand for an integral equation. If, say, $Z_t = f(W_t)$, then of course you'd use Ito's formula, but this isn't the case here.
More generally
$$X_t = X_s + \int_s^t b(t,X_t)\,dt + \int_s^t \sigma(t,X_t)\,dW_t$$
can be written as
$$dX_t = b(t,X_t)\,dt + \sigma(t,X_t)\,dW_t,$$
perhaps with extra information about what the initial value is.
It's used pretty much regardless of how complicated $g$ is, whether $g$ is deterministic, of the form $g_t = h(t,W_t)$, $g_t = h(t, Z_t)$ or even $g_t = h(\omega, t, Z_t)$, where $\omega\in \Omega$ (the sample space), in mean-field equations, etc. etc.