Let us consider a diffusion process defined as $dX_t = g(X_t,t) \, dt + \sigma \, dW_t$ which induces a path measure $Q$ in the time interval $[0,T]$.
Is the following expectation $$ \left\langle \int^T_0 \frac{f(X_t)-g(X_t,t)}{\sigma^2} \, dW_t \right\rangle_Q, $$ constant or zero? Here $f$ is a bounded function.
As mentioned in an answer here and in the blog here,
A sufficient condition for the integral $\int_0^t f(\omega, s)\, dB_s$ to be a martingale on $[0,T]$ is that
- $f(\omega,s)$ is adapted, measurable in s, and
- $\mathbb{E}\left(\int_0^T f^2(\omega,s)\,ds\right) < \infty$. In this case, indeed, $\mathsf{E} \left(\int_0^T f(\omega,s)\, dB_s\right)=0$.
So if those conditions are satisfied for the given $f,g$ (eg. the integrability condition), then yes.
If we don't have the integrability, that integral will not even be well-defined/finite and so we cannot compute its expectation. Also, by taking $f=x^{p}$ for some $0<p<1/2$, we then deal with rough-integrals.
