I am in the following setting.
Let $$Y_T =\int^{X_T}_0\frac{1}{\sigma(x)}dx$$ where $dX_t = \sigma(X_t)dW_t$, with $W_t$ a regular Brownian motion and $\sigma$ an arbitrary positive function. Moreover, we have that $\mathbb{E}[X_T]=0$ ($X_t$ is a martingale).
The problem I am now facing is that I have to find $\mathbb{E}[Y_T]$.
I am not sure how to approach the problem, or if it is even solvable in the current state of the given information, but I attempted the following (very sketchy) solution
$$Y_T =\int^{X_T}_0\frac{1}{\sigma(x)}dx = \int^{X_T}_0\frac{1}{\sigma(x)}\sigma(x)dW=\int^{X_T}_0 dW$$
Which would mean that the expectation is zero. I have a feeling, though, that what I am doing is not allowed.
Any help would be appreciated!