I have to prove the following identity: $\mathbb{E}[C^i_SS_i\hat{z}_i|\hat{z}_0]=\sigma_i\sqrt{\Delta t}\mathbb{E}[C^i_SS_i+C_{SS}^iS_i^2|\hat{z}_0]$.
Is this equivalent to prove: $\mathbb{E}[\frac{C^i_SS_i\hat{z}_i}{C^i_SS_i+C^i_{SS}S_i^2}|\hat{z}_0]=\sigma_i\sqrt{\Delta t}$ where I can axe the $S_i$ or is this step wrong?
This question is equivalent to: Is $\mathbb{E}[XYZ|B]=ab\mathbb{E}[XY+VY^2|B]$ where $X,Y,V$ depend on $B$ and $Z$ is not independent from $X$ and from $Y$ equivalent to
$\mathbb{E}[YZ|B]=ab\mathbb{E}[X+VY|B]$?
Hint: remember that the expectation is an integral. E.g. write $E[\cdot] = \int \cdot \, \mathrm dP$. Your question is similar to asking whether $$ \frac{\int f(x) \, \mathrm dx}{\int g(x)\, \mathrm dx} \stackrel?= \int\frac{f(x)}{g(x)}\, \mathrm dx. $$ Take $f(x) = x^2$ and $g(x) = x$ to see that it does not hold generally.