We have the model
$$\frac{S_n - S_{n-1}}{S_{n-1}} = udt + \sigma dt^a Y_n,$$
with $T=Ndt$, $a =[0,1]$, $\sigma >0$, $P(Y_n=1) = P(Y_n=-1) = 1/2.$
I have found that $$\frac{S_N}{S_0} = \prod_{n=1}^N ( 1 + udt + \sigma dt^a Y_n)$$
How would I go about finding the expectation of this i.e $E(\frac{S_N}{S_0})$ and also $E(\frac{S_N^2}{S_0^2})$?
Do I ignore the product and take the expectation of the terms separately (like a summation) or would I need to take the product into account?