If: $y = \frac{x_0}{n} \sum_{i=1}^{n} e^{rt_i} \;.\;e^{-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}} $
Could someone please explain why squaring y results in this:
$y^2 = \frac{x^2_0}{n^2} \sum_{i,j=1}^{n} \;e^{r(t_i+t_j)+\sigma^2(t_i\wedge t_j)} $
as $e^{-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}} = A_t $ should = 1 as $A_0=1$ and is a martingale so that $A_0 = E_0[A_t]$?
please note:
i < j and $w_t$ is a standard weiner process / brownian motion