my question is to find the mean and variance of the following Ito Stochastic Integral:
$I_T = \int_0^T \sqrt{|W_t|}d W_t$
I know that I need to find the $\mathbb E [\int_0^T \sqrt{|W_t|}d W_t]$
But I don't know how to integrate this, I understand that:
$\int_0^T \sqrt{|W_t|}d W_t = \lim_{|\Delta|\rightarrow 0} \sum_{i=1}^n \sqrt{|{W_t}_{i-1}|}({W_t}_i - {W_t}_{i-1}) $
I am used to a different situation where $\sqrt{|{W_t}_{i-1}|}= {W_t}_{i-1}$ which I substantially easier because I would normally be able to just rewrite $\sqrt{|{W_t}_{i-1}|}({W_t}_i - {W_t}_{i-1})$ as $\frac{1}{2}[{W_t}_i^2 - {W_t}_{i-1}^2-({W_t}_i - {W_t}_{i-1})^2]$.
Since $\sqrt{|{W_t}_{i-1}|}$ is not equal to $ {W_t}_{i-1}$, how would I begin solving this?