Let $$I_{N}^{\alpha}(t)=\sum_{j=0}^{N-1}W(t_{j}^{\alpha})[W(t_{j+1})-W(t_{j})] \ \ \ (*)$$
$$\textrm{where} \ t_{j}=\frac{jt}{N} \ \textrm{and} \ t_{j}^{\alpha}=\alpha t_{j}+(1-\alpha )t_{j+1} \ \ $$
I need to calculate the $\lim_{N\to\infty}\mathbb{E}(I_{N}^{\alpha}(t))$. I've expanded out (*) and found that the first 2 terms go to 0 but I'm struggling to find the final term. Please help.