Applying Ito's Lemma and integrating, show $\int_{t_i}^{t_{i+1}} W_sdW_s = \frac{1}{2}(W_{t_{i+1}}^2 - W_{t_i}^2 - {\Delta}t)$

Question

I need to show

$$\int_{t_i}^{t_{i+1}} W_sdW_s = \frac{1}{2}\left(W_{t_{i+1}}^2 - W_{t_i}^2 - {\Delta}t \right)$$

And then apply the result to show

$$\int_{t_i}^{t_{i+1}} \left [\int_{t_i}^{s}dW_u \right]dW_s = \frac{1}{2}\left({\Delta}W_{i}^2 - {\Delta}t \right)$$

Could someone please advise how I can approach this? I don't know how to do it.

Thanks

Answer 1

Per Ito’s Lemma

$$d(W_s^2)=2W_sdW_s+2ds $$ or $$W_sdW_s= \frac12(d(W_s^2) -ds)$$ Then, integrate to obtain $$\int_{t_i}^{t_{i+1}} W_sdW_s =\frac12(W^2_{t_{i+1}}-W^2_{t_{i}}-\Delta t) $$