Stratonovich integral $\int_0^T W_t \circ dW_t$

Question

I am trying to teach myself some stochastic calculus and am struggling to derive a Stratonovich integral. I am trying to understand how, considering the integral \begin{equation} \int_0^T W_t \circ dW_t \end{equation}

we get the Riemann sum \begin{align} \Rightarrow& \sum_{j=0}^{n-1} W(\frac{t_{j+1} - t_j}{2}) ( W(t_{j+1}) - W(t_{j}))\\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots\\ &=\frac{1}{2} (W^2(T) -W^2(0)) + \sum_{j=0}^{n-1}( W(t_{j+1}) - W(t_{j})). \end{align}

Can anyone provide the identity used to get the last step above (simplifying the Riemann sum) or show how to get to the final step from the initial definition?

Answer 1

Your expression for Stratanovich integral is wrong - the correct sum should be

$$\sum\frac{W(t_{j+1})+W(t_j)}{2}(W(t_{j+1})-W(t_j))=\frac12\sum W^2(t_{j+1})-W^2(t_j)=\frac12 W^2(T)-\frac12 W^2(0).$$