How to calculate the Multiple Stratonovich Integral?

Question

My question is about multiple Stratonovich-Integrals.
I have the following Stratonovich-Integral
$ \int \limits_{t_n}^{t_{n+1}} \int \limits_{t_n}^{s_1}1\,dW(s)dW(s_1).$
How can I calculate it?
Is it right, that $\int \limits_{t_n}^{s_1}1\,dW(s)=W(s_1)-W(t_n)$?
And then I get $ \int \limits_{t_n}^{t_{n+1}} (W(s_1)-W(t_n))dW(s_1)?$
And then I can split the Integral into
$ \int \limits_{t_n}^{t_{n+1}} W(s_1)dW(s_1)=\tfrac{1}{2}(W(t_{n+1}))^2-\tfrac{1}{2}(W(t_{n}))^2$ and
$ \int \limits_{t_n}^{t_{n+1}} W(t_n)dW(s_1)=W(t_n)\int \limits_{t_n}^{t_{n+1}} 1dW(s_1)=W(t_n)(W(t_{n+1}-W(t_n))?$
Hence I would get $ \int \limits_{t_n}^{t_{n+1}} \int \limits_{t_n}^{s_1}1\,dW(s)dW(s_1)=\tfrac{1}{2}(W(t_{n+1}))^2-\tfrac{1}{2}(W(t_{n}))^2-W(t_n)(W(t_{n+1}-W(t_n))$
But that's wrong, I should get $ \int \limits_{t_n}^{t_{n+1}} \int \limits_{t_n}^{s_1}1\,dW(s)dW(s_1)=\tfrac{1}{2}(W(t_{n+1})-W(t_n))^2$
Can you explain we, how to get to this result and what I am doing wrong in my calculation?
Thank you so much!!!

Answer 1

Your calculation is correct. Note that

$$\begin{align*}& \quad\frac{1}{2} W(t_{n+1})^2 - \frac{1}{2} W(t_n)^2 - W(t_n) \cdot (W(t_{n+1})-W_{t_n}) \\ &= \frac{1}{2} W(t_{n+1})^2 + \frac{1}{2} W(t_n)^2 - \frac{1}{2} \cdot 2 \cdot W(t_n) \cdot W(t_{n+1}) \\ &= \frac{1}{2} (W(t_{n+1})-W(t_n))^2 \end{align*}$$