Rewriting a stochastic integral

Question

I have the following stochastic integral

$$ I(t) = \frac{1}{2}\int_{0}^t (t-s)^2 dW_s $$ I wish to rewrite this as a multiple stochastic integral containing only differentials, and I think the following is correct: $$ \frac{1}{2}\int_{0}^t (t-s)^2 dW(s) = \int_{0}^t \int_{0}^s \int_{0}^z dW_u dz ds, $$ But I can't prove it. How can I write $I(t)$ as a multiple stochastic integral containing only differentials with respect to time and brownian motion?

Answer 1

Use $$ \frac{(t-s)^2}{2}=\int_0^{t-s}u\,du=\int_s^t(u-s)\,du $$ which gives $$ \frac{1}{2}\int_{0}^t (t-s)^2 dW(s)=\int_{0}^t \int_s^t(u-s)\,du\, dW(s)\,. $$ Stochastic Fubini allows to write the last double integral as $$ \int_{0}^t \int_0^u(u-s)\, dW(s)\,du\,. $$ Now you can use $$ u-s=\int_s^u\,dz $$ and repeat the exercise.