Stochastic Integral Evaluation

Question

We want to evaluate:

$$ \int_{z=0}^T \int_{s=0}^z dW(s) dW(z) $$

But I don't really understand how to approach the problem. Is this just zero? Why?

Answer 1

Indeed as mentioned in comments we have

$$\int_{z=0}^T \int_{s=0}^z dW(s) dW(z)=\int_{0}^{T}W_{s}dW_{s}$$

and so using Itô-formula we get

$$=\frac{1}{2}(W_{T}^{2}-T).$$

Generally, multiple Itô integrals are given in terms of Hermite polynomials (see notes on The Wiener-Itô Chaos Decomposition and Multiple Wiener integrals)

$$n!H_{n}(W(h))=\int_{T^{n}}h(t_{1})\cdots h(t_{n})dW_{t_{1}}\cdots dW_{t_{n}}.$$