# Integrate Brownian motion with respect to independent Brownian motion

we are wondering what can be said about the distribution of that? We are considering standard Brownian motions and the integral from 0 to 1... More precisely: What can be said about the distribution of $$\int_0^1 W^\epsilon_u dW^\eta_u$$ where $W^\epsilon$ and $W^\eta$ are independent standard Brownian motions. It certainly has mean zero but I don't quite know how I could do anything else with that...

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Please ask a more specific question, otherwise there is not much to say. –  Dan Brumleve Nov 25 '12 at 12:05
And please include fully the question in the body instead of relegating it to the title. –  Did Nov 25 '12 at 12:07
Sorry I hope it is clearer now –  Oliver Nov 25 '12 at 15:22
Centered normal with variance 1/2. –  Did Nov 25 '12 at 15:33
thank you did, could you give me a hint on how you arrive there? –  Oliver Nov 25 '12 at 15:45