# 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
thank you did, could you give me a hint on how you arrive there? –  Oliver Nov 25 '12 at 15:45
I am wondering about the same problem, too. I think conditioned on $W_u^\epsilon$, this will be a Gaussian random variable (as a linear functional of a Gaussian process) with expectation $0$ and Variance $\int_0^t (W_u^\epsilon)^2 du$. But how do I get rid off the conditioning? Did, you say that the integral is normally distributed with expectation $0$ and variance $1/2$. Could you give me a hint at how you get that answer? –  user153868 May 28 at 15:48

## 1 Answer

This might be of interest:

https://samm.univ-paris1.fr/IMG/pdf/pdf_law5.pdf

The characteristic function of the distribution is given on page 2. The proofs can be found in french language in

R. Berthuet (1981): Loi du logharitme itere pour cetaines integrales stochastiques. Ann. Sci. Univ. Clermont-Ferrand Math. 69, pag. 9-18.

and

M. Yor (1978): Remarques sur une formule de Paul Levy. Lecture Notes in Mathemat- ics 784, pag. 343-346.

I am particularly interested in an elementary proof showing that all moments of this random variable are bounded. If I am interested in an even moment I could use Ito's Lemma and Hölders inequality. Is there another way to show this? For instance, using isometry and reflection principle?

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