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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
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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 1

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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