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...
This might be of interest:
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.
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?