The title is a bit long but quite explicit, I am looking for a reference where the moments and cross moment Stratanovitch Iterated Integrals defined as :
$E[J_n(1).J_p(1)]$ with $p\not=n$
With : $J_n(1)_{0,1}=\int_0^1(\int_0^{t_n}...(\int_0^{t_2}o~dW_{t_1})....o~dW_{t_{n-1}}).o~dW_{t_n}$
An extension to multidimensional case would also be appreciated aswell as an extension to the case where some of the integrators are replaced by a $dt$ term. For example, with multi-index notations :
$J({(1,1,0,1)})_{0,1}=\int_0^1(\int_0^{t_4}(\int_0^{t_3}(\int_0^{t_2}o dW_{t_1})o dW_{t_2})dt_3)o dW_{t_4}$
Best regards.
PS 1: By the way I have such general formulas for the Iterated Itô Iterated Integrals (Chapter 5.7 of the Kloeden and Platen's Book Numerical Solution of Stochastic Integrals), but the generalization to the Stratanovitch case is not treated there and I was wondering if anyone has bothered calculating the Stratanovitch case which looks computationaly straithforward but seems an incredibly exhausting task to undertake.
PS 2: I googled for some reference but could find any.