Formulae to about Moment and Cross-moments of Stratanovitch Iterated Integrals 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.
 A: I am sure you are happy to know another answer to this problem 4 years after the question. There is a remarkably elegant theorem that gives you the answer to your question, though in an implicit way. 
The theorem is called "Fawcett Theorem", to be found in page 39 of Fritz and Hairer's "A Course on Rough Paths".
The theorem essentially works by wrapping up the iterated integrals in the tensor algebra $T(\mathbb{R}^d),$ by which you avoid chasing the coordinates (although you have to chase them back if you want an explicit formula).
I avoid writing down the theorem here, since it is written better and thoroughly in the book. I will just rewrite the result:
If we denote by
$$\phi_t := 1+  \sum_{n \ge 1} \mathbb{E} \int^t \int^{t_n} \int \cdots \circ dB^{i_n}_{t_n}$$
We get that:
$$\phi_t = exp \left( \frac{t}{2} \sum_1^d e_i \otimes e_i\right)$$
A: Well,
Using my comment and this article in Arxiv I think I can answer the first part of my question. So first remark that :
$J_n(1)_{(0,1)}=\frac{1}{n!}(W_1)^n$ as the Stratanovitch-Stochastic Integral follows regular differential calculus rules and so by recurrence supposing the result true at order n-1, we have at order $n$:
$\int_0^1(\int_0^{t_n}...(\int_0^{t_2}o~dW_{t_1})....o~dW_{t_{n-1}}).o~dW_{t_n}=\frac{1}{(n-1)!}\int_0^1 (W_{t_n})^{(n-1)}dW_{t_n}=\frac{1}{(n-1)!}[\frac{1}{n}W_{t_n})^n]_0^1=\frac{1}{n!}(W_1)^n$ 
And as the result is trivial at $n=1$, conclusion follows so we have at last :
$E[J_n(1).J_p(1)]=\frac{1}{n!.p!}E[W_1^(n+p)]$ where $W_1$ is a standard gaussian variable. 
So now from formulae (15), (5) in the reference article we have :
$E[J_n(1).J_p(1)]=\frac{(p+n-1)!!}{n!.p!}$ if ($n+p$ is even)
$E[J_n(1).J_p(1)]=0$ if ($n+p$ is odd)
Where 
$(p+n-1)!!=(p+n-1).(p+n-3)....3.1$ if ($n+p$ is even) from formula (5) in the reference.
