# Multiple Wiener integral as Witt polynomial of Brownian motion

I know that if i have a Brownian motion $$W_t$$ the multiple Wiener integral

$$\int_0^t \int_0^{\xi_1}...\int_0^{\xi_n} dW_{\xi_1}...dW_{\xi_n}$$

can be expressed as $$H_n(\int_0^t dW_s)$$ where $$H_n$$ is the nth Hermite polynomial.

If now I have $$m$$ independent Brownian motion there is an analogous formula that allows us to express the integral $$\int_0^t \int_0^{\xi_1}...\int_0^{\xi_n} dW_{\xi_1}^{j_1}...dW_{\xi_n}^{j_n}$$

as a polynomial in the variables $$\int_0^t dW_s^{j_1}...,\int_0^t dW_s^{j_n}$$

I know that the polynomials $$\{T_\alpha(\eta)=\prod_{\alpha_i} H_{\alpha_i}(\eta_i) \ \ \forall \ \ \alpha \text{ multi-index }\}$$ where $$\eta$$ is a Gaussian random variable are widely used in the study of the Wiener chaos so i thought they could be a natural candidate, but i can't find any reference which express this result.

• That's because it's not true. Commented Jan 19, 2023 at 21:07
• @MartinHairer thanks, you spare me a lot of calculations. Do you know if there is any other polynomial which works or if there is any way to at least express $J_{(j_1,..,j_l)}$ in terma of a polynomial in $J_{(j_1,..j_{l-1})}$ Commented Jan 20, 2023 at 5:59
• There is no way of expressing iterated integrals of some given order in terms of iterated integrals of lower order (over the same time increment). If it were the case, stochastic analysis would basically not have much reason to exist... Commented Jan 20, 2023 at 14:41