In the context of Wiener-Ito chaos expansion, I had a look at Ito's paper "Multiple Wiener Integral", 1951.
I am puzzled by his last result, theorem 5.1, that a multiple Wiener integral $$I_p(f)\equiv\int_{[a,b]^p}\ \ f(t_1,t_2,\dots,t_p)d W(t_1)\dots d W(t_p)$$ for $f\in L^2([a,b]^p)$ is related to the iterated stochastic integral $$I_p(f)=p!\int_a^b\int_a^{t_p}\dots\int_a^{t_2} f(t_1,t_2,\dots,t_p)d W(t_1)\dots d W(t_p)$$ I agree when $f$ is a symmetric function in all its variables (a similar result holds for multiple Lebesgue integrals), but in the general case, I was expecting the symmetrization $\tilde{f}$ of $f$ in the iterated integral instead, $$\tilde{f}(t_1,\dots,t_p)\equiv\frac{1}{p!}\sum_{\pi\in S_p}f(t_{\pi(1)}, \dots , t_{\pi(p)}\ )$$ Can somebody confirm that it is a typo in the paper and that we rather have for $f\in L^2([a,b]^p)$ $$I_p(f)=p!\int_a^b\int_a^{t_p}\dots\int_a^{t_2} \tilde{f}(t_1,t_2,\dots,t_p)d W(t_1)\dots d W(t_p)=I_p(\tilde{f})$$ If not, what am I missing then ?
Edit: Following Did's comment, I added the p! in front of iterated integrals to be constistent with the symmetrization.