In page 23 of the book "The Malliavin Calculus and Related Topics" from Nualart one reads:
$\ \ $ Let $f_n:T^n\to\mathbb{R}$ be a symmetric and square integrable function. For these functions the multiple stochastic integral $I_n(f_n)$ with respect to the Gaussian process $\{W(h)=\int_T h_s dW_s,h\in L^2(T)\}$ introduced in Section 1.1.2 coincides with an iterated Itô integral. That is, assuming $T=\mathbb{R}_+$, we have $$I_n(f_n)=n!\int_0^\infty\int_0^{t_n}\cdots\int_0^{t_2}f_n(t_1,\ldots,t_n)\,dW_{t_1}\cdots dW_{t_n}.\tag{1.27}$$
I don't understand this definition. Shoudn't it be instead
$$I_n(f_n) = n! \int_0^\infty \int_0^{t_1} \ldots \int_0^{t_{n-1}} f_n(t_1,\ldots,t_n) dW_{t_n} \ldots dW_{t_2} dW_{t_1}? \tag{*}$$
Let's see how the formula behaves on 2 dimensions. Let $f(t_1,t_2) = 1_{[a,b] \times [c,d]}$ with $a > d$ so that this is an elementary function as defined in page 8, compare
$\ \ $ Fix $m\geq 1$. Set $\mathcal{B}_0=\{A\in\mathcal{B}:\mu(A)<\infty\}$. We want to define the multiple stochastic integral $I_m(f)$ of a function $f\in L^2(T^m,\mathcal{B}^m,\mu^m)$. We denote by $\mathcal{E}_m$ the set of elementary functions of the form $$f(t_1,\ldots,t_m)=\sum_{i_i,\ldots,i_m=1}^na_{i_1\cdots i_m}\mathbf{1}_{A_{i_1}\times\cdots\times A_{i_m}}(t_1,\ldots,t_m),\tag{1.10}$$ where $A_1,A_2,\ldots,A_n$ are pairwise-disjoint sets belongng to $\mathcal{B}_0$, and the coefficients $a_{i_1\cdots i_m}$ are zero if any two of the indices $i_1,\ldots,i_m$ are equal.
The fact that $f$ vanishes on the rectangles that intersect any diagonal subspace $\{t_i=t_j,i\neq j\}$ plays a basic role in the construction of the multiple stochastic integral.
$\ \ $ For a function of the form $(1.10)$ we define $$I_m(f)=\sum_{i_1,\ldots,i_m=1}^n a_{a_1\cdots i_m}W(A_{i_1})\cdots W(A_{i_m}).$$
So for $f$ as above, we have
$$\int_0^\infty \int_0^{t_1} 1_{[a,b]}(t_1) 1_{[c,d]}(t_2)dW_{t_2}dW_{t_1} = \int_a^\infty \int_0^{t_1} 1_{[a,b]}(t_1) 1_{[c,d]}(t_2)dW_{t_2}dW_{t_1} = \int_a^\infty 1_{[a,b]}(t_1) W([c,d])dW_{t_1} = W([a,b])W([c,d]) $$
So then we can see how the identity holds if we use $(*)$. But how do we compute it using (1.27)?
$$\int_0^\infty \int_0^{t_2} 1_{[a,b]}(t_1) 1_{[c,d]}(t_2)dW_{t_1}dW_{t_2}$$