I have the following problem: for a given finite set of smooth and continuously differentiable functions of time $\{f_1(t),f_2(t),f_3(t),\cdots, f_n(t)\}$, I need to find explicit expressions for the mean of the product of Ito stochastic integrals of the form \begin{equation} I_k = \int_{0}^{t} f_k(s) dW(s), \end{equation} where $dW(t)$ is a standard Brownian motion or Wiener process with mean zero, and variance $\mathbb{E}(dW^2) = dt$, assuming all functions $f_k(t)$ are deterministic. Now, we know from the properties of Ito integrals that \begin{equation} \mathbb{E}\left(\int_{0}^{t} f_k(s) dW(s) \right) = 0;\quad \mathbb{E}\left[\left(\int_{0}^{t} f_k(s) dW(s)\right)^2\right] = \int_{0}^{t}\mathbb{E}\left[ f_k^2(s)\right]ds, \end{equation} \begin{equation} \mathbb{E}\left(\int_{0}^{t} f_k(s) dW(s)\int_{0}^{t} f_m(s) dW(s)\right) = \int_{0}^{t}\mathbb{E}\left[f_k(s) f_m(s)\right]ds, \end{equation} Additionally, we know from Isserli's theorem (also known as Wick's theorem) that for a sequence of zero-mean random normal variables $\{X_1, X_2,\cdots, X_n \}$ the following results hold \begin{equation} \mathbb{E}\left(\prod_{k}^N X_k \right) =\begin{cases}\displaystyle \sum_k \prod_{k\neq m} \mathbb{E}(X_k X_m) & N \text{ is even}\\ 0 & N \text{ is odd} \end{cases} \end{equation} As an example, we know from this theorem that \begin{equation} \mathbb{E}\left(X_1 X_2 X_3\right) = 0, \end{equation} \begin{equation} \mathbb{E}\left(X_1 X_2 X_3 X_4\right) = \mathbb{E}\left(X_1 X_2 \right)\mathbb{E}\left(X_3 X_4 \right) + \mathbb{E}\left(X_1 X_3 \right)\mathbb{E}\left(X_2 X_4 \right) + \mathbb{E}\left(X_1 X_4 \right)\mathbb{E}\left(X_2 X_3 \right). \end{equation} Since the stochastic integrals $I_k$ also have mean zero and variance $\int_{0}^{t} f_k^2(s) ds$, they can be thought of as normally distributed processes \begin{equation} I_k = \int_{0}^{t} f_k(s) dW(s) \sim \text{Normal}\left(0,\int_{0}^{t} f_k^2(s) ds\right). \end{equation} My question is, does Isserlis' theorem apply to products of the integrals $I_k$? For example, is it true that \begin{equation} \mathbb{E}\left(\int_{0}^{t} f_1(s) dW(s)\int_{0}^{t} f_2(s) dW(s)\int_{0}^{t} f_3(s) dW(s)\right) = 0, \end{equation} and \begin{align*} &\mathbb{E}\left(\int_{0}^{t} f_1(s) dW(s)\int_{0}^{t} f_2(s) dW(s)\int_{0}^{t} f_3(s) dW(s)\int_{0}^{t} f_4(s)dW(s)\right)\\ &= \mathbb{E}\left(\int_{0}^{t} f_1(s) dW(s)\int_{0}^{t} f_2(s) dW(s)\right) \mathbb{E}\left(\int_{0}^{t} f_3(s) dW(s)\int_{0}^{t} f_4(s) dW(s)\right) \\ &+ \mathbb{E}\left(\int_{0}^{t} f_1(s) dW(s)\int_{0}^{t} f_3(s) dW(s)\right) \mathbb{E}\left(\int_{0}^{t} f_2(s) dW(s)\int_{0}^{t} f_4(s) dW(s)\right)\\ &+\mathbb{E}\left(\int_{0}^{t} f_1(s) dW(s)\int_{0}^{t} f_4(s) dW(s)\right) \mathbb{E}\left(\int_{0}^{t} f_2(s) dW(s)\int_{0}^{t} f_3(s) dW(s)\right)\\ &= \int_{0}^{t} \mathbb{E}\left[f_1(s)f_2(s)\right]ds\int_{0}^{t} \mathbb{E}\left[f_3(s)f_4(s)\right]ds+ \int_{0}^{t} \mathbb{E}\left[f_1(s)f_3(s)\right]ds\int_{0}^{t} \mathbb{E}\left[f_2(s)f_4(s)\right]ds \\ &+ \int_{0}^{t} \mathbb{E}\left[f_1(s)f_4(s)\right]ds\int_{0}^{t} \mathbb{E}\left[f_2(s)f_3(s)\right]ds \end{align*} ? We may assume that all the integrals are taken over the same Wiener increment $dW(t)$.
P.S.: I am a physics student, so I am not well-versed in measure theory or functional analysis. I am mainly looking for ways to simplify the product of several stochastic integrals to extract the mean in explicit form.