I would like to calculate expectation $\mathbb{E}\left[\nu_t\tilde{\nu}_t\right]$ of product of two Cox-Ingersoll-Ross processes $\nu_t$ and $\tilde{\nu}_t$. They are described by the following SDEs:
$$\mathrm{d}\nu_t=\kappa\left(\theta-\nu_t\right)\mathrm{d}t+\sigma\sqrt{\nu_t}\mathrm{d}W_t,$$ $$\mathrm{d}\tilde{\nu}_t=\kappa\left(\theta-\tilde{\nu}_t\right)\mathrm{d}t+\sigma\sqrt{\tilde{\nu}_t}\mathrm{d}W_t,$$ with different but deterministic initial values $\nu_0$ and $\tilde{\nu}_0$ respectively. Here $\kappa$ measures the speed of mean-reverting, $\theta$ is the long-run mean of both processes and $\sigma$ is the standard deviation. $W$ denotes a common Wiener process functioning as source of randomness. Assume that $\kappa$, $\theta$ and $\sigma$ are all constants satisfying Feller's condition. I tried to apply Ito's formula to $\nu_t\tilde{\nu_t}$:
$$\mathrm{d}\left(\nu_t\tilde{\nu_t}\right)=\kappa\theta\left(\nu_t+\tilde{\nu}_t\right)\mathrm{d}t-2\kappa\nu_t\tilde{\nu}_t\mathrm{d}t+\sigma^2\sqrt{\nu_t\tilde{\nu}_t}\mathrm{d}t+*\mathrm{d}W_t.$$
When integrating and taking expectation on both sides, it will introduce a new unknown quantity $\mathbb{E}\left[\sqrt{\nu_t\tilde{\nu}_t}\right]$, which seemingly blocks this approach. I wonder whether there is another approach that works.