Properties of moments of density for nonlocal mean field game equations with a quadratic cost function

We consider mean field game equations with an underlying jump-diffusion process $X_t$ for the case of a quadratic cost function and show that the expectation and variance of $X_t$ obey second-order ordinary differential equations with coefficients depending on the parameters of the cost function. Moreover, for the case of pure diffusion, the characteristic function and the fundamental solution of the equation for the probability density can only be expressed in terms of the expectation $\mathbb{E}$ and the variance $\mathbb{V}$ of the process $X_t$, so that the moments of any order depend only on $\mathbb{E}$ and $\mathbb{V}$.