  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 ${\mathbb V}$.
