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

Understand

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 ${\mathbb V}$.