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

TL;DR

This paper analyzes the properties of moments of the density in nonlocal mean field game equations with quadratic costs, showing that expectation and variance satisfy specific differential equations and that moments depend only on these two quantities.

Contribution

It demonstrates that for certain mean field game equations, moments of the process depend solely on expectation and variance, with explicit differential equations governing these moments.

Findings

Expectation and variance follow second-order ODEs.

Moments of any order depend only on expectation and variance.

Characteristic function and fundamental solution are expressed in terms of expectation and variance.

Abstract

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 ${\mathbb V}$.