On certain analytically solvable problems of mean field games theory

We study the mean field games equations consisting of the coupled Kolmogorov–Fokker–Planck and Hamilton–Jacobi–Bellman equations. The equations are supplemented with initial and terminal conditions. It is shown that for a certain specific choice of data this problem can be reduced to solving a quadratically nonlinear ODE system. This situation occurs naturally in economic applications. As an example, the problem of forming an investor's opinion on an asset is considered.