ИСТИНА

We study the mean field games equations, consisting of the coupled Kolmogorov - Fokker- Planck and Hamilton - Jacobi – Bellman equations. The equations are complemented by initial and terminal conditions. We show that with some specific choice of data, this problem can be reduced to solving a quadratically nonlinear system of ODEs. This situation occurs naturally in economic applications. As an example, we give a problem of forming an investor’s opinion on an asset. Namely, we consider a market in which a large number of investors manage their own portfolio of securities, which includes the asset under consideration. Each investor solves the problem of maximizing a utility function of the capital, which is common for all investors, based on its own ideas about the true characteristics of the asset (trend and volatility). Investors get a penalty both for a deviation from the opinion of the majority and for deviation from the true values of trend and volatility, unknown for investors. The problem is to study the behavior of maxima of distribution of trend and volatility as a function of time, that is, to track how the opinion of the majority of investors about the asset changes in response to the method of control.