Nonlinear Fokker-Planck-Kolmogorov equation in analysis of asset returns

Abstract:
We consider the nonlinear Fokker-Planck-Kolmogorov equation (FPKE) coefficients of which depend on the first moment of the FPKE solution. For the FPKE with constant dispersion and nonlinear drift, an exact evolution operator is constructed and symmetry properties are discussed. The evolution operator is used to construct a solution of the Cauchy problem in the semiclassical approximation for the FPKE with time-dependent dispersion quadratic in coordinates. An example related to distribution of increments of asset prices on financial markets is considered and properties of the distributions with 'power law' and 'heavy tails' are discussed.