Generalized Fokker--Planck Equation with Time-Dependent Transport Coefficients and a Quadratic Potential: Its Application in Econophysics

In order to control non-equilibrium processes and to describe the fat-tail phenomenon in econophysics, we generalize the traditional the Fokker--Planck equation (FPE) by including a quadratic correlation potential, and by making the time-dependent drift--diffusion coefficients. We investigate the su(1,1)⊕u(1) algebraic structure and obtain the exact solutions to the generalized time-dependent FPE by using the algebraic dynamical method. Based on the exact solution, an important issue in modern econophysics, i.e.~the fat-tail distribution in stock markets, is analysed.