Dynamics of Fokker–Planck Equation with Logarithmic Coefficients and Its Application in Econophysics

Abstract

We demonstrate that Fokker–Planck equations with logarithmic factors in diffusion and drift terms can be straightforwardly derived from the class of “constant elasticity of variance" stochastic processes without appealing to any symmetry argument. Analytical closed-form solutions are available for some special cases of this class of Fokker–Planck equations. The dynamics of the underlying stochastic variables are examined. These Fokker–Planck equations have found a rather wide range of applications in various contexts. In particular, in the field of econophysics we have demonstrated their immediate relevance to modelling the exchange rate dynamics in a target zone, e.g. the linked exchange rate system of the Hong Kong dollar. Furthermore, the knowledge of exact solutions in some special cases can be useful as a benchmark to test approximate numerical or analytical procedures.