SIGMA 4 (2008), 038, 10 pages      arXiv:0804.0900      https://doi.org/10.3842/SIGMA.2008.038
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Nonlinear Fokker-Planck Equation in the Model of Asset Returns

Alexander Shapovalov ^{a, b, c}, Andrey Trifonov ^{b, c} and Elena Masalova ^b
a) Tomsk State University, 36 Lenin Ave., 634050 Tomsk, Russia
^b) Tomsk Polytechnic University, 30 Lenin Ave., 634050 Tomsk, Russia
^c) Mathematical Physics Laboratory, Tomsk Polytechnic University, 30 Lenin Ave., 634050 Tomsk, Russia

Received September 30, 2007, in final form March 26, 2008; Published online April 06, 2008

Abstract
The Fokker-Planck equation with diffusion coefficient quadratic in space variable, linear drift coefficient, and nonlocal nonlinearity term is considered in the framework of a model of analysis of asset returns at financial markets. For special cases of such a Fokker-Planck equation we describe a construction of exact solution of the Cauchy problem. In the general case, we construct the leading term of the Cauchy problem solution asymptotic in a formal small parameter in semiclassical approximation following the complex WKB-Maslov method in the class of trajectory concentrated functions.

Key words: Fokker-Planck equation; semiclassical asymptotics; the Cauchy problem; nonlinear evolution operator; trajectory concentrated functions.

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