Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 038, 10 pages      arXiv:0804.0900      http://dx.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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