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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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