Nonlinear Fokker-Planck equation in the model of asset returns

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.

Original language	English
Article number	038
Journal	Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume	4
DOIs	https://doi.org/10.3842/SIGMA.2008.038
Publication status	Published - 6 Apr 2008

Keywords

Fokker-Planck equation
Non-linear evolution operator
Semiclassical asymptotics
The cauchy problem
Trajectory concentrated functions

ASJC Scopus subject areas

Analysis
Geometry and Topology
Mathematical Physics

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10.3842/SIGMA.2008.038

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title = "Nonlinear Fokker-Planck equation in the model of asset returns",

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.",

keywords = "Fokker-Planck equation, Non-linear evolution operator, Semiclassical asymptotics, The cauchy problem, Trajectory concentrated functions",

author = "Shapovalov, {Aleksandr Vasilievich} and Andrey Trifonov and Elena Masalova",

year = "2008",

month = apr,

day = "6",

doi = "10.3842/SIGMA.2008.038",

language = "English",

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AU - Shapovalov, Aleksandr Vasilievich

AU - Trifonov, Andrey

AU - Masalova, Elena

PY - 2008/4/6

Y1 - 2008/4/6

N2 - 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.

AB - 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.

KW - Fokker-Planck equation

KW - Non-linear evolution operator

KW - Semiclassical asymptotics

KW - The cauchy problem

KW - Trajectory concentrated functions

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