一类非线性Black-Scholes方程的最优子代数系
Optimal systems of subalgebras for a nonlinear Black-Scholes equation
摘要：
我们的主要研究对象是描述非线性布莱克-斯科尔斯模型对称性的四维李代数。这个模型实现了一个反馈效应，这是典型的非流动性市场。李代数的结构取决于一个参数，即我们必须处理一个单参数代数族。我们使用Patera-Winternitz方法对这些代数进行了分类。本文对非线性Black-Scholes方程的对称代数族给出了一维、二维和三维子代数的最优系统。最优系统使我们有可能描述方程的一组完整的不变解。
英文摘要：
The main object of our study is a four dimensional Lie algebra which describes the symmetry properties of a nonlinear Black-Scholes model. This model implements a feedback effect which is typical for an illiquid market. The structure of the Lie algebra depends on one parameter, i.e. we have to do with a one-parametric family of algebras. We provide a classification of these algebras using Patera--Winternitz method. Optimal systems of one-, two- and three- dimensional subalgebras are described for the family of symmetry algebras of the nonlinear Black-Scholes equation. The optimal systems give us the possibility to describe a complete set of invariant solutions to the equation.
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