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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (6): 883-892.doi: 10.3969/j.issn.1005-3085.2015.06.009

• • 上一篇    下一篇

Black-Scholes方程的条件Lie-B$\ddot{\rm a}$cklund对称和不变子空间

左苏丽1,   勾  明1,   李吉娜2,   黄  晴1   

  1. 1- 西北大学数学系非线性中心现代物理研究所，西安 710127
    2- 中原工学院理学院，郑州 450007
  • 收稿日期:2014-02-17 接受日期:2014-10-27 出版日期:2015-12-15 发布日期:2016-02-15
  • 基金资助:
    国家自然科学基金数学天元基金 (11226195; 11326167)；陕西省教育厅基金 (JC11217)；河南省自然科学基金 (122300410166)；河南省教育厅自然科学基金 (13A110119).

Conditional Lie-B$\ddot{\rm a}$cklund Symmetry and Invariant Subspace for Black-Scholes Equation

ZUO Su-li1,   GOU Ming1,   LI Ji-na2,   HUANG Qing1   

  1. 1- Department of Mathematics, Center for Nonlinear Studies, Institute of Modern Physics, Northwest University, Xi'an 710127
    2- College of Science, Zhongyuan University Technology, Zhengzhou 450007
  • Received:2014-02-17 Accepted:2014-10-27 Online:2015-12-15 Published:2016-02-15
  • Supported by:
    The Tian Yuan Foundation of National Natural Science Foundation of China (11226195; 11326167); the Natural Science Foundation of Shaanxi Provincial Education Department (JC11217); the Natural Science Foundation of Henan Province (122300410166); the Natural Science Foundation of Henan Provincial Education Department (13A110119).

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摘要： 偏微分方程的精确解蕴含了方程丰富的信息，对于描述各种现象的发展规律起着至关重要的作用．因此偏微分方程的精确解成为了数学、物理、经济等领域研究的热点问题．本文研究了金融数学中最重要的模型之一Black-Scholes方程的广义分离变量解．运用条件Lie-B$\ddot{\rm a}$cklund对称与不变子空间理论相结合的方法，本文得到了形如欧拉方程的条件Lie-B$\ddot{\rm a}$cklund对称．该方程允许的条件Lie-B$\ddot{\rm a}$cklund对称与高阶变系数的常微分方程相对应．同时，我们还得到了该方程允许此特征的所有精确解．

关键词: Black-Scholes方程, 条件Lie-B$\ddot{\rm a}$cklund对称, 不变子空间, 欧拉方程

Abstract:

The exact solution of partial differential equations, which contains rich information for the equations, is very important for describing the development of various phenomena and thus becomes a research focus of scientific fields such as mathematics, physics, economy and so on. In this paper, the generalized separable solutions for Black-Scholes equation, which is one of most important models arising in financial mathematics, are discussed. By using the conditional Lie-B$\ddot{\rm a}$cklund symmetry and invariant subspace theory, we obtain the conditional Lie-B$\ddot{\rm a}$cklund symmetries, which are similar to Euler equation. The conditional Lie-B$\ddot{\rm a}$cklund symmetries, which are admitted by Black-Scholes, are corresponding to high-order variable coefficient ordinary differential equations. At the same time, all of exact solutions associated to the conditional Lie-B$\ddot{\rm a}$cklund symmetries are performed.

Key words: Black-Scholes equation, conditional Lie-B$\ddot{\rm a}$cklund symmetry, invariant subspace, Euler equation

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