对Ivancevic期权定价模型两类孤子解的研究

Research on two kinds of soliton solutions of Ivancevic option pricing model

为证明Hirota双线性方法求解一类基于非线性偏微分方程的金融数学模型的有效性,将其用于求解Ivancevic期权定价模型,以探究该模型是否存在孤子解。首先给出Hirota导数的定义与性质,对该模型做有理变换,引入Hirota导数得到模型的双线性形式;其次从较简单的色散关系出发,把偏微分方程转化为一般的常微分方程,逐步推导出方程的解;最后选取适当的参数,研究两类孤子解的动态特征,并结合图像解释模型精确解中参数的意义。本研究结果可为其他类非线性波动方程的求解提供参考。

In order to prove effectiveness of the Hirota bilinear method in solving a class of financial mathematical models based on nonlinear partial differential equations,it was used to solve the Ivancevic option pricing model to explore whether the model has soliton solutions.Firstly,definition and properties of the Hirota derivative were expounded,and the model was rationally transformed,obtaining the bilinear form of the model by introducing the Hirota derivative.Secondly,starting from the simple dispersion relation,the partial differential equation was transformed into a general ordinary differential equation,with the solution of the equation being derived step by step.Finally,the dynamic characteristics of two kinds of soliton solutions were targeted by selecting appropriate parameters,accounting for the significance of the parameters in the precise solution of the model in light of images.The results of this study can provide a reference for solving other kinds of nonlinear wave equations.