求解非线性Black-Scholes模型的自适应小波精细积分法

Adaptive Wavelet Precise Integration Method on Nonlinear Black-Scholes Equations

针对非线性Black-Scholes方程,基于quasi-Shannon小波函数给出了一种求解非线性偏微分方程的自适应多尺度小波精细积分法.该方法首先利用插值小波理论构造了用于逼近连续函数的多尺度小波插值算子,利用该算子可以将非线性Black-Scholes方程自适应离散为非线性常微分方程组;然后将用于求解常微分方程组的精细积分法和小波变换的动态过程相结合,并利用非线性处理技术(如同伦分析技术)可有效求解非线性Black-Scholes方程.数值结果表明了该方法在数值精度和计算效率方面的优越性.

This paper proposed an adaptive wavelet precise integration method （WPIM） based on quasi-Shannon func tion fornonlinear Black Scholes equations. First, an adaptive wavelet interpolation operator was constructed, which can trans form the nonlinear partial differential equations into a matrix differential equations; Next, the Precise integration method was developd to solve the nonlinear matrix differential equation, which is a new asymptotic analytical method to solve nonlinear differential equations. Third, the exact analytical solution of the system of constant coefficient and nonhomogeneous differential e quations was deduced by this method, which is simpler than the traditional methods. At last, the famous Black-Scholes model was taken as an example to test this new method. The numerical result shows that this method possesses many merits such as high numerical stability and high precision.