Abstract
Interval wavelet numerical method for nonlinear PDEs can improve the calculation precision compared with the common wavelet. A new interval Shannon wavelet is constructed with the general variational principle. Compared with the existing interval wavelet, both the gradient and the smoothness near the boundary of the approximated function are taken into account. Using the new interval Shannon wavelet, a multiscale interpolation wavelet operator was constructed in this paper, which can transform the nonlinear partial differential equations into matrix differential equations; this can be solved by the coupling technique of the wavelet precise integration method (WPIM) and the variational iteration method (VIM). At last, the famous Black-Scholes model is taken as an example to test this new method. The numerical results show that this method can decrease the boundary effect greatly and improve the numerical precision in the whole definition domain compared with Yan’s method.
Highlights
The nonlinear Black-Scholes equations have been increasingly attracting interest over the last two decades, since they provide more accurate values by taking into account more realistic assumptions, such as the transaction costs, risks from an unprotected portfolio, large investor’s preferences, or illiquid markets, which may have an impact on the stock price, the volatility, the drift, and the option price itself [1]
As the matter of fact, the wavelet precise integration method (WPIM) is a simple and effective method for linear partial differential equations proposed by Mei et al [4]
Since the definition domain of wavelet transformation is an infinite interval, the boundary effect would occur when being applied for resolving the engineering problems with bounded interval, for example, ordinary differential equations (ODEs)
Summary
The nonlinear Black-Scholes equations have been increasingly attracting interest over the last two decades, since they provide more accurate values by taking into account more realistic assumptions, such as the transaction costs, risks from an unprotected portfolio, large investor’s preferences, or illiquid markets, which may have an impact on the stock price, the volatility, the drift, and the option price itself [1]. Since the definition domain of wavelet transformation is an infinite interval, the boundary effect would occur when being applied for resolving the engineering problems with bounded interval, for example, ordinary differential equations (ODEs) It will decrease the precision and computational efficiency of the solution. Mei et al [7] proposed a general construction method of interval wavelet based on the restricted variational principle. Mei et al [7] proposed a kind of construction method of the interval interpolation wavelet based on the restricted variational principle. It does not matter with the representation of the wavelet function.
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