Multi-dimensional Legendre wavelets approach on the Black-Scholes and Heston Cox Ingersoll Ross equations

The one dimension Legendre Wavelet is a numerical method to solve one dimension equation. In this paper Black-Scholes equation (B-S), that has applied via single asset American option and Heston Cox-Ingersoll-Ross equation (HCIR), as partial differential equations have been studied in the form of stochastic model at first. The Black-Scholes and Heston Cox-Ingersoll-Ross Stochastic differential equations (SDE) models are converted to partial differential equations with a basic lemma in stochastic differential equation which called Ito lemma including derivatives and integration calculus in stochastic differential equations. Multi-dimensional Legendre wavelets method is based upon the expanded properties of Legendre wavelets from high order that is utilized to reduce these equations in to a system of algebraic equations. In fact the properties of Legendre wavelets are leads to reduce the PDEs problems to solution the ODEs systems. To ability and efficiency of the proposed techniques, numerical results and comparison with the other numerical method named Adomian decomposition method (ADM) for different values of parameters are tabulated and plotted.