چكيده به لاتين
Due to the increasing use of stochastic differential equations in modeling of many phenomena in financial mathematics, insurance, physics, chemistry, engineering, biology, etc., this field has attracted the attention of many researchers. In this thesis, numerical solution of stochastic differential equations and its relation with partial differential equations are investigated. Given that not much work has been done on solving these equations by the method of expansion and the introduction of the basis for the solutions of these equations and there is little success due to the stochastic nature of the solution of these equations, this thesis attempts to present an orthogonal basis to solve a bunch
of these equations, we use two-dimensional Hermite polynomials in this regard. Also, we estimate the parameters of these equations based on the Fokker-Planck equation and Maximum Likelihood Method. Then, with regard to the application of these equations in financial mathematics, we outline modeling stock prices and pricing contracts. Subsequently, under the Black-Scholes model, based on projection methods, a fast and high accurate numerical method for pricing of the discrete European option is presented. In addition, the problem of pricing of discrete European Options with time
dependent parameters and a numerical solution has been proposed to solve these equations. Finally, the accuracy and efficiency of the resulting method are compared with some other numerical and analytical methods in this field.
