Abstract
Based on the reproducing kernel Hilbert space method, a new approach is proposed to approximate the solution of the Black-Scholes equation with Dirichlet boundary conditions and introduce the reproducing kernel properties in which the initial conditions of the problem are satisfied. Based on reproducing kernel theory, reproducing kernel functions with a polynomial form will be constructed in the reproducing kernel spaces spanned by the generalized Jacobi basis polynomials. Some new error estimates for application of the method are established. The convergence analysis is established theoretically. The proposed method is successfully used for solving an option pricing problem arising in financial modelling. The ideas and techniques presented in this paper will be useful for solving many other problems.
Highlights
Boundary value problems of ordinary differential equations play an important role in modelling a wide variety of physical and natural phenomena
Foroutan et al [13] proposed a method based on reproducing kernel Hilbert spaces to obtain approximate solutions of linear and nonlinear four-point boundary value problems
We apply the reproducing kernel method for solving nonlinear third order differential equations that have been extracted from some BlackScholes option pricing problem
Summary
Boundary value problems of ordinary differential equations play an important role in modelling a wide variety of physical and natural phenomena. Foroutan et al [13] proposed a method based on reproducing kernel Hilbert spaces to obtain approximate solutions of linear and nonlinear four-point boundary value problems. Arqub and Rashaideh applied the reproducing kernel method to obtain approximate solutions of integro-differential algebraic systems of temporal two-point boundary value problems in [8]. We apply the reproducing kernel method for solving nonlinear third order differential equations that have been extracted from some BlackScholes option pricing problem. To this end, we introduce a new technique based on reproducing kernel Hilbert space method with generalized Jacobi functions in polynomial space.
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