Chebyshev cardinal wavelets and their application in solving nonlinear stochastic differential equations with fractional Brownian motion

Abstract

In this paper, a new computational method is proposed to solve a class of nonlinear stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm). The method is based on a new class of orthogonal wavelets, namely the Chebyshev cardinal wavelets. These new basis functions have many useful properties such as orthogonality, spectral accuracy, and cardinality. The operational matrices of the derivative and the integration of these new wavelets are derived and used in the implementation of the proposed method. Based on the interpolation property of these new wavelets, a new algorithm is presented for computing nonlinear terms in such problems. The main advantage of the proposed method is the reduction of the problem to a simpler one, which consists of solving a system of nonlinear algebraic equations. The convergence and error analysis of the established method are investigated in the Sobolev space. Moreover, the reliability and applicability of the proposed method are demonstrated by solving several numerical examples. Finally, the application of the proposed method is illustrated by solving two well-known fractional stochastic models in the mathematical finance and the mathematical ecology.