A Collocation Method for Nonlinear Stochastic Differential Equations Driven by Fractional Brownian Motion and its Application to Mathematical Finance

Abstract

The main aim of this article is to demonstrate the collocation method based on the barycentric rational interpolation function to solve nonlinear stochastic differential equations driven by fractional Brownian motion. First of all, the corresponding integral form of the nonlinear stochastic differential equations driven by fractional Brownian motion is introduced. Then, collocation points followed by the Gauss-quadrature formula and Simpson’s quadrature method are used to reduce them into a system of algebraic equations. Finally, the approximate solution is obtained using Newton’s method. The rigorous convergence and error analysis of the presented method has been discussed in detail. The proposed method has been applied to some well-known stochastic models, such as the stock model and a few other examples, to demonstrate the applicability and plausibility of the discussed method. Also, the numerical results of the collocation method based on the barycentric rational interpolation function and barycentric Lagrange interpolation function get compared with an exact solution.