Numerical Methods Based on Polynomials and Orthogonal Basis Functions for Solving Stochastic Differential and Integral Equations Arising in Mathematical Modelling

Abstract

A stochastic process is a mathematical object that is intended to model the evolution in time of a random phenomenon. The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modelling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in the space of particles subject to random forces. Since the 1970s, the Wiener process has been widely applied in financial mathematics and economics to model the evolution of stock prices and bond interest rates. The main objective of this dissertation is to provide various spectral methods, such as the operational matrix method, collocation method, Galerkin method, etc., to solve stochastic equations such as stochastic differential equations, stochastic integral equations, and stochastic integro-differential equations of both integer and fractional order arise virtually in every field of scientific endeavor. Therefore, in the present dissertation shifted Jacobi operational matrix method, the shifted Jacobi Galerkin method is applied to solve both linear and nonlinear stochastic Itô-Volterra integral equations numerically. The computational methods based on the Lucas polynomial, Genocchi polynomial, Lerch polynomial, and Pell polynomial are implemented to solve the multi-dimensional and fractional stochastic Itô-Volterra integral equations. The fractional Brownian motion is a generalization of Brownian motion and was first introduced by A. N. Kolmogorov in 1940 when it was called Wiener Helix. The stochastic equations with fractional Brownian motion are very useful in modelling many problems in biology, physics, mathematical finance, etc. So, in this work, the operational matrix method based on shifted Jacobi polynomial, barycentric rational interpolation collocation method, quintic B-spline collocation method, shifted Chebyshev spectral Galerkin method, and collocation method based on Chebyshev cardinal function to solve stochastic differential equations driven by fractional Brownian motion.