Numerical Solution of a Class of Stochastic Functional Differential Equations with Financial Applications

Abstract

After a brief review of the Euler and Milstein numerical schemes and their convergence results for stochastic differential equations (SDEs) and stochastic functional differential equations (SFDEs), the thesis next proposes two specific SFDEs. The classical Euler and Milstein schemes are developed to find the numerical solutions of these SFDEs, which are then compared with the Ornstein-Uhlenbeck and a modified Ornstein-Uhlenbeck processes. These results are further used to build four different but related stochastic models for stock prices. The fitness of these models is analyzed by comparing real market data. The thesis concludes with a numerical study for option pricing for stock models with path dependent volatilities.