Ekin Baylan, M.Sc.
Department of Financial Mathematics
September 2014, 69 pages

Supervisor: Yeliz Yolcu Okur (Institute of Applied Mathematics, Middle East Technical University, Ankara)
Co-supervisor: Ömür Uğur (Institute of Applied Mathematics, Middle East Technical University, Ankara)

Abstract

The aim of this work is to understand the stochastic Taylor schemes and to measure the accuracy of them by comparing their closeness to the exact solutions at the discretization points. Our assumption is that when we use the stochastic Taylor schemes with higher orders, we obtain better approximation processes to exact solutions of the stochastic differential equations. We give the stochastic Taylor schemes with different orders by regarding the convergence criteria for the stochastic differential equations. While Euler-Maruyama and Milstein schemes are derived by using the derivatives of stochastic Taylor expansion, stochastic Runge-Kutta schemes do not need these derivatives in their constructions. Therefore, we have the chance to get higher order stochastic Taylor schemes with less computational difficulties in Runge-Kutta schemes. Moreover, in the application part of the thesis, we observe that the stochastic Runge-Kutta schemes supply the best approximate processes to the exact solutions, for instance, in simulating Orsntein-Uhlenbeck process and in Monte Carlo method for option pricing models.

Keywords: Stochastic Taylor Schemes, Euler-Maruyama, Milstein, Runge-Kutta, Stochastic Simulations, Monte Carlo