Strong and weak convergence order of numerical methods for stochastic differential equations

Models based on SDEs have applications in many disciplines, but in pratical applications calculating an esplicit solution of an SDE is rare. Hence the development of efficient numerical methods to approximate the solution is a crucial task. Clearly, the method should be chosen in relation to the problem's requirements. If the problem requires the trajectories of the approximation to be close to those of the Ito process, we need a strong or pathwise convergence, with the aim of minimizing the absolute error at the final instant T. On the other hand, in many practical situations the interest focuses on approximating expectations of functionals of the Ito process, like its moments. Hence, it sufficies that the numerical method gives a good approximation of the probability distribution of the random variable X(T) rather than a close approximation of sample path. The first chapter begins with an introductory theorem that underlines the relation between consistency and convergence of a numerical scheme with respect to the strong criterion. Then it follows with the study of the Euler-Maruyama scheme and the Milstein scheme, trying to emphasize why the latter offers such an improvement in the order of strong convergence. Actually, the Milstein scheme is the simplest nontrivial numerical scheme for stochastic ordinary differential equations that achieves a strong order of convergence higher than the Euler-Maruyama one. It was first derived by Grigori N. Milstein using the Ito formula to expand the coefficients. This idea underlies the systematic derivation of higher order scheme, which will be developed in chapter 2. These schemes are based on the so called Ito-Taylor expansion, which is a formal formula based upon the iterated application of the Ito formula and it has analogous properties to the deterministic Taylor formula. We could interpret these stochastic Ito-Taylor expansions as basic numerical schemes. Chapter 3 is dedicated to weak approximations, starting with a theorem about consistency and convergence in the weak sense. This is useful for introducing the link with the solution of the Kolmogorov backward equation. We prove that under specific smoothness assumptions the Euler scheme has a 1 weak order of convergence. Since these assumptions are very restrictive and unlikely in practical cases, we will take advantage of a different version of the Kolmogorov theorem to prove that the Euler scheme converges weakly also with relaxed assumptions, though the order of 1 is no longer guaranteed. Finally we state a convergence theorem for a wide class of time discrete approximations, which allows the retrieval of higher weak order numerical scheme; although in practical applications a weak order of 1 is generally enough. In the last chapter we propose some applications of the numerical methods discussed so far to practical models, mostly arising from financial mathematics, paired with simulations performed with the software R. The purpose is to go through the main results shown in the thesis. We start with the Ornstein-Uhlenbeck-Vasicek process, which is one case in which the Euler scheme is of strong order of convergence 1. Therafter we present the Black-Scholes-Merton model, which is a good example to display the convergence increase from the Euler-Maruyama scheme to the Milstein's one and the difference between the two criteria of convergence. We proceed with the case of holderian drift and a review of recent results on the Cox-Ingersoll-Ross (CIR) model, a case that is both useful and troublesome. The work is concluded with some remarks concerning the current results and possible directions for further research