Numerical Solutions of Stochastic Differential Equations

In this dissertation, we consider the problem of simulation of stochastic differential equations driven by Brownian motions or the general Levy processes. There are two types of convergence for a numerical solution of a stochastic differential equation, the strong convergence and the weak convergence. We first introduce the strong convergence of the tamed Euler-Maruyama scheme under non-globally Lipschitz conditions, which allow the polynomial growth for the drift and diffusion coefficients. Then we prove a new weak convergence theorem given that the drift and diffusion coefficients of the stochastic differential equation are only twice continuously differentiable with bounded derivatives up to order 2 and the test function are third order continuously differentiable with all of its derivatives up to order 3 satisfying a polynomial growth condition. We also introduce the multilevel Monte Carlo method, which is efficient in reducing the total computational complexity of computing the expectation of a functional of the solution of a stochastic differential equation. This method combines the three sides of the simulation of stochastic differential equations: the strong convergence, the weak convergence and the Monte Carlo method. At last, a recent progress of the strong convergence of the numerical solutions of stochastic differential equations driven by Levy processes under non-globally Lipschitz conditions is also presented.