ABSTRACT
This chapter introduces some basic issues concerning discrete time approximations of stochastic differential equations. The stochastic Taylor expansion is a stochastic counterpart of the Taylor expansion in a deterministic framework, and it is essential for the discrete time approximation of stochastic differential equations. The chapter discusses strong and weak Taylor approximations. The simplest strong Taylor approximation is the Euler scheme, also called the Euler-Maryama scheme. As with the strong approximations, the desired order of convergence determines where the Taylor expansion must be truncated. However, the weak convergence criterion only concerns probabilistic aspects of the sample path and not the sample path itself. The chapter examines a simple example to illustrate some aspects of the simulation of a time discrete approximation of an Ito process.
