1. Stochastic differential equations

Abstract

In this chapter, we consider general multidimensional SDEs of the form (1.1) given below. In Section 1.1, we give the standard definitions of various types of the existence and the uniqueness of solutions as well as some general theorems that show the relationship between various properties. Section 1.2 contains some classical sufficient conditions for various types of existence and uniqueness. In Section 1.3, we present several important examples that illustrate various combinations of the existence and the uniqueness of solutions. Most of these examples (but not all) are well known. We also find all the possible combinations of existence and uniqueness. Section 1.4 includes the definition of a martingale problem. We also recall the relationship between the martingale problems and the SDEs. In Section 1.5, we define a solution up to a random time.