4. Classification at Infinity and Global Solutions

Abstract

A classification similar to that given in Chapter 2 can be performed at $+ \infty$. This is the topic of Sections 4.1-4.3.

The results of Chapters 2, 3 apply to local solutions, i.e., solutions up to a random time. In Sections 4.4, 4.5, we study the existence and uniqueness of a global solution, i.e., a solution in the sense of Definition 1.28. This is done for the SDEs that have no more than one singular point.

Throughout this chapter, we assume that $\sigma(x)\neq0$ for all $x\in\mathbb{R}$.