2. One-Sided Classification of Isolated Singular Points

Abstract

In this chapter, we consider SDEs of the form (1).

Section 2.1 deals with the following question: Which points should be called singular for SDE (1)? This section contains the definition of a singular point as well as the reasoning that these points are indeed “singular”.

Several natural examples of SDEs with isolated singular points are given in Section 2.2. These examples illustrate how a solution may behave in the neighbourhood of such a point.

In Section 2.3 we investigate the behaviour of a solution of (1) in the right-hand neighbourhood of an isolated singular point. We present a complete qualitative classification of different types of behaviour.

Section 2.4 contains an informal description of the constructed classification.

The statements that are formulated in Section 2.3 are proved in Section 2.5.

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