Skip to main content

3. Two-Sided Classification of Isolated Singular Points

  • 1187 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1858)

Abstract

In this chapter, we investigate the behaviour of a solution of (1) in the two-sided neighbourhood of an isolated singular point. Many properties related to the “two-sided” behaviour follow from the results of Section 2.3. However, there are some properties that involve both the right type and the left type of a point. The corresponding statements are formulated in Section 3.1.

Section 3.2 contains an informal description of the behaviour of a solution for various types of isolated singular points.

The statements formulated in Section 3.1 are proved in Sect 3.3.

The results of Section 3.1 show that the isolated singular points of only 4 types can disturb uniqueness. These points are called here the branch points. Disturbing uniqueness, the branch points give rise to a variety of “bad” solutions. In particular, one can easily construct a non-Markov solution in the neighbourhood of a branch point. This is the topic of Section 3.4.

However, it turns out that all the strong Markov solutions in the neighbourhood of a branch point admit a simple description. It is given in Section 3.5.

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

Mathematics Subject Classification (2000):

  • 60-02
  • 60G17
  • 60H10
  • 60J25
  • 60J60

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   49.95
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alexander S. Cherny .

Rights and permissions

Reprints and Permissions

Copyright information

© 2005 Springer-Verlag Berlin/Heidelberg

About this chapter

Cite this chapter

Cherny, A.S., Engelbert, HJ. (2005). 3. Two-Sided Classification of Isolated Singular Points. In: Singular Stochastic Differential Equations. Lecture Notes in Mathematics, vol 1858. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31560-5_4

Download citation