H. J. Engelbert

Singular stochastic differential equations

Introduction.- 1. Stochastic Differential Equations.- 2. One-Sided Classification of Isolated Singular Points.- 3. Two-Sided Classification of Isolated Singular Points.- 4. Classification at Infinity and Global Solutions.- 5. Several Special Cases.- Appendix A: Some Known Facts.- Appendix B: Some Auxiliary Lemmas.- Rferences.- Index of Notation.- Index of Terms.

On the theorem of T. Yamada and S. Watanabe

The well-known theorem of T. Yamada and S. Watanabe asserts that (weak) existence and pathwise uniqueness of the solution of a stochastic equation implies the existence of a strong solution. This is the most powerful tool for proving that a stochastic equation possesses a strong solution. However, pathwise uniqueness is far from being a necessary condition for this. Even if the solution is not unique in law it is also of interest to ask for strong solutions. In the present note, we will discuss in more detail the connection between pathwise uniqueness and the existence of a strong solution. We will state a condition which is not only sufficient but also necessary for the existence of a strong solution.

Stochastic differential equations for sticky Brownian motion

We study (i) the stochastic differential equation (SDE) systemfor Brownian motion X in sticky at 0, and (ii) the SDE systemfor reflecting Brownian motion X in sticky at 0, where X starts at x in the state space, is a given constant, is a local time of X at 0 and B is a standard Brownian motion. We prove that both systems (i) have a jointly unique weak solution and (ii) have no strong solution. The latter fact verifies Skorokhods conjecture on sticky Brownian motion and provides alternative arguments to those given in the literature.

One-dimensional stochastic differential equations with generalized and singular drift

Introducing certain singularities, we generalize the class of one-dimensional stochastic differential equations with so-called generalized drift. Equations with generalized drift, well-known in the literature, possess a drift that is described by the semimartingale local time of the unknown process integrated with respect to a locally finite signed measure ν. The generalization which we deal with can be interpreted as allowing more general set functions ν, for example signed measures which are only σ-finite. However, we use a different approach to describe the singular drift. For the considered class of one-dimensional stochastic differential equations, we derive necessary and sufficient conditions for existence and uniqueness in law of solutions.

On Existence and Uniqueness of Reflected Solutions of Stochastic Equations Driven by Symmetric Stable Processes

1 Institut fur Stochastik, Friedrich-Schiller-Universitat, Ernst-Abbe Platz 1–4, D-07743 Jena, Germany (e-mail: engelbert@minet.uni-jena.de) 2 Department of Natural and Applied Sciences, University of Wisconsin-Green Bay, 2420 Nicolet Drive, Green Bay, WI 54311-7001, USA (e-mail: kurenokv@uwgb.edu) 3 Institut fur Stochastik, Friedrich-Schiller-Universitat, Ernst-Abbe Platz 1–4, D-07743 Jena, Germany (e-mail: zalinesc@minet.uni-jena.de)

On a generalization of the theorem of p. levy

Consider A Closed E of the Real Line R and a (Measureble) Function Defined on R and Strictly Concave on Every Component of R/E. The Present Paper Deals with the Following Marticgale Problem: Find a Continuous Local Martingale (X,F), Given on a Family () of Probability Spaces With for every such that x is stopped at the debut of e, and the process is a local martingale up to We show that this martingale problem possesses a unique solution. This solution is a strong Markov continuous local martingale. Furthermore, it is a pure continuous local martingale and therefore satisfies the previsible representation property. Conversely, for every strong Markov continuous local martingale (X,F) we can find a closed subset E of R and a function f strictly concave on every component of R\E such that (X, F) is the unique solution of the above martingale problem. In the particular case that E and f(X) we recover the theorem of P. Levy on the martingale characterization of the Brownian motion.

2. One-Sided Classification of Isolated Singular Points

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.

1. Stochastic Differential Equations

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.

5. Several Special Cases

In Section 5.1, we consider SDEs, for which the coefficients b and \(\sigma\) are power functions in the right-hand neighbourhood of zero or are equivalent to power functions as \(x\downarrow 0\). For these SDEs, we propose a simple procedure to determine the right type of zero.

4. Classification at Infinity and Global Solutions

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