equations, local times, generalized drift, reflection, absorption, non-existence of solutions\
    *2010 MSC:* 60H10, 60J55
author:
- |
    Stefan Blei\
    stefan.blei@uni-jena.de
- |
    Hans-Jürgen Engelbert\
    hans-juergen.engelbert@uni-jena.de
bibliography:
- 'references.bib'
date: |
    Friedrich-Schiller-Universität Jena,\
    Fakultät für Mathematik und Informatik,\
    Institut für Stochastik,\
    D-07743 Jena, Germany\
title: 'A Note on One-dimensional Stochastic Differential Equations with Generalized Drift[^1]'
---

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Introduction and Basic Definitions
==================================

Let be $b$ a measurable real function and $\nu$ a set function which is defined on the bounded Borel sets of the real line $\mathbb{R}$ such that it is a finite signed measure on $\mathscr{B}([-N,N])$ for every $N \in \mathbb{N}$. In the present note, we deal with the one-dimensional stochastic differential equation (SDE) with so-called *generalized drift* introduced as $$\label{eqn:SDE_mvd}
    X_t = X_0 + \int_0^t b(X_s) \, {\mbox{\upshape d}}B_s + \int_\mathbb{R} L^X(t,y) \, \nu({\mbox{\upshape d}}y)\,,$$ where $B$ is a Wiener process and $L^X$ denotes either the right, the left or the symmetric local time of the unknown process $X$. We call $\nu$ appearing in Eq. (\[eqn:SDE\_mvd\]) *drift measure.*\
SDEs of type (\[eqn:SDE\_mvd\]) with generalized drift have been studied previously by many authors. We refer the reader to Harrison and Shepp [@harrison_shepp], N.I. Portenko [@portenko], D.W. Stroock and M. 