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 v. The generalization which we deal with can be interpreted as allowing more general set functions v, for example signed measures which are only sigma-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. (C) 2013 Elsevier B.V. All rights reserved.