One-dimensional Stochastic Differential Equations with Generalized and Singular Drift

TL;DR

This paper extends the class of one-dimensional stochastic differential equations with generalized and singular drifts by allowing more general set functions, providing necessary and sufficient conditions for the existence and uniqueness of solutions.

Contribution

It introduces a new approach to describe singular drifts in SDEs with more general set functions, broadening the scope of existing models.

Findings

Derived necessary and sufficient conditions for existence of solutions.

Established criteria for uniqueness in law of solutions.

Generalized the class of drifts to include -finite signed measures.

Abstract

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 \nu. The generalization which we deal with can be interpreted as allowing more general set functions \nu, 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.