One-dimensional Stochastic Differential Equations with Generalized and Singular Drift

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 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.

Cited by in corpus (5)

• On the Strong Feller Property of Stochastic Delay Differential Equations with Singular Drift
• On general skew Brownian motions
• Strong Existence and Uniqueness for Singular SDEs Driven by Stable Processes
• On some path-dependent SDEs involving distributional drifts
• Pathwise solvability of stochastic integral equations with generalized drift and non-smooth dispersion functions