Existence and uniqueness for singular stochastic differential equations with piecewise well-behaved coefficients

TL;DR

This paper establishes existence and uniqueness results for one-dimensional singular stochastic differential equations with discontinuous or degenerate coefficients, including applications to threshold Cox-Ingersoll-Ross models.

Contribution

It introduces new conditions for pathwise uniqueness without requiring uniform ellipticity or continuity, and develops a novel pasting approach for generalized SDEs.

Findings

Provided sufficient conditions for pathwise uniqueness under weak assumptions.

Developed a pasting theorem transferring properties from local to global solutions.

Established the first existence and uniqueness results for skew sticky threshold diffusions.

Abstract

We study existence and uniqueness for one-dimensional generalized stochastic differential equations with singular coefficients, including distributional drift and degenerate, possibly discontinuous, diffusion coefficients. Such singularities naturally encode changes in the dynamics at thresholds, including reflecting, skew, or sticky interface behavior. We develop two directions. We provide sufficient conditions for pathwise uniqueness, under weak existence and uniqueness in law, without assuming uniform ellipticity or continuity of the diffusion coefficient. We also investigate a pasting approach for generalized stochastic differential equations that transfers strong existence and pathwise uniqueness, as well as weak existence and uniqueness in law, from local component equations to a global solution. To the best of our knowledge, this provides the first explicit pasting theorem yielding pathwise uniqueness in the setting of generalized stochastic differential equations. As an application, we establish the first existence and uniqueness results for a class of skew sticky threshold Cox-Ingersoll-Ross-type diffusions, including the threshold Chan-Karolyi-Longstaff-Sanders process.