Well-posedness for a class of degenerate It\^o-SDEs with fully discontinuous coefficients

TL;DR

This paper establishes the uniqueness in law for a broad class of degenerate and discontinuous stochastic differential equations in multi-dimensional space, under conditions that include measure-zero degeneracy points.

Contribution

It proves uniqueness in law for SDEs with fully discontinuous and degenerate coefficients, extending previous results to more general cases with measure-zero degeneracy points.

Findings

Uniqueness in law holds for the class of SDEs considered.

Weak existence is established for SDEs with less regular drift coefficients.

Degeneracy points have measure zero, allowing for discontinuities and degeneracies in coefficients.

Abstract

We show uniqueness in law for a general class of stochastic differential equations in $\mathbb{R}^d$, $d\ge 2$, with possibly degenerate and/or fully discontinuous locally bounded coefficients among all weak solutions that spend zero time at the points of degeneracy of the dispersion matrix. The points of degeneracy have $d$-dimensional Lebesgue-Borel measure zero. Weak existence is obtained for more general, not necessarily locally bounded drift coefficient.