On the pathwise uniqueness for a class of degenerate It\^{o}-stochastic differential equations

TL;DR

This paper proves pathwise uniqueness for certain degenerate Itô stochastic differential equations, ensuring strong solutions exist and are unique under specific degeneracy conditions using advanced inequalities and regularity estimates.

Contribution

It introduces new techniques involving maximal function inequalities and Krylov estimates to establish pathwise uniqueness for degenerate SDEs with zero-time degeneracy points.

Findings

Pathwise uniqueness holds for the class of degenerate SDEs studied.

Solutions are shown to be strong and unique due to the Yamada-Watanabe Theorem.

The methods involve elliptic regularity and maximal function inequalities.

Abstract

We show pathwise uniqueness for a class of degenerate It\^{o}-SDE among all of its weak solutions that spend zero time at the points of degeneracy of the dispersion matrix. Consequently, by the Yamada-Watanabe Theorem and a weak existence result, the pathwise unique solutions can be shown to be strong and to exist. The main tools to show pathwise uniqueness are inequalities associated with maximal functions and a Krylov type estimate derived from elliptic regularity and uniqueness in law.