Uniqueness for the Skorokhod problem in an orthant: critical cases

Abstract

Consider the Skorokhod problem in the closed non-negative orthant: find a solution $(g(t),m(t))$ to \[ g(t)= f(t)+ Rm(t),\] where $f$ is a given continuous vector-valued function with $f(0)$ in the orthant, $R$ is a given $d\times d$ matrix with 1's along the diagonal, $g$ takes values in the orthant, and $m$ is a vector-valued function that starts at 0, each component of $m$ is non-decreasing and continuous, and for each $i$ the $i^{th}$ coordinate of $m$ increases only when the $i^{th}$ coordinate of $g$ is 0. The stochastic version of the Skorokhod problem replaces $f$ by the paths of Brownian motion. It is known that there exists a unique solution to the Skorokhod problem if the spectral radius of $|Q|$ is less than 1, where $Q=I-R$ and $|Q|$ is the matrix whose entries are the absolute values of the corresponding entries of $Q$. The first result of this paper shows pathwise uniqueness for the stochastic version of the Skorokhod problem holds if the spectral radius of $|Q|$ is equal to 1. The second result of this paper settles the remaining open cases for uniqueness for the deterministic version when the dimension $d$ is two.