Reflected Diffusions Defined via the Extended Skorokhod Map

This work introduces the extended Skorokhod problem (ESP) and associated extended Skorokhod map (ESM) that enable a pathwise construction of reflected diffusions that are not necessarily semimartingales. Roughly speaking, given the closure $G$ of an open connected set in ${\mathbb R}^J$, a non-empty convex cone $d(x) \subset {\mathbb R}^J$ specified at each point $x$ on the boundary $\partial G$, and a cadlag trajectory $\psi$ taking values in ${\mathbb R}^J$, the ESM $\bar \Gamma$ defines a constrained version $\phi$ of $\psi$ that takes values in $G$ and is such that the increments of $\phi - \psi$ on any interval $[s,t]$ lie in the closed convex hull of the directions $d(\phi(u)), u \in (s,t]$. When the graph of $d(\cdot)$ is closed, the following three properties are established: (i) given $\psi$, if $(\phi,\eta)$ solve the ESP then $(\phi,\eta)$ solve the corresponding Skorokhod problem (SP) if and only if $\eta$ is of bounded variation; (ii) given $\psi$, any solution $(\phi,\eta)$ to the ESP is a solution to the SP on the interval $[0,\tau_0)$, but not in general on $[0,\tau_0]$, where $\tau_0$ is the first time that $\phi$ hits the set ${\cal V}$ of points $x \in \partial G$ such that $d(x)$ contains a line; (iii) the graph of the ESM $\bar \Gamma$ is closed on the space of cadlag trajectories (with respect to both the uniform and the $J_1$-Skorokhod topologies). The paper then focuses on a class of multi-dimensional ESPs on polyhedral domains with a non-empty ${\cal V}$-set. Uniqueness and existence of solutions for this class of ESPs is established and existence and pathwise uniqueness of strong solutions to the associated stochastic differential equations with reflection is derived. The associated reflected diffusions are also shown to satisfy the corresponding submartingale problem. Lastly, it is proved that these reflected diffusions are semimartingales on $[0,\tau_0]$. One motivation for the study of this class of reflected diffusions is that they arise as approximations of queueing networks in heavy traffic that use the so-called generalised processor sharing discipline.