  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 R^J, a non-empty convex cone d(x) in
R^J, specified at each point x on the boundary of G, and a cadlag trajectory
\psi taking values in R^J, the ESM 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]. General deterministic properties of the ESP are first established under
the only assumption that the graph of d(.) is closed. Next, for a class of
multi-dimensional ESPs on polyhedral domains, pathwise uniqueness and existence
of strong solutions to the associated stochastic differential equations is
established. In addition, it is also proved that these reflected diffusions are
semimartingales on [0,\tau_0], where \tau_0 is the time to hit the set of
points x on the boundary for which d(x) contains a line. 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.
