Let \(D[0, \infty)\) be the space of càdlàg functions (i.e. continuous on the right with finite left limits) that are defined on \([0, \infty)\) and take values in \((-\infty, \infty).\) The space of right continuous functions whose left limits (at all points in \((0, \infty)\)) and values both lie in \([-\infty, \infty)\) (respectively, \((-\infty, \infty]\)) will be denoted as \(D^-[0, \infty)\) (respectively \(D^+[0, \infty)\)). The authors consider Skorokhod problem and extended Skorokhod problem.
Definition. Extended Skorokhod Problem (ESP) on \([l(\cdot), r(\cdot)]).\) Suppose that \(l \in D^-[0,\infty),\) \(r \in D^+[0, \infty)\) and \(l \leq r.\) Given any \(\psi \in D[0, \infty)\), a pair of functions \((\phi, \eta) \in D[0, \infty)\times D[0, \infty)\) is said to solve the ESP on \([l(\cdot), r(\cdot)])\) for \(\psi\) if and only if it satisfies the following properties: 5mm

1.

For every \(t \in [0, \infty)\), \(\phi (t) = \psi (t) + \eta (t) \in [l(t), r(t)].\)

2.

For every \(0\leq s <t<\infty,\)
\(\eta(t) - \eta(s) \geq 0\quad \text{if}\quad \phi(u) < r(u) \text{ for all} \quad u \in (s, t]\)
\(\eta(t) - \eta(s) \leq 0 \quad \text{if}\quad \phi(u) > l(u) \text{ for all}\quad u \in (s, t]\)

3.

For every \(0\leq t < \infty\),
\(\eta(t) - \eta(s) \geq 0\)if \(\phi(t) < r(t)\).
\(\eta(t) - \eta(s) \leq 0\)if \(\phi(t) > l(t)\),
where \(\eta (0-)\) is to be interpreted as \(0.\)

The following theorem is the main result in this paper.
Theorem. Suppose that \(l \in D^-[0,\infty)\), \(r \in D^+[0, \infty\) and \(l\leq r.\) Then for each \(\psi \in D[0, \infty,\) there exists a unique pair \((\phi, \eta) \in D[0,\infty) \times D[0, \infty)\) that solves the ESP on \([l(\cdot), r(\cdot)]\) for \(\psi.\) Moreover, the ESM \(\bar\Gamma_{l,r}\) admits the following explicit representation: \(\bar\Gamma_{l,r} =\psi - \Xi_{l, r} (\psi),\) where the mapping \(\Xi_{l, r} : D[0, \infty) \mapsto D[0, \infty)\) is defined as following: for each \(t \in [0,\infty),\)
\[ \begin{aligned} \Xi_{l, r} (\psi(t) \dot=& \max \Biggl( \Biggl[ (\psi(0) - r(0))^+ \wedge \inf_{ u\in [0, t]} (\psi (u) - l(u)) \Biggr],\\ & \sup_{s\in [0,t]}\Biggl[ (\psi(s) - r(s))^+ \wedge \inf_{ u\in [s, t]} (\psi (u) - l(u)) \Biggr]\Biggr),\end{aligned} \]
Furthermore, the map \((l, r, \psi) \mapsto \bar\Gamma_{l, r}\) is a continuous map on \(D^-[0, \infty) \times D^+[0,\infty)\times D[0,\infty)\) (with respect to topology of uniform convergence on compact sets). Lastly, if \(\inf_{t \geq 0} (r(t) - l(t)) > 0\) then \(\Gamma_{l, r} = \bar\Gamma_{l,r}\).
The authors apply analysis of the one-dimensional reflected Brownian motion in a time-dependent interval to study the behavior of the constraining process and, in particular, the semimartingale property of a class of two-dimensional reflected Brownian motions in a fixed domain. Reflecting Brownian motions in time-dependent domains arise in queueing theory, statistical physics, control theory and finance.