Skorokhod problem

Find a constrained path and a boundary regulator which together modify a given driving path. In the half-line case, write $x=g+k\ge 0$, require $k(0)=0$ and $k$ nondecreasing, and require its increases to occur only where $x=0$. For multidimensional orthants the regulator acts through a specified reflection matrix. The Skorokhod reflection map on the half-line solves the one-dimensional continuous-path case explicitly.

For a continuous driving path with $g(0)\ge 0$, the running supremum supplies the minimal nondecreasing regulator keeping the path nonnegative. It increases only at a new negative running minimum, where the reflected path is zero. Any regulator satisfying the Skorokhod problem constraints must coincide with it. On a finite time interval, $\Vert \Gamma (g)-\Gamma (h){\Vert }_{\infty }\le 2\Vert g-h{\Vert }_{\infty }$, so the map is continuous and transfers a functional central limit theorem to reflected workloads.