On the completion of Skorokhod space

Abstract

We consider the classical Skorokhod space ${\mathbb {D}}[0,1]$ and the space of continuous functions ${\mathbb {C}}[0,1]$ equipped with the standard Skorokhod distance $\rho $.

It is well known that neither $({\mathbb {D}}[0,1],\rho )$ nor $({\mathbb {C}}[0,1],\rho )$ is complete. We provide an explicit description of the corresponding completions. The elements of these completions can be regarded as usual functions on $[0,1]$ except for a countable number of instants where their values vary “instantly".