A Non-Skorohod Topology on the Skorohod Space

Abstract

A new topology (called $S$) is defined on the space $D$ of functions $x: [0,1] \to R^1$ which are right-continuous and admit limits from the left at each $t > 0$. Although $S$ cannot be metricized, it is quite natural and shares many useful properties with the traditional Skorohod's topologies $J_1$ and $M_1$. In particular, on the space $P(D)$ of laws of stochastic processes with trajectories in $D$ the topology $S$ induces a sequential topology for which both the direct and the converse Prohorov's theorems are valid, the a.s. Skorohod representation for subsequences exists and finite dimensional convergence outside a countable set holds.