The Knothe-Rosenblatt distance and its induced topology

Abstract

A basic and natural coupling between two probabilities on ${\mathbb{R}}^N$ is given by the Knothe-Rosenblatt coupling. It represents a multiperiod extension of the quantile coupling and is simple to calculate numerically. We consider the distance on $\mathcal{P}({\mathbb{R}}^N)$ that is induced by considering the transport costs associated to the Knothe-Rosenblatt coupling. We show that this Knothe-Rosenblatt distance metrizes the adapted weak topology which is a stochastic process version of the usual weak topology and plays an important role, e.g. concerning questions on stability of stochastic control and probabilistic operations. We also establish that the Knothe-Rosenblatt distance is a geodesic distance, give a Skorokhod representation theorem for the adapted weak topology, and provide multi-dimensional versions of our results.

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