The Knothe-Rosenblatt distance and its induced topology
arxiv(2023)
摘要
A basic and natural coupling between two probabilities on 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 𝒫 (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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