The decomposition of stretched Brownian motion into Bass martingales

TL;DR

This paper extends the decomposition of stretched Brownian motion into Bass martingales to the general case without irreducibility, using a convex paving of Euclidean space.

Contribution

It introduces a decomposition of stretched Brownian motion into Bass martingales over a convex paving, generalizing previous irreducible cases.

Findings

Decomposition into Bass martingales on convex sets

Identification of a convex paving related to irreducibility

Extension of previous irreducibility results

Abstract

In previous work J. Backhoff-Veraguas, M. Beiglb\"ock and the present authors showed that the notions of stretched Brownian motion and Bass martingale between two probability measures on Euclidean space coincide if and only if these two measures satisfy an irreducibility condition. Now we consider the general case, i.e. a pair of measures which are not necessarily irreducible. We show that there is a paving of Euclidean space into relatively open convex sets such that the stretched Brownian motion decomposes into a (possibly uncountable) family of Bass martingales on these sets. This paving coincides with the irreducible convex paving studied in previous work of H. De March and N. Touzi.