Existence of $q$-Bass martingales in the semidiscrete setting

2026-07-17

AI Summary

This study addresses the construction of martingales—termed $q$-Bass martingales—that, under prescribed initial and terminal marginal distributions, are as close as possible to a given reference measure $q$. Focusing on the semi-discrete setting where the initial marginal has finite atomic support, the authors establish the existence of such $q$-Bass martingales for the first time by leveraging a geometric parametrization via convex polygonal chains and tools from martingale optimal transport theory. Moreover, they demonstrate the uniqueness of the associated Bass measure under additive translations. These results provide foundational theoretical guarantees for the existence of $q$-Bass martingales and the uniqueness of Bass measures in the semi-discrete framework.

Abstract

The class of $q$-Bass martingales provides a natural answer to a central question in martingale optimal transport: how to construct martingales with prescribed initial and terminal marginals whose transition kernel remains as close as possible to a given reference measure $q$. We prove the existence of $q$-Bass martingales when the initial marginal is supported on finitely many atoms, and establish uniqueness, up to an additive translation constant, of the associated Bass measure. Our approach is geometric and relies on the analysis of a suitable parametrization of convex polygonal chains.

Problem

• martingale optimal transport
• q-Bass martingales
• prescribed marginals
• reference measure
• semidiscrete setting

Innovation

• q-Bass martingales
• martingale optimal transport
• convex polygonal chains
• semidiscrete setting
• Bass measure