Convergence of Sinkhorn's Algorithm for Entropic Martingale Optimal Transport Problem

TL;DR

This paper proves that Sinkhorn's algorithm converges exponentially for the Entropic Martingale Optimal Transport problem, providing a solid theoretical foundation for its use in financial modeling and no-arbitrage pricing.

Contribution

It establishes the dual formulation of EMOT, proves exponential convergence of Sinkhorn's algorithm without assuming optimal potentials, and confirms the absence of a primal-dual gap.

Findings

Sinkhorn's algorithm converges exponentially for EMOT.

Dual formulation of EMOT is rigorously established.

No primal-dual gap exists in the EMOT problem.

Abstract

In this paper, we study the Entropic Martingale Optimal Transport (EMOT) problem on \mathbb{R}. The investigation of the EMOT problem arises in the calibration problem of the Stochastic Volatility Models, where martingale constraints reflect no-arbitrage pricing conditions under the risk-neutral measure, as originally proposed by Henry-Labordere. We first establish the dual formulation of the EMOT problem and prove that Sinkhorn's algorithm achieves an exponential convergence rate under mild conditions. Notably, our analysis does not presuppose the existence of optimal potentials and rigorously confirms the absence of a primal-dual gap. These results provide a theoretical foundation for solving EMOT via Sinkhorn's method and constructing the optimal distribution from dual coefficients.