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

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.

💡 Research Summary

The paper investigates the Entropic Martingale Optimal Transport (EMOT) problem on the real line, a formulation that combines the classical optimal transport framework with a martingale constraint and an entropic regularization term. The authors motivate EMOT through the calibration of stochastic volatility models, where the martingale condition encodes the no‑arbitrage requirement under the risk‑neutral measure.

Problem formulation.
Given probability measures μ, ν on ℝ (the asset price at two future dates) and an auxiliary measure ρ on a third space ℝ (e.g., stochastic volatility), the EMOT problem is
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