Generalized specific entropy on Wiener space with application to Martingale Optimal Transport

TL;DR

This paper introduces a novel specific-entropy framework for continuous-time martingale transport using Poisson jump approximations, enabling explicit entropy functionals and efficient numerical schemes.

Contribution

It develops a new Poissonization approach to define generalized entropy on Wiener space, avoiding high-dimensional Sinkhorn problems and allowing local volatility reflection.

Findings

Proves weak convergence of Poisson approximations.

Identifies limiting entropy functionals for continuous martingales.

Demonstrates numerical schemes in one and two dimensions.

Abstract

Classical entropy regularization is poorly suited to continuous-time martingale transport, since relative entropy between diffusion laws typically forces their volatility characteristics to coincide. We introduce a specific-entropy framework based on Poisson jump approximations of continuous martingales. In the Gaussian-mark case, this yields explicit generalized specific entropy functionals on Wiener space, whose limiting costs depend not only on the limiting martingale laws but also on the microscopic approximation mechanism. This Poissonization approach avoids deterministic grid refinement and the associated high-dimensional multimarginal Sinkhorn problems, while allowing jump intensities to reflect local volatility. We prove weak convergence of the Poisson approximations and identify the limiting entropy functionals. For a trace-normalized Poisson scheme, the resulting cost defines a continuous-time specific-entropic martingale optimal transport problem, called SEMOT. This cost yields compactness, existence, and strong duality, and leads formally to a coupled Hamilton-Jacobi-Bellman/Fokker-Planck system. The resulting structure suggests Sinkhorn type numerical schemes, which we implement in one and two dimensions.