From entropic transport to martingale transport, and applications to model calibration

TL;DR

This paper introduces a discrete-time semi-martingale optimal transport formulation using multi-marginal entropic transport, enabling efficient numerical calibration and connecting to continuous-time processes as the time step diminishes.

Contribution

It develops a novel discrete-time semi-martingale transport model based on multi-marginal entropic transport, extending Sinkhorn algorithms for calibration tasks.

Findings

Provides a new numerical approach for model calibration.

Recovers semi-martingale processes in the continuous limit.

Connects entropic transport with semi-martingale optimal transport.

Abstract

We propose a discrete time formulation of the semi martingale optimal transport problembased on multi-marginal entropic transport. This approach offers a new way to formulate and solve numerically the calibration problem proposed by Guo et al. 2022, using a multi-marginal extension of Sinkhorn algorithm as in Benamou, Carlier, and Nenna 2019; Carlier et al. 2017; Benamou et al. 2019. In the limit when the time step goes to zero we recover, as detailed in the companion paper Benamou et al. 2024, a semi-martingale process, solution to a semi-martingale optimal transport problem, with a cost function involving the so-called specific entropy introduced in Gantert 1991, see also F{\"o}llmer 2022 and Backhoff-Veraguas and Unterberger 2023.