Entropy Martingale Optimal Transport and McKean-Vlasov equations

Funded by the European Union - NextGenerationEU under the National Recovery and Resilience Plan (PNRR) - Mission 4 Education and research - Component 2 From research to business - Investment 1.1 Notice Prin 2022 - DD N. 104 del 2/2/2022

This project will contribute to the development of two major research areas: Optimal Transport (OT) theory and McKean-Vlasov (MKV) equations and control problems. Our project enters these fields of research from different perspectives and aims at building new bridges between these two areas.

A) We will introduce the Entropy Martingale Optimal Transport (EMOT) problem that differs from the classical optimal transport in two ways. Firstly, a martingale constraint is imposed on the admissible solutions in order to address a number of relevant applications in Mathematical Finance, where martingales play a crucial role. Secondly, instead of requiring an exact match of the two prescribed marginal distributions, a penalization functional of entropic form discriminates those solutions for which the marginals are "close to the target". Our focus will be on:
(i) the EMOT - robust pricing hedging-duality;
(ii) stability properties;
(iii) the entropy counterpart of the Weak Martingale OT and of the Causal Martingale OT.

B1) We first stress that EMOT and MKV optimal control problems are naturally related. Indeed, the former can be interpreted as a stochastic control optimization over a class of martingale models (for example Brownian diffusions with controlled volatility) with the objective of minimizing a value function that depends on the terminal distribution of the controlled process. We thus aim at studying the novel class of minimum-entropy stochastic optimal control problems related to the EMOT by various methodologies, as dynamic programming and BSDEs methods.

B2) A further link between EMOT and MKV equations is given by the problem of constructing martingales with given fixed-time marginals through inversion of Markovian projection theorems: this problem leads to conditional MKV equations. This approach is widely used in the financial industry for calibrating local-stochastic volatility models. Results on this class of equations are not fully comprehensive especially due to their conditional aspect which makes the available techniques largely inapplicable. MKV optimization and EMOT problems can both benefit from a coordinated analysis, as proposed in this project. The two approaches could, in fact, provide reciprocal sketches of a unitary painting, towards a deeper understanding of the complexity of mathematical methods for financial modeling.

In addition, we will present other remarkable applications, ranging from Machine Learning to renewable Energy Markets.