Abstract
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. Achieved Results The project "Entropy Martingale Optimal Transport and McKean-Vlasov equations" achieved a substantial and coherent advancement of its scientific objectives. It strengthened the connection between Optimal Transport (OT) theory, McKean-Vlasov (MKV) equations, stochastic control and mathematical finance, and produced new tools for stochastic modelling under uncertainty. The work was carried out by the University of Milan unit, coordinated by Marco Frittelli, and the University of Bologna unit, coordinated by Andrea Pascucci, with complementary expertise in convex duality, risk measures, robust pricing, stochastic analysis, partial differential equations and control theory. A central result was the development and consolidation of the Entropy Martingale Optimal Transport (EMOT) framework. The project extended classical optimal transport by combining martingale constraints, which are essential in financial modelling, with entropic penalization, which makes it possible to handle situations in which the available marginal distributions are uncertain or only approximately known. This led to a new class of stochastic optimization problems and to nonlinear pricing-hedging dualities, where the usual linear expectation is replaced by nonlinear utility-based functionals. These results provide a flexible mathematical framework for robust pricing, hedging and model calibration, and connect optimal transport with utility maximization, convex duality and risk measurement. The project also produced significant progress on McKean-Vlasov systems, mean field control and related stochastic control problems. Results were obtained on singular limits of backward stochastic differential equations, the control of two-scale stochastic systems with jumps in infinite-dimensional spaces, mean field optimal control and optimal stopping problems, and convergence properties of Wasserstein-type distances. These contributions improve the understanding of high-dimensional and interacting stochastic systems, which are central in probability theory and increasingly relevant for applications in finance, energy markets and data-driven modelling. Another important line of results concerns path-dependent and non-local partial differential equations, in particular Hamilton-Jacobi-Bellman equations arising from stochastic control problems. The project addressed settings with time-measurable Hamiltonians and path-dependent dynamics, extending classical PDE techniques to more general and irregular stochastic frameworks. In parallel, the research advanced stochastic calculus tools through new work on weak Dirichlet processes and generalized martingale problems, with relevance for models involving jumps and irregular trajectories. A further achievement was the development of the emerging area of collective finance. The project deepened the analysis of cooperation among agents, collective arbitrage, collective and systemic risk measures, and pricing-hedging duality under interaction between market participants. This refinement did not change the original objectives, but broadened and strengthened them within the same scientific fields. The results connect rigorous mathematical finance with new computational approaches, including the use of neural networks for risk sharing and risk allocation problems, and open perspectives for applications to energy markets and machine learning. The scientific output of the project includes a substantial body of peer-reviewed publications and recent preprints produced in the period 2024-2026. The publications cover entropy martingale optimal transport, convex duality, robust financial modelling, systemic and collective risk measures, stochastic control, mean field systems, optimal stopping, path-dependent PDEs, Wasserstein analysis and generalized martingale problems. The presence of recent preprints and ongoing work confirms that the project has generated a strong pipeline of future publications in international journals and has opened further research directions in mean field systems, stochastic control, Wasserstein-type analysis and cooperative mathematical finance. Dissemination was an integral part of the results achieved. Members of the research units presented their work at international conferences and workshops, including events in mathematical finance, stochastic control and probability theory, and gave invited seminars at national and international institutions. The project team also contributed directly to the organization of scientific events and advanced training activities. In particular, the two international winter schools on "Optimal transport: from robust pricing to model calibration" (January 2025) and "Mean-field systems in finance, neuroscience and AI" (January 2026) involved more than 100 PhD students, postdoctoral researchers, researchers and professors overall, with grants supporting young participants in the first school. These initiatives strengthened the international visibility of the project and supported the training of early-career researchers. The activities were implemented without significant deviations from the approved plan and in compliance with the principles of Open Access, DNSH, gender equality, generational balance and equal opportunities. Since the research was mainly theoretical and computational, it did not involve activities with negative environmental impact. Results were disseminated through international journals, preprint repositories, conferences, seminars and collaborative events, promoting accessibility, transparency and scientific exchange. Overall, the project achieved its main objectives, advanced the state of the art in optimal transport, McKean-Vlasov equations, stochastic control and mathematical finance, and provided new theoretical and methodological foundations for future research.
Project details
Unibo Team Leader: Andrea Pascucci
Unibo involved Department/s:
Dipartimento di Matematica
Coordinator:
Università degli Studi di MILANO(Italy)
Total Unibo Contribution: Euro (EUR) 93.298,00
Project Duration in months: 24
Start Date:
28/09/2023
End Date:
28/02/2026
