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RoFiRM Report Summary

Project ID: 321111
Funded under: FP7-IDEAS-ERC
Country: France

Final Report Summary - ROFIRM (Mathematical Methods for Robust Financial Risk Management)

This project is focused on the newly introduced problem of martingale optimal transport and risk management under mean field interaction. The research conducted is strongly motivated by relevant problems in financial risk management, and makes beautiful connections to optimal transport and stochastic analysis.

An important and significant progress was achieved in all the research directions of the project. Together with the excellent team of collaborators recruited within the project, we have obtained remarkable results on the one-dimensional martingale optimal transport, and solved some questions in the corresponding connection to the Skorohod embedding problem, which remained open during the last three decades. We have also made a significant progress in the multi-dimensional martingale transport problem, which raises very challenging technical difficulties.

A general solution method for a class of problems in continuous time contract theory was introduced by means of a remarkable connection with second order backward stochastic differential equations. Contract theory plays a central role for the modeling of moral hazard in economics and lies at the heart of the analysis of economic interactions. Our general solution method opens the door for many interesting applications to modern problems related to the risk management of decentralized online technologies (Fintech), which require to solve appropriate multi-agent optimization problems.

Finally, an important progress has also been achieved in the numerical approximation side of the project. We have introduced a new class of Monte Carlo approximation for nonlinear PDEs based on an original representation in terms of marked branching diffusions. This is the very first genuinely high-dimensional numerical method for nonlinear PDEs. Remarkably, the method works beyond the context of elliptic and parabolic PDEs, and cover a large class of nonlinear PDEs including hyperbolic ones.

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