Optimal control of path-dependent McKean-Vlasov SDEs in infinite dimension

Titel

Optimal control of path-dependent McKean-Vlasov SDEs in infinite dimension

Föreläsare

Professor Fausto Gozzi, Luiss University, Rom, Italien

Abstract

In this talk, we introduce first a family of controlled path-dependent McKean-Vlasov equations valued in Hilbert spaces motivated by non Markovian mean-field models driven by stochastic PDEs. The theory for such problems is still largely missing, hence we will give an outline of the basic results which we recently obtained trying to focus on the main ideas and the future goals in applications.

The basic results provided in the paper that will be recalled here are:

• the well-posedness of the state equation;
• the Dynamic Programming Principle (DPP) in a general framework;
• the crucial law invariance property of the value function $V$, this means that $V$ can be viewed as a function on the Wasserstein space of probability measures on the set of continuous functions valued in Hilbert space;
• a notion of pathwise measure derivative, which extends the Wasserstein derivative due to P.L.Lions (2006), and prove a related functional Ito formula in the spirit of Dupire (2009) and Wu-Zhang (2020);
• the proof that, using the DPP, the value function $V$ is a viscosity solution
  (with an appropriate notion) of the so-called Master Bellman equation in a suitable form.

(Based on a recent submitted paper with A. Cosso, I. Kharroubi, H. Pham, M. Rosestolato)

Arrangör

Webbinariet arrangeras av forskningsområdet Deterministic and Stochastic Modelling inom Linnaeus University Centre for Data Intensive Sciences and Applications (DISA).