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This lecture presents Jean-Claude Zambrini of the University of Lisbon discussing the "Integrability of Hamilton-Jacobi-Bellman equation" at IPAM's Dynamics of Density Operators Workshop at UCLA. Explore how the Hamilton-Jacobi-Bellman equation has become central in stochastic Optimal Control, Mass transportation, Mean Field games, and Machine Learning. Discover a stochastic version of Jacobi's integration Theorem for Hamilton-Jacobi equation, which is fundamental in classical and quantum mechanics. Learn about the existence of martingales of the diffusions optimal for a regularized Action functional and Lagrangian (Running cost), and understand how when the diffusion parameter of random paths approaches zero, the presented theorem and its hypotheses reduce to the classical Jacobi's Theorem. This 45-minute presentation was recorded on April 30, 2025, as part of IPAM's workshop series at UCLA.
