Recorded 30 April 2025. Jean-Claude Zambrini of the University of Lisbon presents "Integrability of Hamilton-Jacobi-Bellman equation" at IPAM's Dynamics of Density Operators Workshop. Abstract: Hamilton-Jacobi-Bellman equation became central in stochastic Optimal Control, Mass transportation, Mean Field games, Machine Learning... We present a stochastic version of Jacobi's integration Theorem for Hamilton-Jacobi equation, fundamental in classical (and quantum) mechanics.Our Theorem predicts the existence of martingales of the diffusions optimal for a regularized Action functional and Lagrangian (Running cost ).When the diffusion parameter of random paths tends to zero, our Theorem and its hypotheses reduce to the classical Jacobi's Theorem. (J.C. Zambrini, with Qiao Huang) Learn more online at: https://www.ipam.ucla.edu/programs/workshops/workshop-ii-dynamics-of-density-operators/