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.
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