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We develop a non-anticipating calculus of variations for functionals on aspace of laws of continuous semi-martingales, which extends the classical one.We extend Hamilton's least action principle and Noether's theorem to thisgeneralized stochastic framework. As an application we obtain, under mildconditions, a stochastic Euler-Lagrange condition and invariants for thecritical points of recent problems in stochastic control, namely for thesemi-martingale optimal transportation problems.

Weak calculus of variations for functionals of laws of semi-martingales