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International audienceWe study several aspects of covariance control problems over martingale processes in $\RR^d$ with constraints on the terminal distribution, arising from the theory of repeated games with incomplete information. We show that these control problems are the limits of discrete-time stochastic optimization problems called problems of maximal variation of martingales. Optimal solutions are then characterized using convex duality techniques and the dual problem is shown to be an unconstrained stochastic control problem characterized by an HJB equation. We deduce from this relationship that solutions of the control problem are the images by the gradient of the solution of the HJB equation of the solutions of the dual stochastic control problem using tools from optimal transport theory

Covariance Control Problems over Martingales with Fixed Terminal Distribution Arising from Game Theory