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We describe an abstract control-theoretic framework in which the validity of the dynamic programming principle can be established in continuous time by a verification of a small number of structural properties. As an application we treat several cases of interest, most notably the lower-hedging and utility-maximization problems of financial mathematics, both of which are naturally posed over “sets of martingale measures.”

Dynamic Programming for Controlled Markov Families: Abstractly and over Martingale Measures