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A decision maker fears that data are generated by a statistical perturbation of an approximating model that is either a controlled diffusion or a controlled measure over continuous functions of time. A perturbation is constrained in terms of its relative entropy. Several different two-player zero-sum games that yield robust decision rules are related to one another, to the max–min expected utility theory of Gilboa and Schmeidler [Maxmin expected utility with non-unique prior, J. Math. Econ. 18 (1989) 141–153], and to the recursive risk-sensitivity criterion described in discrete time by Hansen and Sargent [Discounted linear exponential quadratic Gaussian control, IEEE Trans. Automat. Control 40 (5) (1995) 968–971]. To represent perturbed models, we use martingales on the probability space associated with the approximating model. Alternative sequential and nonsequential versions of robust control theory imply identical …