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We develop a theory for continuous-time non-Markovian stochastic controlproblems which are inherently time-inconsistent. Their distinguishing featureis that the classical Bellman optimality principle no longer holds. Ourformulation is cast within the framework of a controlled non-Markovian forwardstochastic differential equation, and a general objective functional setting.We adopt a game-theoretic approach to study such problems, meaning that we seekfor sub-game perfect Nash equilibrium points. As a first novelty of this work,we introduce and motivate a refinement of the definition of equilibrium thatallows us to establish a direct and rigorous proof of an extended dynamicprogramming principle, in the same spirit as in the classical theory. This inturn allows us to introduce a system consisting of an infinite family ofbackward stochastic differential equations analogous to the classical HJBequation. We prove that this system is fundamental, in the sense that itswell-posedness is both necessary and sufficient to characterise the valuefunction and equilibria. As a final step we provide an existence and uniquenessresult. Some examples and extensions of our results are also presented.

Me, myself and I: a general theory of non-Markovian time-inconsistent stochastic control for sophisticated agents