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

We develop a theory for continuous-time non-Markovian stochastic control problems which are inherently time-inconsistent. Their distinguishing feature is that the classical Bellman optimality principle no longer holds. Our formulation cast within framework of controlled forward differential equation, and general objective functional setting. adopt game-theoretic approach to study such problems, meaning we seek subgame perfect Nash equilibrium points. As first novelty this work, introduce motivate refinement definition allows us establish direct rigorous proof an extended dynamic programming principle, in same spirit as theory. This turn system consisting infinite family backward equations analogous HJB equation. prove fundamental, sense its well-posedness both necessary sufficient characterise value function equilibria. final step, provide existence uniqueness result. Some examples extensions our results also presented.