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

arXiv: https://arxiv.org/abs/2002.12572 · HTML: https://mlsanity.com/paper/2002.12572/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2020.

## Built on by (most cited first, 2 of 2)

- [Inference of Utilities and Time Preference in Sequential Decision-Making](https://mlsanity.com/paper/2405.15975/) (2024)
- [Mean-Field Generalisation Bounds for Learning Controls in Stochastic Environments](https://mlsanity.com/paper/2508.16001/) (2025)

## Abstract (arXiv)

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 is cast within the framework of a controlled non-Markovian forward stochastic differential equation, and a general objective functional setting. We adopt a game-theoretic approach to study such problems, meaning that we seek for sub-game perfect Nash equilibrium points. As a first novelty of this work, we introduce and motivate a refinement of the definition of equilibrium that allows us to establish a direct and rigorous proof of an extended dynamic programming principle, in the same spirit as in the classical theory. This in turn allows us to introduce a system consisting of an infinite family of backward stochastic differential equations analogous to the classical HJB equation. We prove that this system is fundamental, in the sense that its well-posedness is both necessary and sufficient to characterise the value function and equilibria. As a final step we provide an existence and uniqueness result. Some examples and extensions of our results are also presented.