# On the Well-posedness of Hamilton-Jacobi-Bellman Equations of the Equilibrium Type

arXiv: https://arxiv.org/abs/2307.01986 · HTML: https://mlsanity.com/paper/2307.01986/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jul 2023.

## Built on by (most cited first, 1 of 1)

- [Beyond the Bellman Recursion: A Pontryagin-Guided Framework for Non-Exponential Discounting](https://mlsanity.com/paper/2605.20996/) (2026)

## Abstract (arXiv)

This paper studies the well-posedness of a class of nonlocal parabolic partial differential equations (PDEs), or equivalently equilibrium Hamilton-Jacobi-Bellman equations, which has a strong tie with the characterization of the equilibrium strategies and the associated value functions for time-inconsistent stochastic control problems. Specifically, we consider nonlocality in both time and space, which allows for modelling of the stochastic control problems with initial-time-and-state dependent objective functionals. We leverage the method of continuity to show the global well-posedness within our proposed Banach space with our established Schauder prior estimate for the linearized nonlocal PDE. Then, we adopt a linearization method and Banach's fixed point arguments to show the local well-posedness of the nonlocal fully nonlinear case, while the global well-posedness is attainable provided that a very sharp a-priori estimate is available. On top of the well-posedness results, we also provide a probabilistic representation of the solutions to the nonlocal fully nonlinear PDEs and an estimate on the difference between the value functions of sophisticated and na"ive controllers. Finally, we give a financial example of time inconsistency that is proven to be globally solvable.