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arxiv: 2307.01986 · v2 · submitted 2023-07-05 · 🧮 math.AP · math.OC· q-fin.MF

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

Qian Lei , Chi Seng Pun This is my paper

Pith reviewed 2026-05-24 08:13 UTC · model grok-4.3

classification 🧮 math.AP math.OCq-fin.MF
keywords nonlocal parabolic PDEsequilibrium Hamilton-Jacobi-Bellman equationstime-inconsistent stochastic controlwell-posednessSchauder estimatesmethod of continuityBanach fixed point theorem
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The pith

Nonlocal equilibrium HJB equations admit global well-posedness in a Banach space when a Schauder estimate holds for the linear case.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves well-posedness results for nonlocal parabolic PDEs that serve as equilibrium Hamilton-Jacobi-Bellman equations arising from time-inconsistent stochastic control problems with initial-time-and-state dependent objectives. It establishes global existence for the linear version by applying the method of continuity inside a chosen Banach space once a Schauder prior estimate is available for the linearized equation. Local well-posedness for the fully nonlinear case follows from linearization plus Banach fixed-point arguments, with global nonlinear results available only if a sharp a-priori bound is supplied. These results also yield a probabilistic representation of solutions and a quantitative comparison between sophisticated and naive value functions, illustrated by a solvable financial example.

Core claim

The authors show that nonlocal parabolic PDEs modeling equilibrium strategies in time-inconsistent control problems are well-posed. Specifically, global well-posedness holds for the linear case in a proposed Banach space using the method of continuity and an established Schauder estimate for the linearized equation. For the fully nonlinear case, local well-posedness follows from linearization and Banach fixed point theorem, while global well-posedness requires a sharp a priori estimate. They also provide a probabilistic representation of solutions and an estimate on the difference between sophisticated and naive value functions, with an example of global solvability in finance.

What carries the argument

Method of continuity for the linearized nonlocal PDE with Schauder prior estimate in a Banach space, combined with linearization and Banach fixed-point theorem for the nonlinear problem.

If this is right

  • Global solutions exist for linear nonlocal equilibrium HJB equations in the Banach space.
  • Local solutions exist for the nonlocal fully nonlinear equations via linearization and fixed-point arguments.
  • Solutions to the nonlinear PDEs admit a probabilistic representation.
  • The difference between value functions of sophisticated and naive controllers can be estimated.
  • A concrete financial model of time inconsistency is globally solvable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same continuity argument could be tested on other nonlocal operators that appear in time-inconsistent problems with different dependence structures.
  • If numerical schemes can be shown to inherit the Schauder estimate, they would inherit the global existence result for the linear case.
  • The gap between local and global nonlinear results suggests that sharpening the a-priori bound is the main remaining analytic task.

Load-bearing premise

The Schauder prior estimate for the linearized nonlocal PDE holds inside the chosen Banach space.

What would settle it

A specific instance of the linearized nonlocal PDE in the Banach space where the Schauder estimate fails to hold, or an explicit time-inconsistent control problem whose associated equilibrium equation has no solution.

read the original abstract

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\"{i}ve controllers. Finally, we give a financial example of time inconsistency that is proven to be globally solvable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper studies well-posedness of nonlocal parabolic PDEs (equilibrium HJB equations) with nonlocality in both time and space, arising from time-inconsistent stochastic control. It claims global well-posedness for the linear case via the method of continuity in a custom Banach space, relying on an established Schauder a-priori estimate for the linearized nonlocal operator; local well-posedness for the fully nonlinear case via linearization and Banach fixed-point arguments, with global well-posedness under an additional sharp a-priori bound; plus a probabilistic representation of solutions, an estimate comparing sophisticated and naïve value functions, and a solvable financial example.

Significance. If the Schauder estimate and continuity argument close rigorously, the results would supply a functional-analytic framework for a broad class of time-and-space nonlocal equilibrium HJB equations, extending existing theory for time-inconsistent problems and enabling analysis of equilibrium strategies with initial-time-and-state dependent objectives.

major comments (2)
  1. [Abstract / linear case] Abstract and the linear well-posedness section: the global well-posedness claim for the linear problem rests on the method of continuity in a Banach space whose norm is constructed from the Schauder a-priori estimate; the manuscript must verify that this estimate remains uniform in the continuity parameter and absorbs the time-nonlocal kernel without loss of regularity or parameter dependence, otherwise the a-priori bound fails to close the argument.
  2. [Nonlinear case] Nonlinear well-posedness section: the local well-posedness via linearization and Banach fixed point inherits the same requirement on the linear Schauder estimate; any gap in controlling the time-nonlocal term for the linearized operator propagates directly to the contraction mapping.
minor comments (1)
  1. Notation for the time-nonlocality kernel and the precise definition of the Banach space norm should be stated explicitly at the first appearance to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to confirm uniformity of the Schauder estimates. The concerns are addressed by adding explicit statements on parameter independence; the core arguments remain unchanged.

read point-by-point responses

  1. Referee: [Abstract / linear case] Abstract and the linear well-posedness section: the global well-posedness claim for the linear problem rests on the method of continuity in a Banach space whose norm is constructed from the Schauder a-priori estimate; the manuscript must verify that this estimate remains uniform in the continuity parameter and absorbs the time-nonlocal kernel without loss of regularity or parameter dependence, otherwise the a-priori bound fails to close the argument.

    Authors: We agree that uniformity must be stated explicitly. The Schauder a-priori estimate (Theorem 3.1) is obtained via a perturbation argument around the local operator; the constants depend only on the ellipticity constants, the Hölder norms of the coefficients, and the integrability properties of the time-nonlocal kernel, all of which are independent of the continuity parameter λ. The Banach-space norm is deliberately constructed to absorb the time-nonlocal term uniformly, so the a-priori bound closes for every λ ∈ [0,1]. We will insert a short paragraph after the statement of Theorem 3.1 and a remark in the continuity-method proof (Section 4) that records this independence. revision: yes

  2. Referee: [Nonlinear case] Nonlinear well-posedness section: the local well-posedness via linearization and Banach fixed point inherits the same requirement on the linear Schauder estimate; any gap in controlling the time-nonlocal term for the linearized operator propagates directly to the contraction mapping.

    Authors: The local well-posedness argument (Section 5) linearizes around a given function and applies the Banach fixed-point theorem in a small ball of the same Banach space used for the linear theory. Because the linearized operator satisfies the same uniform Schauder estimate (with constants independent of the linearization point inside the ball), the contraction constant is strictly less than one for sufficiently small time intervals. The same explicit uniformity statement added for the linear case therefore closes the nonlinear argument as well. We will add a sentence in the proof of Theorem 5.1 referencing the uniform bound. revision: yes

Circularity Check

0 steps flagged

No circularity: standard PDE existence proofs via continuity and fixed-point methods.

full rationale

The derivation relies on the method of continuity applied to a linearized nonlocal PDE after establishing a Schauder a-priori estimate, followed by linearization and Banach fixed-point for the nonlinear case. These are self-contained functional-analytic arguments with no fitted parameters, no self-definitional reductions, and no load-bearing self-citations that collapse the central claim to prior unverified work by the same authors. The Schauder estimate is presented as proven within the paper to close the continuity path, not assumed or renamed from the target result. This matches the default case of non-circular mathematical proofs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard results from functional analysis (Banach fixed-point theorem, method of continuity) and on the authors' derivation of a Schauder estimate; no free parameters or new postulated entities are introduced.

axioms (2)
  • standard math Banach fixed-point theorem applies to the linearized map in the chosen function space
    Invoked for local well-posedness of the nonlinear problem
  • standard math Method of continuity yields global well-posedness once a uniform a-priori estimate is available
    Central to the linear global result

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Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages · cited by 1 Pith paper

  1. [1]

    Basak and G

    S. Basak and G. Chabakauri , Dynamic mean-variance asset allocation , Review of Financial Studies, 23 (2010), pp. 2970–3016, https://doi.org/10.1093/rfs/hhq028

    work page doi:10.1093/rfs/hhq028 2010

  2. [2]

    Bj ¨ork, M

    T. Bj ¨ork, M. Khapko, and A. Murgoci , On time-inconsistent stochastic control in con- tinuous time , Finance and Stochastics, 21 (2017), pp. 331–360, https://doi.org/10.1007/ s00780-017-0327-5

    work page 2017

  3. [3]

    Bj ¨ork and A

    T. Bj ¨ork and A. Murgoci , A theory of Markovian time-inconsistent stochastic control in discrete time, Finance and Stochastics, 18 (2014), pp. 545–592, https://doi.org/10.1007/ s00780-014-0234-y

    work page 2014

  4. [4]

    Bj ¨ork, A

    T. Bj ¨ork, A. Murgoci, and X. Y. Zhou , Mean-variance portfolio optimization with state- dependent risk aversion , Mathematical Finance, 24 (2014), pp. 1–24, https://doi.org/10. 1111/j.1467-9965.2011.00515.x

    work page arXiv 2014

  5. [5]

    Cheridito, H

    P. Cheridito, H. M. Soner, N. Touzi, and N. Victoir , Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs , Communications on Pure and Applied Mathematics, 60 (2007), pp. 1081–1110, https://doi.org/10.1002/cpa.20168

    work page doi:10.1002/cpa.20168 2007

  6. [6]

    S. D. ˙Eˇidel’man, Parabolic Systems, North-Holland and Pub. Co.Wolters-Noordhoff, Amster- dam, Groningen, 1969

    work page 1969

  7. [7]

    Friedman , Partial Differential Equations of Parabolic Type , Prentice-Hall, Englewood Cliffs, N.J., 1st ed., 1964

    A. Friedman , Partial Differential Equations of Parabolic Type , Prentice-Hall, Englewood Cliffs, N.J., 1st ed., 1964

    work page 1964

  8. [8]

    Y. Hamaguchi, Extended backward stochastic Volterra integral equations and their applications to time-inconsistent stochastic recursive control problems, Mathematical Control & Related Fields, 11 (2021), pp. 197–242, https://doi.org/10.3934/mcrf.2020043

    work page doi:10.3934/mcrf.2020043 2021

  9. [9]

    Hamaguchi , Small-time solvability of a flow of forward-backward stochastic differential equations, Applied Mathematics & Optimization, 84 (2021), pp

    Y. Hamaguchi , Small-time solvability of a flow of forward-backward stochastic differential equations, Applied Mathematics & Optimization, 84 (2021), pp. 567–588

    work page 2021

  10. [10]

    J. Han, A. Jentzen, and W. E , Solving high-dimensional partial differential equations using deep learning, Proceedings of the National Academy of Sciences, 115 (2018), pp. 8505–8510, https://doi.org/10.1073/pnas.1718942115

    work page doi:10.1073/pnas.1718942115 2018

  11. [11]

    X. D. He and Z. L. Jiang , On the equilibrium strategies for time-inconsistent problems in continuous time, SIAM Journal on Control and Optimization, 59 (2021), pp. 3860–3886, https://doi.org/10.1137/20m1382106

    work page doi:10.1137/20m1382106 2021

  12. [12]

    X. D. He and X. Y. Zhou , Who are I: Time inconsistency and intrapersonal conflict and reconciliation, in Stochastic Analysis, Filtering, and Stochastic Optimization, G. Yin and T. Zariphopoulou, eds., Springer International Publishing, 2022. A Commemorative Vol- ume to Honor Mark H. A. Davis’s Contributions

    work page 2022

  13. [13]

    Hern ´andez and D

    C. Hern ´andez and D. Possama ¨ı, Me, myself and i: A general theory of non-Markovian time-inconsistent stochastic control for sophisticated agents , arXiv: 2002.12572, (2020), https://arxiv.org/abs/2002.12572

    work page arXiv 2002

  14. [14]

    N. E. Karoui, S. Peng, and M.-C. Quenez , Backward stochastic differential equations in finance, Mathematical Finance, 7 (1997), pp. 1–71, https://doi.org/10.1111/1467-9965. 00022

    work page doi:10.1111/1467-9965 1997

  15. [15]

    T. Kong, W. Zhao, and T. Zhou, Probabilistic high order numerical schemes for fully nonlin- ear parabolic PDEs, Communications in Computational Physics, 18 (2015), p. 1482–1503, https://doi.org/10.4208/cicp.240515.280815a

    work page doi:10.4208/cicp.240515.280815a 2015

  16. [16]

    N. V. Krylov , Nonlinear Elliptic and Parabolic Equations of the Second Order , vol. 7 of Mathematics and its Applications, Springer Netherlands, 1 ed., 1987

    work page 1987

  17. [17]

    O. A. Lady ˇzenskaja, V. A. Solonnikov, and N. N. Ural’ceva , Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, 1st ed., 1968

    work page 1968

  18. [18]

    Laibson, Golden eggs and hyperbolic discounting, The Quarterly Journal of Economics, 112 (1997), pp

    D. Laibson, Golden eggs and hyperbolic discounting, The Quarterly Journal of Economics, 112 (1997), pp. 443–478, https://doi.org/10.1162/003355397555253

    work page doi:10.1162/003355397555253 1997

  19. [19]

    Lei and C

    Q. Lei and C. S. Pun , Nonlocality, nonlinearity, and time inconsistency in stochastic dif- ferential games , arXiv: 2112.14409, (2021), https://doi.org/10.48550/arXiv.2112.14409, https://arxiv.org/abs/2112.14409

    work page doi:10.48550/arxiv.2112.14409 2021

  20. [20]

    Lei and C

    Q. Lei and C. S. Pun , Nonlocal fully nonlinear parabolic differential equations arising in time-inconsistent problems, Journal of Differential Equations, 358 (2023), pp. 339–385, https://doi.org/10.1016/j.jde.2023.02.025

    work page doi:10.1016/j.jde.2023.02.025 2023

  21. [21]

    G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 1 ed., Nov. 1996, https://doi.org/10.1142/3302

    work page doi:10.1142/3302 1996

  22. [22]

    Lindensj ¨o, A regular equilibrium solves the extended HJB system , Operations Research Letters, 47 (2019), pp

    K. Lindensj ¨o, A regular equilibrium solves the extended HJB system , Operations Research Letters, 47 (2019), pp. 427–432, https://doi.org/10.1016/j.orl.2019.07.011

    work page doi:10.1016/j.orl.2019.07.011 2019

  23. [23]

    Lunardi , Maximal space regularity in nonhoomogeneous initial boundary value parabolic problem, Numerical Functional Analysis and Optimization, 10 (1989), pp

    A. Lunardi , Maximal space regularity in nonhoomogeneous initial boundary value parabolic problem, Numerical Functional Analysis and Optimization, 10 (1989), pp. 323–349, https: 66 Q. LEI AND C. S. PUN //doi.org/10.1080/01630568908816306

    work page doi:10.1080/01630568908816306 1989

  24. [24]

    Lunardi , Analytic Semigroups and Optimal Regularity in Parabolic Problems , Springer Basel, 1st ed., 1995

    A. Lunardi , Analytic Semigroups and Optimal Regularity in Parabolic Problems , Springer Basel, 1st ed., 1995

    work page 1995

  25. [25]

    Ma and J

    J. Ma and J. Yong , Forward-Backward Stochastic Differential Equations and their Appli- cations, Lecture Notes in Mathematics, Springer, 1st ed., 1999, https://doi.org/10.1007/ 978-3-540-48831-6

    work page 1999

  26. [26]

    Ma and J

    J. Ma and J. Zhang , Representation theorems for backward stochastic differential equations , The Annals of Applied Probability, 12 (2002), pp. 1390–1418, https://doi.org/10.1214/ aoap/1037125868

    work page arXiv 2002

  27. [27]

    Portfolio Selection

    H. Markowitz, Portfolio selection, The Journal of Finance, 7 (1952), pp. 77–91, https://doi. org/10.1111/j.1540-6261.1952.tb01525.x

    work page doi:10.1111/j.1540-6261.1952.tb01525.x 1952

  28. [28]

    J. L. Pedersen and G. Peskir , Optimal mean-variance portfolio selection, Mathematics and Financial Economics, 11 (2016), pp. 137–160, https://doi.org/10.1007/s11579-016-0174-8

    work page doi:10.1007/s11579-016-0174-8 2016

  29. [29]

    Peng, A nonlinear Feynman–Kac formula and applications , in Proceedings of Symposium of System Sciences and Control Theory, World Scientific, 1992, pp

    S. Peng, A nonlinear Feynman–Kac formula and applications , in Proceedings of Symposium of System Sciences and Control Theory, World Scientific, 1992, pp. 173–184

    work page 1992

  30. [30]

    R. A. Pollak , Consistent planning , The Review of Economic Studies, 35 (1968), p. 201, https://doi.org/10.2307/2296548

    work page doi:10.2307/2296548 1968

  31. [31]

    G. D. Prato and L. Tubaro , Fully nonlinear stochastic partial differential equations , SIAM Journal on Mathematical Analysis, 27 (1996), pp. 40–55, https://doi.org/10.1137/ s0036141093256769

    work page 1996

  32. [32]

    Sinestrari , On the abstract Cauchy problem of parabolic type in spaces of continuous functions, Journal of Mathematical Analysis and Applications, 107 (1985), pp

    E. Sinestrari , On the abstract Cauchy problem of parabolic type in spaces of continuous functions, Journal of Mathematical Analysis and Applications, 107 (1985), pp. 16–66, https://doi.org/10.1016/0022-247x(85)90353-1

    work page doi:10.1016/0022-247x(85)90353-1 1985

  33. [33]

    H. M. Soner, N. Touzi, and J. Zhang , Wellposedness of second order backward SDEs , Probability Theory and Related Fields, 153 (2011), pp. 149–190, https://doi.org/10.1007/ s00440-011-0342-y

    work page 2011

  34. [34]

    R. H. Strotz , Myopia and inconsistency in dynamic utility maximization , The Review of Economic Studies, 23 (1955), pp. 165–180, https://doi.org/10.2307/2295722

    work page doi:10.2307/2295722 1955

  35. [35]

    Tversky and D

    A. Tversky and D. Kahneman , Advances in prospect theory: Cumulative representation of uncertainty, Journal of Risk and Uncertainty, 5 (1992), pp. 297–323, https://doi.org/10. 1007/bf00122574

    work page 1992

  36. [36]

    Wang, Extended backward stochastic Volterra integral equations, quasilinear parabolic equa- tions, and Feynman–Kac formula , Stochastics and Dynamics, 21 (2020), p

    H. Wang, Extended backward stochastic Volterra integral equations, quasilinear parabolic equa- tions, and Feynman–Kac formula , Stochastics and Dynamics, 21 (2020), p. 2150004, https://doi.org/10.1142/s0219493721500040

    work page doi:10.1142/s0219493721500040 2020

  37. [37]

    Wang and J

    T. Wang and J. Yong , Backward stochastic Volterra integral equations—representation of adapted solutions, Stochastic Processes and their Applications, 129 (2019), pp. 4926–4964, https://doi.org/10.1016/j.spa.2018.12.016

    work page doi:10.1016/j.spa.2018.12.016 2019

  38. [38]

    Q. Wei, J. Yong, and Z. Yu, Time-inconsistent recursive stochastic optimal control problems, SIAM Journal on Control and Optimization, 55 (2017), pp. 4156–4201, https://doi.org/ 10.1137/16m1079415

    work page doi:10.1137/16m1079415 2017

  39. [39]

    Yan and J

    W. Yan and J. Yong, Time-inconsistent optimal control problems and related issues, in Mod- eling, Stochastic Control, Optimization, and Applications, G. Yin and Q. Zhang, eds., vol. 164, Springer International Publishing, 2019, pp. 533–569, https://doi.org/10.1007/ 978-3-030-25498-8 22

    work page 2019

  40. [40]

    Yong, Time-inconsistent optimal control problems and the equilibrium HJB equation, Math- ematical Control & Related Fields, 2 (2012), pp

    J. Yong, Time-inconsistent optimal control problems and the equilibrium HJB equation, Math- ematical Control & Related Fields, 2 (2012), pp. 271–329, https://doi.org/10.3934/mcrf. 2012.2.271

    work page doi:10.3934/mcrf 2012

  41. [41]

    Yong and X

    J. Yong and X. Y. Zhou , Stochastic Controls : Hamiltonian Systems and HJB Equations , vol. 43 of Stochastic Modelling and Applied Probability, Springer New York, New York, NY, 1st ed., 1999, https://doi.org/10.1007/978-1-4612-1466-3

    work page doi:10.1007/978-1-4612-1466-3 1999

  42. [42]

    X. Y. Zhou and D. Li , Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), pp. 19–33, https://doi.org/ 10.1007/s002450010003

    work page doi:10.1007/s002450010003 2000