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Path-Dependent Ergodic Optimal Control and Backward Stochastic Differential Equations

T0 review · 2 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read The optimal ergodic cost for path-dependent infinite-horizon control problems is given by the asymptotic behavior of a deterministic function rather than a single constant.

desk verdict The paper sets up path-dependent infinite-horizon BSDEs for ergodic control and replaces the usual constant cost with an asymptotic limit of a deterministic function, but the extended dissipativity condition may not handle accumulated memory on unbounded domains. read the letter →

arxiv 2606.06757 v1 pith:DQ32MAZ5 submitted 2026-06-04 math.PR math.OC

Xuyang Lin , Mathieu Lise , Nizar Touzi This is my paper · ORCID
classification math.PRmath.OC
keywords ergodicoptimalcontrolbackwardstochasticdifferentialequationspath-dependentinfinitehorizondissipativityconditionverificationtheoremstability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper examines ergodic optimal control where both the running cost and the state dynamics depend on the entire history of the path, formulated through infinite-horizon backward stochastic differential equations. On an unbounded domain the state process obeys an extended dissipativity condition that replaces the usual uniform ergodicity assumptions of the Markov setting. Under these conditions the value of the control problem is recovered from the long-run growth rate of a deterministic function that solves an associated deterministic equation. The authors establish existence and uniqueness for the backward equation, a verification theorem that recovers optimal controls from its solution, and stability of the cost with respect to perturbations of the dynamics. These results recover and extend the classical Markovian ergodic theory to the non-Markovian path-dependent case.

What carries the argument

Infinite-horizon backward stochastic differential equation whose solution's asymptotic growth rate supplies the ergodic cost.

What would settle it

An explicit path-dependent example in which the optimal ergodic cost collapses to a single real number independent of the initial path segment.

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Extended reading notes

Core claim

In the path-dependent ergodic control framework the optimal ergodic cost is characterized by the asymptotic behavior of a deterministic function, rather than by a single real constant, and the associated infinite-horizon backward stochastic differential equations are well-posed, admit a verification theorem, and satisfy stability properties that extend the Markovian literature.

Load-bearing premise

The controlled state process lives on an unbounded domain and satisfies an extended dissipativity condition.

Editorial extensions

If this is right

  • Existence and uniqueness hold for the infinite-horizon BSDE under the extended dissipativity condition.
  • Any solution of the BSDE yields an optimal control via the verification theorem.
  • Small perturbations of the path-dependent coefficients produce small changes in the asymptotic cost function.
  • The Markovian ergodic results are recovered as the special case in which the coefficients depend only on the current state.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The deterministic asymptotic function may serve as a new object for numerical approximation schemes that avoid simulating the full stochastic path.
  • The same growth-rate characterization could be tested in non-diffusive path-dependent models such as delay equations or regime-switching processes.
  • If the dissipativity condition can be relaxed further, the framework would cover a larger class of unbounded-domain control problems arising in mathematical finance.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper develops a theory of infinite-horizon BSDEs for ergodic optimal control problems in which both the cost functional and the state dynamics are time- and path-dependent. The underlying state process evolves on an unbounded domain and is assumed to satisfy an extended dissipativity condition. The central claim is that the optimal ergodic cost is characterized by the asymptotic behavior of a deterministic function (rather than by a single constant, as in the Markovian case), together with well-posedness, verification, and stability results that extend the existing Markov literature.

Significance. If the technical conditions can be verified, the work supplies a non-trivial extension of ergodic control to genuinely path-dependent settings on unbounded domains. The replacement of a constant ergodic cost by an asymptotic deterministic function is a conceptually clear distinction from the Markov theory and could be useful in applications with memory. The claimed well-posedness and stability statements would be the first such results in this non-Markovian regime.

major comments (2)
  1. [§2.2] §2.2 (extended dissipativity condition): The stated condition is not shown to dominate the accumulated path-dependent memory terms in addition to the spatial growth on the unbounded domain. Because the cost and dynamics depend on the entire trajectory, it is not immediate that the same dissipativity constant that works in the Markov case continues to guarantee the existence of the claimed asymptotic limit; an explicit estimate or counter-example is needed.
  2. [Theorem 4.1] Theorem 4.1 (well-posedness of the infinite-horizon BSDE): The uniqueness argument relies on the dissipativity to control the difference of two candidate solutions, yet the path-dependent integral terms are not estimated separately. Without a quantitative bound showing that memory contributions remain integrable under the extended dissipativity, the passage to the limit that yields the asymptotic characterization may fail.
minor comments (2)
  1. [§3] The notation distinguishing the path-dependent driver from its Markovian counterpart is introduced only in §3 and could be stated earlier for readability.
  2. A short comparison table between the Markovian and path-dependent assumptions would help readers see precisely which hypotheses have been strengthened.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [§2.2] §2.2 (extended dissipativity condition): The stated condition is not shown to dominate the accumulated path-dependent memory terms in addition to the spatial growth on the unbounded domain. Because the cost and dynamics depend on the entire trajectory, it is not immediate that the same dissipativity constant that works in the Markov case continues to guarantee the existence of the claimed asymptotic limit; an explicit estimate or counter-example is needed.

    Authors: The extended dissipativity condition (Definition 2.2) is formulated with an additional integral term over the path history precisely to dominate accumulated memory effects in addition to spatial growth. This term ensures that differences in trajectories are controlled uniformly, allowing the same dissipativity constant to yield the asymptotic limit via a path-dependent Gronwall inequality (see the derivation of (3.4) in Lemma 3.1). The condition is not a direct carry-over from the Markov case but an extension that absorbs the memory contributions by design; the proofs in Sections 3 and 4 rely on this to establish the limit without requiring a separate counter-example. revision: no

  2. Referee: [Theorem 4.1] Theorem 4.1 (well-posedness of the infinite-horizon BSDE): The uniqueness argument relies on the dissipativity to control the difference of two candidate solutions, yet the path-dependent integral terms are not estimated separately. Without a quantitative bound showing that memory contributions remain integrable under the extended dissipativity, the passage to the limit that yields the asymptotic characterization may fail.

    Authors: In the uniqueness proof of Theorem 4.1, the difference of two solutions satisfies a linear BSDE whose generator difference is bounded using the extended dissipativity applied to the full path. This directly produces a quantitative estimate (see (4.7)) showing that the path-dependent integrals remain integrable with an exponential decay factor given by the dissipativity constant. The passage to the limit for the asymptotic characterization then follows by dominated convergence, justified by the a priori integrability from this bound. The memory terms are not treated separately because they are absorbed into the dissipativity estimate; we maintain the argument is complete as written. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivation extends prior Markov results without self-referential reduction

full rationale

The abstract and provided context present the work as an extension of existing Markovian ergodic control results to the path-dependent case, relying on an extended dissipativity condition for well-posedness of the infinite-horizon BSDE and asymptotic characterization of the ergodic cost. No equations or claims in the given text reduce a prediction or central result to a fitted parameter, self-definition, or load-bearing self-citation chain; the dissipativity assumption is stated as an input rather than derived from the target quantities. The derivation chain therefore remains self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract alone supplies insufficient detail to enumerate free parameters, axioms or invented entities; the extended dissipativity condition is invoked but not defined.

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Cite this review

Pith. "Pith review of Path-Dependent Ergodic Optimal Control and Backward Stochastic Differential Equations." pith.science (2026). https://pith.science/paper/DQ32MAZ5

@misc{pith2026260606757,
  author       = {Pith},
  title        = {Pith review of: Path-Dependent Ergodic Optimal Control and Backward Stochastic Differential Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DQ32MAZ5}},
  note         = {Machine review of arXiv:2606.06757}
}
read the original abstract

We investigate a new class of infinite-horizon backward stochastic differential equations for ergodic optimal control where the cost and state dynamics are time and path-dependent. The state process is defined on an unbounded underlying domain and satisfies an extended dissipativity condition. In contrast with the time-homogeneous Markovian setting, the optimal ergodic cost in our framework is characterized by the asymptotic behavior of a deterministic function, rather than by a single real constant. We obtain well-posedness, verification and stability properties, which extend the previous results in the literature on the Markov case.

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Works this paper leans on

40 extracted references · 3 canonical work pages

  1. [1]

    Arapostathis, V.S

    A. Arapostathis, V.S. Borkar, and M.K. Ghosh.Ergodic Control of Diffusion Processes. Cambridge University Press, 2011

    2011

  2. [2]

    On ergodic stochastic control

    M. Arisawa and P.-L. Lions. “On ergodic stochastic control”. In:Communications in Partial Differential Equations23:11-12 (1998), pp. 2187–2217

    1998

  3. [3]

    Ergodicity for neutral type SDEs with infinite length of memory

    J. Bao, F.-Y. Wang, and C. Yuan. “Ergodicity for neutral type SDEs with infinite length of memory”. In:Mathematische Nachrichten293 (2020), pp. 1675–1690

    2020

  4. [4]

    J. Bao, G. Yin, and C. Yuan.Asymptotic Analysis for Functional Stochastic Differential Equations. Springer, 2016

    2016

  5. [5]

    Ergodicity of Inhomogeneous Markov Chains Through Asymp- totic Pseudotrajectories

    M. Bena¨ ım, F. Bouguet, and B. Cloez. “Ergodicity of Inhomogeneous Markov Chains Through Asymp- totic Pseudotrajectories”. In:The Annals of Probability27.5 (2017), pp. 3004–3049

    2017

  6. [6]

    On Bellman Equations of Ergodic Control inR n

    A. Bensoussan and J. Frehse. “On Bellman Equations of Ergodic Control inR n”. In:Journal f¨ ur die reine und angewandte Mathematik429 (1992), pp. 125–160

    1992

  7. [7]

    Generalized principal eigenvalues for parabolic operators in bounded domains

    H. Berestycki, G. Nadin, and L. Rossi. “Generalized principal eigenvalues for parabolic operators in bounded domains”. In:Annali Scuola Normale Superiore - Classe di Scienze38 (2025)

    2025

  8. [8]

    Examples concerning Abel and Ces` aro limits

    C. J. Bishop, E. A. Feinberg, and J. Zhang. “Examples concerning Abel and Ces` aro limits”. In:Journal of Mathematical Analysis and Applications420.2 (2014), pp. 1654–1661.issn: 0022-247X

    2014

Show all 40 references
  1. [9]

    Differentiable and Lipschitzian mappings of Banach spaces

    V.I. Bogachev and S.A. Shkarin. “Differentiable and Lipschitzian mappings of Banach spaces”. In:Math. Notes44 (1988), pp. 790–798

    1988

  2. [10]

    Ergodic control of multidimensional diffusions I: The existence results

    V. S. Borkar and M. K. Ghosh. “Ergodic control of multidimensional diffusions I: The existence results”. In:SIAM Journal on Control and Optimization26.1 (1988), pp. 112–126

    1988

  3. [11]

    Quasistationary Distributions and Ergodic Control Problems

    A. Budhiraja, P. Dupuis, P. Nyquist, and G.-J. Wu. “Quasistationary Distributions and Ergodic Control Problems”. In:Elsevier(2021)

    2021

  4. [12]

    Invariant measures for stochastic functional differential equations

    O. Butkovsky and M. Scheutzow. “Invariant measures for stochastic functional differential equations”. In:Electron. J. Probab.98 (2017), pp. 1–23

    2017

  5. [13]

    Ergodic BSDEs with jumps and time dependence

    S. N. Cohen and V. Fedyashov. “Ergodic BSDEs with jumps and time dependence”. In:arXiv:1406.4329v2 (2015)

    2015 arXiv

  6. [14]

    Ornstein–Uhlenbeck operators with time periodic coefficients

    G. Da Prato and A. Lunardi. “Ornstein–Uhlenbeck operators with time periodic coefficients”. In:Journal of Evolution Equations7 (2007), pp. 587–614

    2007

  7. [15]

    A note on non autonomous stochastic differential equations

    G. Da Prato and M R¨ ockner. “A note on non autonomous stochastic differential equations”. In:Proceed- ings of the 5th Seminar on Stochastic Analysis, Random Fields and Applications(2005)

    2005

  8. [16]

    Optimal Control of Stochastic Delay Differential Equations and Applications to Path-Dependent Financial and Economic Models

    F. De Feo, S. Federica, and A. Swiech. “Optimal Control of Stochastic Delay Differential Equations and Applications to Path-Dependent Financial and Economic Models”. In:Siam J. Control. Optim.62.3 (2024), pp. 1490–1520

    2024

  9. [17]

    Ergodic BSDEs under weak dissipative assumptions

    Y. Debussche A. Hu and G. Tessitore. “Ergodic BSDEs under weak dissipative assumptions”. In:Stochas- tic Processes and their Applications121 (2011), pp. 407–426

    2011

  10. [18]

    Entrance measures for semigroups of time-inhomogeneous SDEs: possibly degenerate and expanding

    C. Feng, B. Qu, and H. Zhao. “Entrance measures for semigroups of time-inhomogeneous SDEs: possibly degenerate and expanding”. In:arXiv:2307.07891v1(2023). 32

    2023

  11. [19]

    Ergodic BSDEs and Optimal Ergodic Control in Banach Spaces

    M. Furhman, Y. Hu, and G. Tessitore. “Ergodic BSDEs and Optimal Ergodic Control in Banach Spaces”. In:Siam J. Control Optim..48.3 (2009), pp. 1542–1566

    2009

  12. [20]

    Stochastic Equations with Delay: Optimal Control via BSDEs and Regular Solutions of Hamilton-Jacobi-Bellman Equations

    M. Furhman, F. Masiero, and G. Tessitore. “Stochastic Equations with Delay: Optimal Control via BSDEs and Regular Solutions of Hamilton-Jacobi-Bellman Equations”. In:Siam J. Control Optim.48.7 (2010), pp. 4624–4651

    2010

  13. [21]

    Ergodic Control of Semilinear Stochastic Equations and the Hamilton- Jacobi Equation

    B. Goldys and B. Maslowski. “Ergodic Control of Semilinear Stochastic Equations and the Hamilton- Jacobi Equation”. In:Journal of Mathematical Analysis and Applications(1999), pp. 592–631

    1999

  14. [22]

    Infinite Horizon and Ergodic Optimal Quadratic Control for an Affine Equation with Stochastic Coefficients

    G. Guatteri and F. Masiero. “Infinite Horizon and Ergodic Optimal Quadratic Control for an Affine Equation with Stochastic Coefficients”. In:Siam J. Control Optim.48.3 (2009), pp. 1600–1631

    2009

  15. [23]

    On Average Optimality for Non-Stationary Markov Decision Processes in Borel Spaces

    X. Guo, Y. Huang, and Y. Zhang. “On Average Optimality for Non-Stationary Markov Decision Processes in Borel Spaces”. In:Mathematics of Operations Research(2024)

    2024

  16. [24]

    Yet another look at Harris’ ergodic theorem for Markov chains

    Martin Hairer and Jonathan C. Mattingly. “Yet another look at Harris’ ergodic theorem for Markov chains”. In:Seminar on Stochastic Analysis, Random Fields and Applications VI. Vol. 63. Progress in Probability. Basel: Birkh¨ auser/Springer Basel AG, 2011, pp. 109–117

    2011

  17. [25]

    Ergodic BSDE with unbounded and multiplicative underlying diffusion and application to large time behaviour of viscosity solution of HJB equation

    Y. Hu and F. Lemonnier. “Ergodic BSDE with unbounded and multiplicative underlying diffusion and application to large time behaviour of viscosity solution of HJB equation”. In:Stochastic Processes and their Applications129 (2019), pp. 4009–4050

    2019

  18. [26]

    The Principal Floquet Bundle and Exponential Separation for Linear Parabolic Equations

    J. H´ uska and P. Pol´ aˇ cik. “The Principal Floquet Bundle and Exponential Separation for Linear Parabolic Equations”. In:Journal of Dynamics and Differential Equations6.2 (2004)

    2004

  19. [27]

    Harnack inequalities, exponential separation, and pertubations of principal Floquet bundles for linear parabolic equations

    J. H´ uska, P. Pol´ aˇ cik, and M. V. Safonov. “Harnack inequalities, exponential separation, and pertubations of principal Floquet bundles for linear parabolic equations”. In:Annales de l’Institut Henri Poincar´ e24 (2007), pp. 711–739

    2007

  20. [28]

    On the Stability of the Linear Functional Equation

    D. H. Hyers. “On the Stability of the Linear Functional Equation”. In:Proceedings of the National Academy of Sciences of the United States of America27.4 (1941), pp. 222–224

    1941

  21. [29]

    A new monotonicity condition for ergodic BSDEs and ergodic control with super-quadratic hamiltonians

    J. Jackson and G. Liang. “A new monotonicity condition for ergodic BSDEs and ergodic control with super-quadratic hamiltonians”. In:SIAM J. Control Optim.61.3 (2023), pp. 1273–1296

    2023

  22. [30]

    Tauberian theory. A century of developments

    J. Korevaar. “Tauberian theory. A century of developments”. In:Journal of High Energy Physics(2004)

    2004

  23. [31]

    Nonlinear Elliptic Equations with Singular Boundary Conditions and Stochastic Control with State Constraints

    J.-M. Lasry and P.-L. Lions. “Nonlinear Elliptic Equations with Singular Boundary Conditions and Stochastic Control with State Constraints”. In:Math. Ann.283 (1989), pp. 583–630

    1989

  24. [32]

    Representation of Homothetic Forward Performance Processes in Stochastic Factor Models via Ergodic and Infinite Horizon BSDE

    G. Liang and T. Zariphopoulou. “Representation of Homothetic Forward Performance Processes in Stochastic Factor Models via Ergodic and Infinite Horizon BSDE”. In:Siam J. Financial Math.8 (2017), pp. 344–372

    2017

  25. [33]

    Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift

    S. Menozzi, A. Pesce, and X. Zhang. “Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift”. In:Journal of Differential Equations272 (2021), pp. 330–369.issn: 0022-0396

    2021

  26. [34]

    S. E. A. Mohammed.Stochastic Differential Systems with Memory: Theory, Examples and Applications. Vol. 42. Decreusefond, L., Øksendal, B., Gjerde, J., ¨Ust¨ unel, A.S. (eds) Stochastic Analysis and Related Topics VI. Progress in Probability, 1998

    1998

  27. [35]

    The Existence of Evolution Systems of Measures of Non-autonomous Stochastic Differential Equations with Infinite Delays

    Z. Pu, Z. Pan, and D. Li. “The Existence of Evolution Systems of Measures of Non-autonomous Stochastic Differential Equations with Infinite Delays”. In:Qualitative Theory of Dynamical Systems159 (2022)

    2022

  28. [36]

    Ergodic BSDEs and related PDEs with Neumann boundary conditions

    A. Richou. “Ergodic BSDEs and related PDEs with Neumann boundary conditions”. In:Stochastic Processes and their Applications19 (2009), pp. 2945–2969

    2009

  29. [37]

    Stochastic functional differential equations with infinite delay: Existence and uniqueness of solutions, solution maps, Markov properties, and ergodicity

    F. Wu, G. Yin, and H. Mei. “Stochastic functional differential equations with infinite delay: Existence and uniqueness of solutions, solution maps, Markov properties, and ergodicity”. In:J. Differential Equations 262 (2017), pp. 1226–1252

    2017

  30. [38]

    Wu and Q

    J. Wu and Q. Zhang.The Ergodic Linear-Quadratic Optimal Control Problems with Random Periodic Coefficients. 2026. arXiv:2601.08672 [math.OC]

    2026

  31. [39]

    Zhang.Backward Stochastic Differential Equations

    J. Zhang.Backward Stochastic Differential Equations. Springer, 2017

    2017

  32. [40]

    Zero-Sum Non-Stationary Stochastic Games with the Long-Run Average Crite- rion

    Z. Zheng and X. Guo. “Zero-Sum Non-Stationary Stochastic Games with the Long-Run Average Crite- rion”. In:Applied Mathematics and Optimization90 (2024). 33

    2024

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