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REVIEW 5 major objections 4 minor 47 references

Mean-Field Games with two-sided singular controls for L\'evy processes

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Pith's one-line read A mean-field game with a Lévy-driven state and two-sided singular controls admits discounted and ergodic threshold equilibria, with discounted equilibria converging to the ergodic one.

desk verdict A genuinely new MFG existence result for Lévy-driven two-sided singular control, but the main proof rests on a false integrability claim and needs a fix before it can be trusted. read the letter →

arxiv 2505.22936 v1 pith:VRAPFM4N submitted 2025-05-28 math.OC math.PR

Facundo Oliú This is my paper
classification math.OCmath.PR MSC 49N8091A1660G51
keywords mean-fieldgamessingularcontrolLévyprocessesDynkinergodicdiscountedtwo-sidedreflectionNashequilibrium
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

Mean-field games model many identical players whose costs depend on the aggregate behavior of the population; here the paper treats the case where each player controls a finite-mean Lévy process by pushes upward and downward at proportional costs, so the optimal policy is reflection inside an interval. The paper proves that for every sufficiently small discount rate there is an equilibrium pair of barriers $a^*_{\varepsilon} < 0 < b^*_{\varepsilon}$, that the same construction produces an ergodic equilibrium $a^* \le 0 \le b^*$ in the long-run-average limit, and that discounted equilibria converge along a subsequence to the ergodic one. The value function is characterized through an integro-differential equation, uniqueness is shown to fail in a constructed example, and the mean-field equilibrium is shown to be an approximate Nash equilibrium for large finite-player games. A sympathetic reader would care because these are easy-to-check conditions for equilibrium existence in a class of stochastic control games that includes cash-management and dividend problems, with only continuity, convexity, and growth conditions on the cost and interaction map.

What carries the argument

The load-bearing mechanism is the Dynkin game attached to the first derivative of the value function: for a fixed aggregate expectation $p$, the marginal value of pushing the state is $V_{\varepsilon,p}(x) = \sup_{\tau} \inf_{\sigma} E_x \int_0^{\tau\wedge\sigma} e^{-\varepsilon s} c_x(X_s,p)\,ds + e^{-\varepsilon\sigma}q_d 1_{\tau\ge\sigma} - e^{-\varepsilon\tau}q_u 1_{\sigma>\tau}$, a zero-sum stopping-time game between the lower exit time $\tau(a)$ and the upper exit time $\sigma(b)$. Its saddle point gives the optimal barriers $(a^*_{\varepsilon,p}, b^*_{\varepsilon,p})$, and the paper defines the best-response map $F(a,b) = (a^*_{\varepsilon,p_{a,b}}, b^*_{\varepsilon,p_{a,b}})$ on a rectangle $[-L,0]\times[0,L]$. Continuity of $F$ is proved from uniform barrier bounds and Lipschitz dependence of the Dynkin value on $p$; Brouwer's fixed point theorem then yields the discounted equilibrium. For the ergodic theorem, regenerative and renewal theory for the reflected process converts discounted costs into long-run average costs.

What would settle it

Look for a finite-mean Lévy process that is neither a subordinator nor the opposite of one and whose expected first exit time $\sup_{x\in[-L,L]} E_x(\tau_{-L}\wedge\sigma_L)$ is infinite; for such a process the bound in Proposition 3.4 would not be finite, so the continuity of $F$ in Lemma 3.8 would fail and the existence proof via Brouwer's theorem would not apply.

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Extended reading notes

Core claim

The central claim is stated as Theorem 2.10 and Theorem 2.12: the discounted mean-field game always has an $\varepsilon$-discounted equilibrium of threshold form with $a^*_{\varepsilon} < 0 < b^*_{\varepsilon}$, and there is an ergodic equilibrium $a^* \le 0 \le b^*$ obtained as a subsequential limit of discounted equilibria as the discount rate vanishes. The equilibrium is two-sided reflection of the Lévy process at the lower barrier $a^*$ and upper barrier $b^*$, and the value function in the continuation interval solves the integro-differential inequality $Lu - \varepsilon u + c(\cdot,p) = 0$ with variational inequalities outside. The paper also claims that an MFG equilibrium is an approximate Nash equilibrium for the symmetric $N$-player game, and gives a concrete family where two different equilibria coexist, so uniqueness is not part of the picture.

Load-bearing premise

The proof depends on the undisplayed assertion, made just before Proposition 3.4, that every non-null Lévy process exits an interval in finite expected time; if a non-monotone Lévy process could have infinite expected interval-exit time, the uniform continuity bound for the best-response map and the fixed-point existence proof would break.

Editorial extensions

If this is right

  • For any sufficiently small discount rate, a threshold equilibrium exists and the equilibrium barriers can be located by solving the associated Dynkin game.
  • As the discount rate goes to zero, a subsequence of discounted equilibria converges to an ergodic equilibrium, so approximating long-run problems by discounted ones is justified in this setting.
  • Equilibria satisfy a variational characterization: the marginal value equals $-q_u$ at the lower barrier and $q_d$ at the upper barrier, giving a first-order condition that can be checked numerically.
  • Several equilibria can coexist for the same cost and interaction data, so algorithms or comparative statics that presume uniqueness may miss solutions.
  • The mean-field equilibrium is an approximate Nash equilibrium for a symmetric $N$-player game once $N$ is large, either for reflecting controls or under uniform second-moment bounds.

Reading between the lines

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

  • The same fixed-point route should work for interaction terms other than $E f(X^\eta_\infty)$, such as smooth functions of the stationary law, as long as the uniform barrier bound of Proposition 3.1 and the continuity estimate of Proposition 3.4 survive.
  • The uniqueness counterexample suggests that non-uniqueness is driven by oscillation of the interaction function; adding monotonicity or a small Lipschitz constant to the mean-field coupling would plausibly restore uniqueness, though the paper does not prove this.
  • The Abelian-limit theorem suggests a computational route: solve the discounted fixed-point problem for several small $\varepsilon$ values and extrapolate the barriers to the ergodic equilibrium, avoiding a direct solution of the ergodic variational problem.
  • If the interval-exit-time integrability assumption is replaced by a uniform Cesàro estimate, the continuity proof might extend to processes with heavy-tailed jumps, for which the paper's proof does not currently apply.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The paper studies a mean-field game in which a representative agent controls a finite-mean Lévy process by adding and subtracting nondecreasing cádlág processes, with the running cost depending on the stationary law of the aggregate controlled process. The main claims are existence of an ε-discounted mean-field game equilibrium via a Brouwer fixed point on the best-response map given by reflection barriers (Theorem 2.10), existence of an ergodic equilibrium as an Abelian limit of discounted equilibria (Theorem 2.12), characterization by an integro-differential equation, a non-uniqueness remark (Remark 5.1), and approximation by finite-player Nash equilibria (Section 6). The proofs use an adjoint Dynkin game from Mordecki and Oliú (2024), regenerative-process theory, and a continuity argument for the best-response map F.

Significance. If the existence theorem and the convergence theorem were correctly proved, this would be a useful contribution: it extends mean-field singular control MFGs from diffusion settings to general Lévy processes, connects two-sided singular control to an adjoint Dynkin game, and treats discounted and ergodic equilibria in one framework with explicit examples. The Brouwer fixed-point strategy on barrier pairs is natural, and the regenerative-process argument for the Abelian limit is an elegant idea. However, the proof as written contains a false auxiliary assertion on exit-time integrability, a notational/argument error in the continuity lemma, an invalid non-uniqueness counterexample, and an incomplete step in the finite-player approximation. These gaps currently leave the main claims unsupported despite the plausibility of the overall approach.

major comments (5)
  1. [Section 3.2, Proposition 3.4 and equation (8)] The proof uses the sentence 'the exit time of intervals for non-null Lévy processes have finite mean' as an unproven fact, and that fact is false. Let X be a spectrally negative α-stable process with α∈(1,2), Lévy measure Π(dx)=c(-x)^{-α-1}dx for x<0, and no drift or Gaussian part. This process has finite mean, is neither a subordinator nor the opposite of one, and does not violate the standing structural assumptions for suitable c. Starting from 0, it has no positive jumps and no positive drift, so it never reaches +L; hence τ_{-L}∧σ_L=τ_{-L}, and the Laplace transform E_0[e^{-λτ_{-L}}]=exp(-c'Lλ^{1/α}) gives E_0[τ_{-L}]=∞. Therefore the constant sup_{x∈[-L,L]} E_x(τ_{-L}∧σ_L) in (8) is infinite, Proposition 3.4 supplies no quantitative continuity, and the subsequent inequalities in Lemma 3.8 collapse. The existence proof via Brouwer in Theorem 2.10(iii) is unsupported as written. This can be repaired by explicitly assuming or proving a finite modulus for interval exit times, but the necessary hypothesis is not present.
  2. [Lemma 3.8] The continuity proof for F argues about the wrong coordinate. From V_{p_{a,b}}(x)<q_d for x∈(a*_p,b*_p), the information obtained for nearby p_n is that x lies to the left of b*_{p_n}; this gives a statement about F2, not F1. The displayed chain 'limsup F1(a_n,b_n) ≥ liminf F1(a_n,b_n) ≥ b*_{p_{a,b},ǫ}' is therefore unjustified, and the contradiction assumption that limsup F1(a_n,b_n)=\hat b>b*_{p_{a,b},ǫ} is inconsistent with F1 taking values in [-L,0] while b*≥0. The subsequent argument with b1<b2<b3 inside (b*,\hat b) is coherent only if F1 is replaced by F2 throughout. As written, the continuity of F is not proved.
  3. [Remark 5.1] The non-uniqueness counterexample does not satisfy the hypotheses of Section 5.1. The function h(y)=0.01+e^y|cos(y)| is not convex and vanishes at infinitely many points, so c(x,y)=x^2h(y) does not satisfy the strict positivity of inf_{(-r,r)×R} c_{xx} required in Assumption 2.5(i), and it is not covered by the convexity and positivity assumptions stated at the beginning of Section 5.1. Thus the two displayed approximate equilibria do not establish the claimed failure of uniqueness.
  4. [Proposition 4.4] The proof that E D^{0,b}_{τ1}<∞ is incomplete. After quoting the inequality from Andersen et al. (2015), the text says 'we deduce (??) holds', where (??) is a literal placeholder and no argument is given for controlling the stochastic-integral and jump terms on the random horizon τ1. This step is needed to verify the hypotheses of Theorem 4.3 for the cumulative control cost, and hence for the Abelian limit (12). Please provide a complete argument or a precise reference.
  5. [Section 6, Theorem 6.4] In the proof, the passage from Hoeffding's inequality for the average \bar f^{a,b,-i} to the bound on E[(h(\bar f)-h(E f))^2] requires a Lipschitz or bounded-difference property of h, but h is only assumed continuous. The displayed estimate '|h(x)-h(y)| ≤ b^2+a^2' is not justified by the stated hypotheses, and the same issue affects the discounted case. As a result, the r-Nash bound is not established. This is a separate gap in the finite-player approximation claim.
minor comments (4)
  1. [Lemma 3.5] The proof interchanges the limits over n and T without a uniform-in-n estimate; please justify the equality of the two limits. The degenerate case a=b should also be written out more carefully.
  2. [Section 5.1] In the equation for the roots of the characteristic exponent, '(ǫ+λ2+λ2)z²' appears to be a typo for '(ǫ+λ1+λ2)z²'.
  3. [Proposition 3.6] The word 'quaternion' should be 'quadruple' or 'four points'.
  4. [Notation] The notation c'(x,p) is used interchangeably with c_x(x,p) without comment; please standardize the notation throughout.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the MFG equilibrium is obtained as a genuine Brouwer fixed point, and the heavy reliance on Mordecki-Oliú (2024) is independent one-agent support rather than a self-referential reduction.

full rationale

The derivation chain is not circular. The equilibrium (a*, b*) is defined by the fixed-point equation F(a, b) = (a*_{p_{a,b},\epsilon}, b*_{p_{a,b},\epsilon}), and existence is proved by Brouwer's theorem after establishing continuity of F (Lemma 3.8). This is a self-consistency condition in the usual MFG sense, not a definition of the conclusion. The single-agent Dynkin-game results (value function regularity, existence and characterization of optimal reflection barriers) are imported from Mordecki-Oliú (2024) via explicit theorem references; those results are parameter-free, state their own assumptions, and do not contain the MFG equilibrium being constructed, so the citation is independent evidence and does not raise the circularity score. The worked examples solve the equilibrium equations from first principles (Section 5), and Remark 5.1's non-uniqueness counterexample is constructed independently. The only flagged weakness is the assertion before Proposition 3.4 that 'the exit time of intervals for non-null Lévy processes have finite mean', used to bound (8) by r sup_x E_x(τ_-L ∧ σ_L); this is a potentially false correctness claim (spectrally negative α-stable processes can have infinite mean interval exit times) and would undermine the proof of Lemma 3.8, but it is a missing justification or error, not a circular reduction of the theorem to its own assumptions. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the author's prior single-agent theory (Mordecki-Oliú 2024), standard results on reflected Lévy processes (Andersen et al. 2015), and renewal theory (Asmussen 2008). No free parameters are fitted to data. The only ad hoc element is the unproved assertion of finite-mean exit times. No new entities are postulated.

assumptions (5)
  • domain assumption The single-agent two-sided singular control problem for Lévy processes is characterized by Dynkin games, with the value function in the domain of the generator, as in Mordecki-Oliú (2024).
    Invoked in Section 3.1 and Theorem 2.10(i), importing the full structural theory from the author's earlier paper.
  • ad hoc to paper Exit times from intervals for non-null finite-mean Lévy processes have finite expectation.
    Stated implicitly before Proposition 3.4 and used to prove Lipschitz properties of the value function; no proof or reference is given.
  • standard math Lévy processes with two-sided reflection have a unique strong solution and a stationary distribution, as in Andersen et al. (2015).
    Used throughout to define reflecting controls and stationary measures.
  • standard math The regenerative process theorem (Asmussen 2008) applies to the reflected Lévy process and its cumulative costs.
    Used in Section 4.1 to obtain the abelian limit of discounted costs.
  • domain assumption Assumption 2.5: c(·,y) is strictly convex, has positive cxx lower bound on compact neighborhoods, grows at least linearly, and satisfies an integrability condition.
    Core hypothesis for the single-agent control and the fixed-point argument.

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Pith. "Pith review of Mean-Field Games with two-sided singular controls for L\'evy processes." pith.science (2026). https://pith.science/paper/VRAPFM4N

@misc{pith2026250522936,
  author       = {Pith},
  title        = {Pith review of: Mean-Field Games with two-sided singular controls for L\'evy processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VRAPFM4N}},
  note         = {Machine review of arXiv:2505.22936}
}
read the original abstract

In a probabilistic mean field game driven by a L\'evy process an individual player aims to minimize a long run discounted/ergodic cost by controlling the process through a pair of increasing and decreasing c\`adl\`ag processes, while he is interacting with an aggregate of players through the expectation of a controlled process by another pair of c\`adl\`ag processes. With the Brouwer fixed point theorem, we provide easy to check conditions for the existence of mean field game equilibrium controls for both the discounted and ergodic control problem, characterize them as the solution of an integro-differential equation and show with a counterexample that uniqueness does not always holds. Furthermore, we study the convergence of equilibrium controls in the abelian sense. Finally, we treat the convergence of a finite-player game to this problem to justify our approach.

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Works this paper leans on

47 extracted references · 43 canonical work pages

  1. [1]

    A Stationary Mean-Field Equilibrium Model of Irreversible Investment in a Two-Regime Economy

    A \" d, R., Basei, M. & Ferrari, G. A Stationary Mean-Field Equilibrium Model of Irreversible Investment in a Two-Regime Economy. (2023). https://doi.org/10.48550/arXiv.2305.00541

    work page Pith review arXiv doi:10.48550/arxiv.2305.00541 2023

  2. [2]

    Alvarez, L.H.R. (2018). A Class of Solvable Stationary Singular Stochastic Control Problems, (2018). arXiv preprint arXiv:1803.03464v1 \ https://doi.org/10.48550/arXiv.1803.03464

    work page Pith review arXiv doi:10.48550/arxiv.1803.03464 2018

  3. [3]

    & Pihlsgård, M

    Andersen, L.N., Asmussen, S., Glynn, P.W. & Pihlsgård, M. L\'evy Processes with Two-Sided Reflection. L\'evy Matters V. Lecture Notes in Mathematics(), vol 2149. Springer, Cham. (2015). https://doi.org/10.1007/978-3-319-23138-9_2

    work page doi:10.1007/978-3-319-23138-9_2 2015

  4. [4]

    & Zheng, Y

    Arapostathis, A., Caffarelli, L., Pang, G. & Zheng, Y. Ergodic Control of a Class of Jump Diffusions with Finite Lévy Measures and Rough Kernels, SIAM Journal on Control and Optimization, 57,(2),1516-1540, (2019). https://doi.org/10.1137/18M1166717

    work page doi:10.1137/18m1166717 2019

  5. [5]

    Applied Probability and Queues, Second Edition, Stochastic Modelling and Applied Probability, 51

    Asmussen, S. Applied Probability and Queues, Second Edition, Stochastic Modelling and Applied Probability, 51. Springer Sciene & Business Media, (2008)

    work page 2008

  6. [6]

    and Taksar, M

    Asmussen, S. and Taksar, M. Controlled diffusion models for optimal dividend pay-out. Insurance: Mathematics and Economics, 20, 1–15, (1997). \ https://doi.org/10.1016/S0167-6687(96)00017-0

    work page doi:10.1016/s0167-6687(96)00017-0 1997

  7. [7]

    & Sosa, A

    Aspirot, L., Mordecki, E. & Sosa, A. Dynkin Games for L\'evy Processes. preprint arXiv:2410.23509, (2024) https://arxiv.org/abs/2410.23509

    work page arXiv 2024

  8. [8]

    & Di Persio L

    Benazzoli C., Campi L. & Di Persio L. Mean field games with controlled jump–diffusion dynamics: Existence results and an illiquid interbank market model, Stochastic Processes and their Applications, 130, Issue 11, 6927-6964, (2020). https://doi.org/10.1016/j.spa.2020.07.004

    doi:10.1016/j.spa.2020.07.004 2020

Show all 47 references
  1. [9]

    & Yam, P., (2020)

    Bensoussan A., Djehiche B., Tembine H. & Yam, P., (2020). Mean-Field-Type Games with Jump and Regime Switching, Dynamic Games and Applications, Springer, 10(1), 19-57, (2020). https://doi.org/10.1007/s13235-019-00306-2

    2020 doi

  2. [10]

    L\'evy processes

    Bertoin, J. L\'evy processes. Cambridge University Press, Series Number 121, (1996)

    1996

  3. [11]

    Bounded variation singular stochastic control and Dynkin game

    Boetius, F. Bounded variation singular stochastic control and Dynkin game. SIAM J. CONTROL OPTIM. Society for Industrial and Applied Mathematics 44, 1289–1321, (2005)

    2005

  4. [12]

    & Guo, X

    Cao, H. & Guo, X. MFGs for partially reversible investment, Stochastic Processes and their Applications 150, 995--1014, (2022). https://doi.org/10.1016/j.spa.2020.09.006

    2022 doi

  5. [13]

    Stationary Discounted and Ergodic Mean Field Games of Singular Control

    Cao, H., Dianetti, J., Ferrari, G. Stationary Discounted and Ergodic Mean Field Games of Singular Control. Mathematics of Operations Research 48, 1871--1898, (2023)

    2023

  6. [14]

    Applications of mean field games in financial engineering and economic theory

    Carmona, R. Applications of mean field games in financial engineering and economic theory. Mean field games, Proceedings of Symposia in Applied Mathematics, 78, 165--219, (2021)

    2021

  7. [15]

    & Delarue, F

    Carmona, R. & Delarue, F. Probabilistic theory of mean field games with applications. Springer, (2018)

    2018

  8. [16]

    , Fouque J

    Carmona, R. , Fouque J. & Sun, L. Mean field games & systemic risk. Communications in Mathematical Sciences, 13(4), 911–933, (2013). https://doi.org/10.4310/CMS.2015.v13.n4.a4

    2013 doi

  9. [17]

    & Sohr, T

    Christensen, S., Anne, B. & Sohr, T. Competition versus Cooperation: A class of solvable mean field impulse control problems. SIAM J. Control. Optim., 59, 3946-3972, (2021). https://arxiv.org/abs/2010.06452

    2021 arXiv

  10. [18]

    & Oli\'u, F

    Christensen, S., Mordecki, E. & Oli\'u, F. Two sided ergodic singular control and mean field game for diffusions, Decisions Econ Finan., (2023). https://doi.org/10.1007/s10203-024-00464-y

    2023 doi

  11. [19]

    Clark. W. Mathematical Bioeconomics: The Mathematics of Conservation. Third edition. John Wiley & Sons, (2010)

    2010

  12. [20]

    & G\"ottlich, S

    Festa, A. & G\"ottlich, S. A Mean Field Game approach for multi-lane traffic management. IFAC-PapersOnLine 51(32), 793--798, (2018)

    2018

  13. [21]

    Fu, G. & Horst. U. Mean field games with singular controls, SIAM Journal on Control and Optimization, 55(6), 3833–3868, (2017). https://doi.org/10.1137/17M1123742

    2017 doi

  14. [22]

    Guo, X. & Xu, R. Stochastic games for fuel followers problem: N vs MFG, SIAM Journal on Control and Optimization, 57(1):659–692, (2019). https://doi.org/10.1137/17M1159531

    2019 doi

  15. [23]

    & Tomecek P

    Guo X. & Tomecek P. CONNECTIONS BETWEEN SINGULAR CONTROL AND OPTIMAL SWITCHING. Society for Industrial and Applied Mathematics, no 1, 47, 421-433, (2008). https://doi.org/10.1137/060669024

    2008 doi

  16. [24]

    & Tomecek P

    Guo X. & Tomecek P. A CLASS OF SINGULAR CONTROL PROBLEMS AND THE SMOOTH FIT PRINCIPLE, Society for Industrial and Applied Mathematics, no 6, 47, 3076-3099, (2009). https://doi.org/10.1137/070685336

    2009 doi

  17. [25]

    & Taksar, M

    H jgaard, B. & Taksar, M. Optimal risk control for a large corporation in the presence of returns on investments, Finance & Stochastics, 5, 527–547, (2001). https://doi.org/10.1007/PL00000042

    2001 doi

  18. [26]

    & Mihail, Z

    Andrew, J. & Mihail, Z. A singular control problem with an expected and a pathwise ergodic performance criterion, Journal of Applied Mathematics and Stochastic Analysis, (2006). http://eudml.org/doc/53397

    2006

  19. [27]

    & Shiryaev, A.N

    Jacod, J. & Shiryaev, A.N. Limit Theorems for Stochastic Processes. 2nd. ed. Springer-Verlag, Berlin, (2003)

    2003

  20. [28]

    & Shiryaev, A

    Jeanblanc-Picqué, M. & Shiryaev, A. N. Optimization of the flow of dividends, Russian Math. Surveys, 50, 257 – 277, (1995). https://doi.org/10.1070/RM1995v050n02ABEH002054

    1995 doi

  21. [29]

    A class of singular stochastic control problems, Adv

    Karatzas. A class of singular stochastic control problems, Adv. Appl. Prob., 15(2):225-254, (1983). https://doi.org/10.2307/1426435

    1983 doi

  22. [30]

    & Wang, H

    Karatzas, I. & Wang, H. Connections between bounded-variation control and Dynkin games, (2003)

    2003

  23. [31]

    Introductory Lectures on Fluctuations of L\'evy Processes with Applications

    Kyprianou, A.E. Introductory Lectures on Fluctuations of L\'evy Processes with Applications. Springer, (2006)

    2006

  24. [32]

    & Zhu, C

    Kunwai, K., Xi, F., Yin, G. & Zhu, C. On an Ergodic Two-Sided Singular Control Problem, Appl. Math Optim., 86, (2022). https://doi.org/10.1007/s00245-022-09881-0

    2022 doi

  25. [33]

    Mean field games via controlled martingale problems: existence of Markovian equilibria, Stochastic Processes and their Applications, 125:2856– 2894, (2015)

    Lacker, D. Mean field games via controlled martingale problems: existence of Markovian equilibria, Stochastic Processes and their Applications, 125:2856– 2894, (2015). https://doi.org/10.1016/j.spa.2015.02.006

    2015 doi

  26. [34]

    & Zariphopoulou, T

    Lacker, D. & Zariphopoulou, T. Mean field and N-agent games for optimal investment under relative performance criteria, Mathematical Finance, 0(0), (2019). https://doi.org/10.1111/mafi.12206

    2019 doi

  27. [35]

    & Lions P-L

    Lasry, J-M. & Lions P-L. Jeux à champ moyen. I – Le cas stationnaire, Comptes Rendus Mathematique, 343, Issue 9, (2006). https://doi.org/10.1016/j.crma.2006.09.019

    2006 doi

  28. [36]

    & Robin, M

    Menaldi, J.L. & Robin, M. Some singular control problem with long term average criterion. Lecture Notes in Control and Information Sciences, vol 59, (1984). https://api.semanticscholar.org/CorpusID:124873068

    1984

  29. [37]

    L., & Robin, M

    Menaldi, J. L., & Robin, M. Singular ergodic control for multidimensional Gaussian–Poisson processes, Stochastics, 85(4), 682–691, (2013). https://doi.org/10.1080/17442508.2013.795569

    2013

  30. [38]

    & Oli\'u, F

    Mordecki, E. & Oli\'u, F. Two sided long-time optimization singular control problems for Lévy processes and Dynkin games, arXiv preprint arXiv:2403.05731 , (2024). \ https://doi.org/10.48550/arXiv.2403.05731

    2024 doi arXiv

  31. [39]

    Optimal dividend payments and reinvestments of diffusion processes with fixed and proportional costs, SIAM Journal on Control and Optimization, 47, 2201–2226, (2008)

    Paulsen, J. Optimal dividend payments and reinvestments of diffusion processes with fixed and proportional costs, SIAM Journal on Control and Optimization, 47, 2201–2226, (2008). https://doi.org/10.1137/070691632

    2008 doi

  32. [40]

    & Keppo, J

    Peura, S. & Keppo, J. S. Optimal Bank Capital with Costly Recapitalization, The Journal of Business, 79, No. 4, 2163-2201, (2006). https://doi.org/10.1086/503660

    2006 doi

  33. [41]

    Ninety years of the Brouwer fixed point theorem, Vietnam Journal of Mathematics, 27, (1999)

    Park & Sehie. Ninety years of the Brouwer fixed point theorem, Vietnam Journal of Mathematics, 27, (1999)

    1999

  34. [42]

    L\'evy processes and infinitely divisible distributions

    Sato K. L\'evy processes and infinitely divisible distributions. Cambridge University Press, English edition, (1999)

    1999

  35. [43]

    & Gaver, D

    Shreve, S., Lehoczky, J. & Gaver, D. Optimal consumption for general diffusion with absorbing and reflecting barriers, SIAM Journal on Control and Optimization, 22, 55–75, (1984)

    1984

  36. [44]

    & Mahajan, A

    Subramanian, J. & Mahajan, A. Reinforcement Learning in Stationary Mean-field Games, AAMAS '19: Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems, (2019). DOI: 10.5555/3306127.3331700

    2019

  37. [45]

    Stationary Stochastic Control for Itô Processes, Advances in Applied Probability, 34(1), 128–140, (2002)

    Ananda P.N Weerasinghe. Stationary Stochastic Control for Itô Processes, Advances in Applied Probability, 34(1), 128–140, (2002). http://www.jstor.org/stable/1428375

    2002

  38. [46]

    An Abelian Limit Approach to a Singular Ergodic Control Problem, SIAM Journal on Control and Optimization

    Ananda P.N Weerasinghe. An Abelian Limit Approach to a Singular Ergodic Control Problem, SIAM Journal on Control and Optimization. 46(2), 714-737, (2007). https://doi.org/10.1137/050646998

    2007 doi

  39. [47]

    On the Solutions of the Problem for a Singular Ergodic Control, Journal of Optimization Theory and Applications, Springer, 173(3), 746-762, (2017)

    Wu, Y.L & Chen, Z.Y. On the Solutions of the Problem for a Singular Ergodic Control, Journal of Optimization Theory and Applications, Springer, 173(3), 746-762, (2017). https://ideas.repec.org/a/spr/joptap/v173y2017i3d10.1007_s10957-017-1099-y.html

    2017

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.