Pith. sign in

REVIEW 4 major objections 4 minor 50 references

This paper establishes a simultaneous-perturbation equilibrium notion that fully characterizes time-consistent strategies for mixed regular-singular control under a mean-variance objective, and proves an explicit reinsurance equilibrium wit

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 11:48 UTC pith:NL4WMHDE

load-bearing objection Genuinely new equilibrium framework for mixed regular-singular MV control, but the flagship explicit solution hinges on a monotone free boundary the paper never proves. the 4 major comments →

arxiv 2607.26635 v1 pith:NL4WMHDE submitted 2026-07-29 math.OC

Equilibrium for Regular-Singular Control under Mean-Variance Criterion

Zongxia Liang , Xiaodong Luo , Jiayu Zhang This is my paper
classification math.OC MSC 93E2049L2091B30
keywords mixed regular-singular controlmean-variance criteriontime-consistent equilibriumsimultaneous perturbationextended HJB systemfree boundary switchingreinsurancecoupled control
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Mean-variance objectives make decisions time-inconsistent: a strategy that looks best today will not look best from a future self's perspective. This paper treats the successive selves as players in an intrapersonal game and asks for time-consistent equilibria when the controller has both a regular (continuous) control and a singular (jump/impulse) control that act jointly on the state. The authors propose an equilibrium notion that perturbs both controls simultaneously, prove both sufficient and necessary characterizations via an extended Hamilton-Jacobi-Bellman system, and construct explicit equilibrium strategies for a reinsurance problem in which the singular free boundary switches depending on the regular-control regime. The result matters because it supplies a rigorous framework for time-inconsistent mixed control problems and shows that the two controls can be intrinsically coupled.

Core claim

The paper's central claim is that a closed-loop equilibrium for the mixed regular-singular mean-variance problem can be defined by simultaneous small perturbations of both controls, and that equilibrium laws are exactly characterized by an extended HJB system coupling a value function V and an auxiliary function g (the expected terminal state under the equilibrium). Under regularity and integrability conditions, Theorem 3.1 verifies that solutions of this system are equilibria, and Theorem 4.1 shows any equilibrium must satisfy it. In the reinsurance specialization, the paper constructs explicit V* and g* with a quadratic ansatz and derives an equilibrium free boundary k(t) that is piecewise

What carries the argument

The central object is the extended HJB system (3.5)-(3.9), consisting of a variational inequality for V and a linear equation for the auxiliary function g; the equilibrium singular law is a partition of the state space into a waiting region and an action region separated by a free boundary k(t), where the gradient of V satisfies a smooth-pasting condition. The simultaneous perturbation equilibrium definition (Definition 2.3) is what makes the characterization possible: it treats the regular and singular controls as coupled and compares the equilibrium against all infinitesimal perturbations of both. In the explicit reinsurance solution, the free boundary k(t) must be monotonically decreasing

Load-bearing premise

The verification argument for the explicit reinsurance equilibrium assumes the singular free boundary k(t) never increases; if a parameter choice makes k(t) rise on some interval, the proof that singular purchases occur only at the initial time fails and the candidate may not be an equilibrium.

What would settle it

Compute the explicit free boundary k(t) from (5.18), (5.13), (5.17) for parameters satisfying Assumption 5.1 and Conditions (1), (2), (4), (5) of Theorem 5.1 but with k'(t)>0 somewhere. If such parameters exist, the candidate (Pi*, Xi*) cannot be verified as an equilibrium by the paper's Step 2, and the theorem's Condition (3) is not merely technical.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Time-consistent equilibria now have a rigorous sufficient-and-necessary characterization for mixed regular-singular mean-variance problems, not just for purely regular or purely singular ones.
  • The explicit reinsurance equilibrium shows that the singular control's free boundary can depend on the active regime of the regular control, producing a piecewise-smooth boundary that switches at the intersection of the two control regimes.
  • When the coupling parameter alpha2 vanishes, the mixed equilibrium decouples into the equilibrium of the regular-control-only and singular-control-only problems, confirming that the coupling is the source of the regime-switching structure.
  • The verification theorem gives a concrete recipe: solve for V and g from the extended HJB system, then read off the equilibrium control laws from the minimizer and the waiting/action partition.

Where Pith is reading between the lines

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

  • The monotonicity condition on k(t) (Condition 3 of Theorem 5.1) is likely not intrinsic; the paper's Remark 3.1 suggests a change-of-variable formula with local time on surfaces could relax it. One testable extension is to repeat the verification under a non-monotone boundary using that generalized formula.
  • The same framework should apply to other time-inconsistent mixed problems, such as mean-variance dividend control with transaction costs or investment with irreversible consumption; the key data is the coupling of the controls in the state dynamics and objective.
  • A numerical implementation could search the parameter space for the first occurrence of k'(t)>0, which would either produce a counterexample to the current verification proof or identify the boundary of the theorem's validity.

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

4 major / 4 minor

Summary. The paper develops a game-theoretic equilibrium framework for continuous-time mixed regular-singular control problems under a mean-variance objective. It introduces a simultaneous-perturbation equilibrium notion (Definition 2.3), proves a verification theorem (Theorem 3.1) and necessary conditions (Theorem 4.1), and then specializes to a two-line reinsurance model. In the reinsurance application, the paper constructs an explicit candidate equilibrium with a piecewise free boundary k(t) that switches according to whether the regular control is interior or saturated, and it analyzes the decoupled case α2=0. The claimed contribution is an explicit, mutually coupled regular-singular equilibrium with a regime-switching free boundary, which is new relative to earlier work.

Significance. If the verification results are correct, the paper makes a useful contribution: it extends the time-inconsistent equilibrium toolkit from purely regular or purely singular control to mixed regular-singular problems, and it provides a concrete reinsurance example in which the singular free boundary is endogenously determined by the active regular-control regime. The explicit formulas in Section 5 are nontrivial and the degeneracy analysis at α2=0 is a valuable check. The general verification and necessary-condition theorems are stated in appropriate generality and could be reused beyond the reinsurance application. However, the explicit equilibrium claim is only conditionally established: the key verification step relies on an assumed monotonicity of the free boundary, and several central computations are either delegated to 'straightforward computation' or omitted entirely. Thus the significance is real but contingent on closing these gaps.

major comments (4)
  1. [Theorem 5.1, Condition (3) and Step 2 of the proof] Condition (3) assumes k'(t)≤0 and k(T)≤0, and this monotonicity is load-bearing. In Step 2, the paper uses k decreasing to assert that singular purchases occur only at the initial time, so that the trajectory after the initial projection stays in W* and Itô's formula can be applied despite g*_y being discontinuous at y=k(t). The same monotonicity is used in Step 3 to argue that perturbed paths remain in a smooth region for small h. Yet no argument is given that Conditions (1), (2), (4), (5) imply k'(t)≤0. Remark 5.2 exhibits only a small-α2 parameter regime where k' is proportional to -α2. If a non-monotone k occurs, the reduction g(x,t,y)=g(x-(max{y,k(t)}-y),t,max{y,k(t)}-y) and the subsequent Itô argument fail. This is not a cosmetic regularity issue; it blocks the claimed explicit equilibrium for the full stated parameter set. The authors should either prove monotonicity from the stat
  2. [Theorem 5.1, Step 1; Theorems 5.2 and 5.4] The core verification of the explicit solution is incomplete. Step 1 of Theorem 5.1 states that 'through a straightforward computation' the pair (V*,g*) satisfies the HJB system (3.5)-(3.9), but the actual verification across the three regions W*∩B, W*∩B^c, and P* is not shown. Since the piecewise formulas involve f(y), y*(t), y*_1(t), k(t), and boundary conditions, the HJB check is non-trivial; the reader cannot easily confirm it without reproducing the algebra. Similarly, Theorems 5.2 and 5.4 omit proofs entirely with 'similar arguments.' Given that these theorems contain the paper's central explicit results, the omitted computations should be supplied in an appendix or in the main text.
  3. [Theorem 4.1, Step 2 of the proof] There is a sign/consistency problem in the derivation of the necessary conditions. After equation (4.3), the text reads: 'Using Definition 2.3, we have lim inf_{h↓0} {J(x,t,y;u,η)-J(x,t,y;bπ,bξ)} ≤ 0 yielding ∫... ≥0.' But Definition 2.3 and equation (2.13) require the lim inf of the divided difference to be ≥0, not ≤0. If the displayed sign is a typo, it should be corrected; if not, the derivation of the nonnegativity of θe^{ρ(T-t)}c - V_x + V_y is not justified. As written, this step is internally inconsistent and must be fixed for the necessary conditions to be reliable.
  4. [Lemma 5.1 and Theorem 5.1 verification of Condition (4)] Lemma 5.1 claims V* ∈ C^{2,1,1}(Q), but the proof only checks continuity of V and its first derivatives at the free boundaries. The second derivative checks, especially V*_yy and V*_xx across y=k(t) and y=ŷ(t), are not shown. Since Condition (1) of Theorem 3.1 requires V,g∈C^{2,1,1} (or the relaxed regularity in Remark 3.1), this is a gap in the verification. Additionally, Theorem 5.1 Step 2 asserts without detailed argument that Condition (4) of Theorem 3.1 holds for all admissible perturbations, citing only linear growth and Proposition 2.1. Because the perturbed dynamics can cross the singular barrier, the required uniform integrability of the Itô terms is not immediate and deserves a concrete proof.
minor comments (4)
  1. [Theorem 5.1, Step 2] The text says g*_y is discontinuous 'solely at the points where y=y_1(t)', but the relevant discontinuity is at y=k(t). This appears to be a typographical error and should be corrected.
  2. [Theorem 5.1, Step 1 / equation after (5.24)] The line 'combining with θe^{ρ(T-t)}(c0(t)+α1c1(t)y)-V*_x(x,t,y)-V*_y(x,t,y)=0' has the wrong sign before V*_y. The singular-regime condition is θe^{ρ(T-t)}c - V_x + V_y = 0, as used in (3.3)-(3.5). Please correct the sign.
  3. [Theorem 3.1, proof / Step 2] In the sentence 'Condition (4) of Theorem 3.1 holids' there is a typo ('holids' for 'holds'). More substantively, the expression in the proof for g(x,t,y) after the projection uses the third coordinate 'max{y,k(t)}-y'; this is not the correct singular-state coordinate after projection, which should be k(t). Please clarify the notation.
  4. [Section 5, notation] Several functions depend on both t and y but the arguments are sometimes suppressed, e.g., θc(t,y) versus θ(c0(t)+α1c1(t)y). This makes the already complex formulas harder to check. A table of notation or a consistent explicit-argument convention would improve readability.

Circularity Check

0 steps flagged

No significant circularity: the equilibrium framework, verification theorem, and explicit reinsurance solution are self-contained; the k'<=0 condition is a stated sufficient hypothesis, not a fitted input or renamed prediction.

full rationale

The paper derives its results from a self-contained chain: Definition 2.3 defines equilibrium via simultaneous perturbations; Theorem 3.1 proves a verification theorem from the extended HJB system; Theorem 4.1 proves the converse necessary conditions under regularity assumptions; and the explicit reinsurance solution in Theorem 5.1 is constructed by solving the resulting HJB equations (5.4)-(5.6), with no parameter fitted to force the target result. The central claims do not reduce by construction to their inputs: the equilibrium condition (2.13) is not the same as the HJB equations (3.5)-(3.9), and the verification proof supplies the missing argument rather than assuming the conclusion. The paper's own prior work (Z. Liang, Luo, and Yuan 2024; Z. Liang and Luo 2025) is cited for terminology and as background in Remarks 2.3, 2.5, and 2.6, but the load-bearing definitions and theorems here are proved within the paper, so these self-citations are not load-bearing. The skeptical concern about Condition (3) of Theorem 5.1, namely k'(t)<=0, is a genuine a-posteriori regularity condition and is explicitly stated as a hypothesis; the proof uses it to justify applying Ito's formula across the discontinuous g*_y. This makes the theorem conditional, not circular: the paper does not claim k'<=0 follows from the other assumptions, nor does it rename a fitted quantity as a prediction. Remark 5.2 exhibits only a small-alpha_2 parameter regime where the condition holds, which is a limitation of scope, not a circular step. Thus there is no self-definitional step, no fitted input called a prediction, and no reduction of the core result to a citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

No parameters are fitted to data. The model uses exogenous constants and functions (a_L, a_H, σ_H, b, γ, θ, ρ, m, α1, α2, c0, c1) as inputs; the illustrative parameter specification in Remark 5.2 is a consistency check, not a fitted quantity. No new particles, forces, or ad hoc state variables are introduced; the free boundary k(t) is derived from the solution of the HJB system. The most notable added structure is the tie-breaking rule of Assumption 2.1 and the monotonicity condition on k(t), both of which are axioms of the construction rather than consequences of the model.

axioms (6)
  • standard math Standard Itô calculus and SDE well-posedness; Lipschitz/growth conditions (2.11) ensure unique strong solution (Proposition 2.1).
    Invoked throughout the proof of the verification theorem to justify Itô's formula, martingale arguments, and integrability.
  • ad hoc to paper Tie-breaking rule Assumption 2.1: on indifference, the controller selects the maximal admissible intervention.
    This is not derived from preferences; it is a convention needed to define a unique terminal condition and closed-form equilibria. A different tie-break would change the boundary condition.
  • domain assumption Compact upper bound m on the singular control is necessary to avoid unbounded terminal descent (Remark 2.1).
    Without m, the cost functional may be unbounded below at T, making the HJB boundary condition ill-posed.
  • domain assumption Assumption 5.1: σ_L=0, a_L-a_H≥α2 m, positive cost coefficients, and θc0(T)≥1.
    These restrictions make the reinsurance example tractable and exclude trivial regimes; they are not consequences of the general framework.
  • domain assumption Smoothness assumptions C^{2,1,1} in Theorems 3.1 and 4.1; the explicit solution relaxes this to C^{2,1,0} with a discontinuous g_y at the free boundary.
    The necessary and sufficient characterizations require smoothness in x, t, y. The explicit solution only satisfies weaker regularity, requiring a separate patch argument.
  • ad hoc to paper Condition k'(t)≤0 in Theorem 5.1(3) is assumed, not derived from primitives.
    This monotonicity is used to argue that singular purchases occur only at the initial time, making Itô's formula valid despite the discontinuous g_y. If k is not monotone, the verification in Step 2 of Theorem 5.1 breaks down.

pith-pipeline@v1.3.0-daily-deepseek · 34774 in / 12631 out tokens · 117729 ms · 2026-08-01T11:48:25.354289+00:00 · methodology

0 comments
read the original abstract

This paper studies a class of mixed regular-singular control problems under mean-variance criteria. We seek time-consistent equilibrium strategies in an intrapersonal game setting and propose a novel equilibrium notion. Under which, we derive a verification theorem and necessary conditions providing a full mathematical characterization of the equilibrium. To illustrate the theory, we construct explicit coupled equilibrium solutions for a reinsurance problem, where the regular control depends on the singular control state, and the free boundary of the singular control switches dynamically in accordance with the variation of the regular control expression, yielding nontrivial coupling. In the degenerate case $\alpha_2=0$, the coupled solution reduces to the combination of two independent single-control equilibria and coincides with the limit as the parameter tends to zero.

Figures

Figures reproduced from arXiv: 2607.26635 by Jiayu Zhang, Xiaodong Luo, Zongxia Liang.

Figure 1
Figure 1. Figure 1: Space partitions jointly induced by the regular switching boundary ˆy [PITH_FULL_IMAGE:figures/full_fig_p025_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Time-singular control space partition exhibiting a structural transition at the intersection epoch [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

50 extracted references · 1 canonical work pages

  1. [1]

    SIAM Journal on Control and Optimization , volume=

    Equilibria for time-inconsistent singular control problems , author=. SIAM Journal on Control and Optimization , volume=. 2024 , publisher=

    2024

  2. [2]

    Mathematical Finance , volume=

    Weak equilibria for time-inconsistent control: With applications to investment-withdrawal decisions , author=. Mathematical Finance , volume=. 2023 , publisher=

    2023

  3. [3]

    SIAM Journal on Financial Mathematics , volume=

    Stackelberg reinsurance and premium decisions with MV criterion and irreversibility , author=. SIAM Journal on Financial Mathematics , volume=. 2025 , publisher=

    2025

  4. [4]

    Finance and Stochastics , volume=

    On time-inconsistent stochastic control in continuous time , author=. Finance and Stochastics , volume=. 2017 , publisher=

    2017

  5. [5]

    2021 , publisher=

    Time-inconsistent Control Theory with Finance Applications , author=. 2021 , publisher=

    2021

  6. [6]

    Journal of Risk and Insurance , volume=

    Moral hazard in reinsurance markets , author=. Journal of Risk and Insurance , volume=

  7. [7]

    Martin, Catherine Donati and Lejay, Antoine and Rouault, Alain , year=. S

  8. [8]

    Applied Mathematics and Optimization , volume=

    Generalized solution in singular stochastic control: the nondegenerate problem , author=. Applied Mathematics and Optimization , volume=. 1992 , publisher=

    1992

  9. [9]

    2003 , publisher=

    Stochastic Differential Equations: An Introduction with Applications , author=. 2003 , publisher=

    2003

  10. [10]

    Applied Mathematics and Optimization , volume=

    Continuous-time mean-variance portfolio selection: A stochastic LQ framework , author=. Applied Mathematics and Optimization , volume=. 2000 , publisher=

    2000

  11. [11]

    Available at SSRN 1694759 , year=

    A general theory of Markovian time inconsistent stochastic control problems , author=. Available at SSRN 1694759 , year=

  12. [12]

    The 2002 International Conference on Control and Automation, 2002

    A diffusion model for optimal dividend distribution for a company with constraints on risk control , author=. The 2002 International Conference on Control and Automation, 2002. ICCA. Final Program and Book of Abstracts. , pages=. 2002 , organization=

    2002

  13. [13]

    Mathematics of Operations Research , volume=

    Dynamic optimal reinsurance and dividend payout in finite time horizon , author=. Mathematics of Operations Research , volume=. 2023 , publisher=

    2023

  14. [14]

    arXiv preprint arXiv:2604.02249 , year=

    Analytic Optimal Control for a Class of Driftless x-Flat Systems , author=. arXiv preprint arXiv:2604.02249 , year=

    Pith/arXiv arXiv

  15. [15]

    Applied Mathematics & Optimization , volume=

    Dynamic programming principle for classical and singular stochastic control with discretionary stopping , author=. Applied Mathematics & Optimization , volume=. 2023 , publisher=

    2023

  16. [16]

    SIAM Journal on Control and Optimization , volume=

    Equilibrium strategies for singular dividend control problems under the mean-variance criterion , author=. SIAM Journal on Control and Optimization , volume=

  17. [17]

    SIAM Journal on Control and Optimization , year=

    He, Xiangyu and Jiang, Zhaoli , title=. SIAM Journal on Control and Optimization , year=

  18. [18]

    The Review of Economic Studies , volume=

    Myopia and inconsistency in dynamic utility maximization , author=. The Review of Economic Studies , volume=. 1955 , publisher=

    1955

  19. [19]

    arXiv preprint math/0604264 , year=

    Being serious about non-commitment: subgame perfect equilibrium in continuous time , author=. arXiv preprint math/0604264 , year=

    Pith/arXiv arXiv

  20. [20]

    arXiv preprint arXiv:0808.3790 , year=

    Equilibrium policies when preferences are time inconsistent , author=. arXiv preprint arXiv:0808.3790 , year=

    Pith/arXiv arXiv

  21. [21]

    AIMS Math , volume=

    Robust optimal investment-reinsurance strategies with the preferred reinsurance level of reinsurer , author=. AIMS Math , volume=

  22. [22]

    The Review of Financial Studies , volume=

    Dynamic mean-variance asset allocation , author=. The Review of Financial Studies , volume=. 2010 , publisher=

    2010

  23. [23]

    Management Science , volume=

    A model of casino gambling , author=. Management Science , volume=. 2012 , publisher=

    2012

  24. [24]

    The Journal of Finance , year =

    Markowitz, Harry , title =. The Journal of Finance , year =. doi:10.1111/j.1540-6261.1952.tb01525.x , issn =

    arXiv 1952

  25. [25]

    Management Science , year =

    Bajeux-Besnainou, Isabelle and Portait, Roland , title =. Management Science , year =. doi:10.1287/mnsc.44.11.S79 , issn =

    work page doi:10.1287/mnsc.44.11.s79

  26. [26]

    Mathematical Finance , volume=

    Optimal dynamic portfolio selection: Multiperiod mean-variance formulation , author=. Mathematical Finance , volume=. 2000 , publisher=

    2000

  27. [27]

    Operations Research Letters , pages=

    Dynamic mean-variance hybrid portfolio optimization with spectral risk regulation , author=. Operations Research Letters , pages=. 2026 , publisher=

    2026

  28. [28]

    Journal of the operations research society of china , volume=

    Survey on multi-period mean--variance portfolio selection model , author=. Journal of the operations research society of china , volume=. 2022 , publisher=

    2022

  29. [29]

    SIAM Journal on Control and Optimization , volume=

    Time-inconsistent stochastic linear--quadratic control , author=. SIAM Journal on Control and Optimization , volume=. 2012 , publisher=

    2012

  30. [30]

    SIAM Journal on Control and Optimization , volume=

    Time-inconsistent stochastic linear-quadratic control: Characterization and uniqueness of equilibrium , author=. SIAM Journal on Control and Optimization , volume=. 2017 , publisher=

    2017

  31. [31]

    SIAM Journal on Financial Mathematics , volume=

    Time-consistent mean-variance pairs-trading under regime-switching cointegration , author=. SIAM Journal on Financial Mathematics , volume=. 2019 , publisher=

    2019

  32. [32]

    The ANZIAM Journal , volume=

    Mean--variance equilibrium asset-liability management strategy with cointegrated assets , author=. The ANZIAM Journal , volume=. 2020 , publisher=

    2020

  33. [33]

    Scandinavian Actuarial Journal , volume=

    Time-consistent mean-variance reinsurance-investment strategy for insurers under CEV model , author=. Scandinavian Actuarial Journal , volume=. 2016 , publisher=

    2016

  34. [34]

    arXiv preprint arXiv:2502.11052 , year=

    Time-consistent portfolio selection with strictly monotone mean-variance preference , author=. arXiv preprint arXiv:2502.11052 , year=

    Pith/arXiv arXiv

  35. [35]

    Mathematical Finance , volume=

    Mean--variance portfolio optimization with state-dependent risk aversion , author=. Mathematical Finance , volume=. 2014 , publisher=

    2014

  36. [36]

    arXiv preprint arXiv:2507.04836 , year=

    Time-inconsistent singular control problems: Reflection and Absolutely continuous controls with exploding rates , author=. arXiv preprint arXiv:2507.04836 , year=

    Pith/arXiv arXiv

  37. [37]

    , title =

    Liang, Xiaoqing and Young, Virginia R. , title =. Journal of Applied Probability , year =

  38. [38]

    2026 , journal =

    Time-consistent annuitization and asset allocation under the mean-variance criterion , author =. 2026 , journal =

    2026

  39. [39]

    2024 , note=

    Dynamic Mean-Variance Portfolio Selection with Transaction Costs , author=. 2024 , note=

    2024

  40. [40]

    Neurocomputing , volume=

    A McKean--Vlasov optimal mixed regular-singular control problem for nonlinear stochastic systems with Poisson jump processes , author=. Neurocomputing , volume=. 2016 , publisher=

    2016

  41. [41]

    Stochastic Processes and their Applications , volume=

    A constrained non-linear regular-singular stochastic control problem, with applications , author=. Stochastic Processes and their Applications , volume=. 2004 , publisher=

    2004

  42. [42]

    arXiv preprint arXiv:2606.25461 , year=

    Equilibrium singular dividend control under ambiguity aggregation of heterogeneous discount rates , author=. arXiv preprint arXiv:2606.25461 , year=

    Pith/arXiv arXiv

  43. [43]

    arXiv preprint arXiv:2605.19914 , year=

    Time-Inconsistent Singular Control Problems with a Running Minimum Process , author=. arXiv preprint arXiv:2605.19914 , year=

    Pith/arXiv arXiv

  44. [44]

    SIAM Journal on Financial Mathematics , volume=

    Equilibrium mean-variance dividend rate strategies , author=. SIAM Journal on Financial Mathematics , volume=. 2025 , publisher=

    2025

  45. [45]

    Applied Mathematics & Optimization , volume=

    Inconsistent investment and consumption problems , author=. Applied Mathematics & Optimization , volume=. 2015 , publisher=

    2015

  46. [46]

    2014 , journal =

    On dividend strategies with non-exponential discounting , author =. 2014 , journal =

    2014

  47. [47]

    2016 , journal =

    Equilibrium dividend strategy with non-exponential discounting in a dual model , author =. 2016 , journal =

    2016

  48. [48]

    2021 , journal =

    A free boundary problem of liquidity management for optimal dividend and insurance in finite horizon , author =. 2021 , journal =

    2021

  49. [49]

    2023 , journal =

    Optimal risk sharing and dividend strategies under default contagion: A semi-analytical approach , author =. 2023 , journal =

    2023

  50. [50]

    Insurance: Mathematics and Economics , volume=

    Optimal risk and dividend control for a company with debt liability , author=. Insurance: Mathematics and Economics , volume=