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REVIEW 2 major objections 4 minor 80 references

For a reflected max-payoff stopping problem the gain is a signed measure with a negative diagonal surface term, and the value is a killed resolvent of that measure.

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2026-07-14 01:13 UTC pith:OONZ7YUG

load-bearing objection Clean technical corrections on measure-valued stopping gains and killed resolvents for reflected max-payoff stopping; conditional but carefully framed. the 2 major comments →

arxiv 2607.09987 v1 pith:OONZ7YUG submitted 2026-07-10 math.AP math.PR

Reflected Optimal Stopping with a Max-Type Payoff: Measure-Valued Stopping Gains and Killed Resolvent Representation

Ye Liang This is my paper
classification math.AP math.PR MSC 60G4060J6060J5535R3549L25
keywords optimal stoppingreflected diffusionobstacle problemviscosity solutionsigned measurekilled resolventmax-type payofffree boundary
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.

The paper studies when to stop a two-dimensional diffusion that is pushed back into the positive quadrant, when the reward is the maximum of the two coordinates (scaled). Reflection, a genuinely two-dimensional exercise region, and the kink of the max payoff together break the usual free-boundary shortcuts. The author shows that the natural stopping-gain object is not a function but a signed measure whose singular part lives on the kink line and is always nonpositive. Because of that singularity, pointwise sign conditions on the gain must be read carefully, and the value admits a representation only as a resolvent killed at the first entry into the stopping set, not as the unrestricted reflected resolvent. A verification theorem and a conditional epigraph description of the stopping set are proved under explicit measure-superharmonicity and admissibility hypotheses, so the singular geometry and the killing are treated as first-class objects rather than formalities.

Core claim

For the infinite-horizon problem of optimally stopping a normally reflected diffusion in the quadrant with payoff G = x1 ∨ α x2, the stopping-gain object Γ = c + rG - LG is the signed measure Γ = Γ_ac dx + Γ^Δ, where the diagonal component is the nonpositive surface measure Γ^Δ(dx) = -(n⊤ a(x)n)/(2√(1+α²)) σ_Δ(dx) with n = (1,-α). The value is given by the killed-resolvent formula V = G - R_C^r Γ rather than by any unrestricted reflected resolvent of the gain.

What carries the argument

The signed stopping-gain measure Γ = Γ_ac dx + Γ^Δ together with the killed resolvent R_C^r Γ accumulated only up to the first hitting time of the stopping set; these two objects replace ordinary pointwise gain conditions and unrestricted resolvents in the verification and representation theorems.

Load-bearing premise

The verification and candidate-boundary results rest on an explicit admissibility assumption that a generalized Itô formula holds for the candidate, together with global measure-superharmonicity and the requirement that the contact set meets the kink line only on a null set.

What would settle it

In the constant-coefficient reflected Brownian case, compute the distributional second derivative of G across the diagonal and check whether the resulting surface measure equals exactly -(q/(2√(1+α²)))σ_Δ with q = n⊤ a n; any different coefficient or a vanishing singular part would refute the claimed decomposition.

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

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 / 4 minor

Summary. The paper studies infinite-horizon optimal stopping for a normally reflected diffusion in the quadrant with max-type payoff G=x1∨αx2. It formulates the reflected obstacle problem, proves a verification theorem under explicit Itô–Krylov–Tanaka admissibility and measure-superharmonicity (Theorem 3.2), and derives a conditional epigraph structure for the stopping set from monotonicity of H=V−G (Theorem 3.7). The central technical contributions are the signed-measure decomposition of the stopping gain Γ=c+rG−LG, with explicit nonpositive diagonal surface measure Γ^Δ(dx)=−(n⊤a(x)n)/(2√(1+α^{2}))σ_Δ(dx) (Theorem 3.10), and the killed-resolvent representation V=G−R_C^r Γ rather than the unrestricted reflected resolvent (Theorem 3.13). A candidate-boundary verification theorem and a constant-coefficient RBM example complete the development.

Significance. If the conditional claims hold under the stated hypotheses, the paper supplies a useful structural correction for multi-dimensional reflected optimal stopping with nonsmooth max-type rewards: the kink of G produces a signed surface measure that must be separated from the absolutely continuous part, and the potential must be killed at first entry into the stopping set. The explicit Tanaka–co-area formula for Γ^Δ and the resolvent decomposition (Remark 3.14) are clean and reusable. The verification and candidate-boundary theorems are carefully framed so that regularity is not smuggled in from viscosity status alone. The constant-coefficient example shows the singular term is intrinsic, not an artifact of variable coefficients. The work is conditional rather than unconditional free-boundary theory, but the conditional structure is a genuine contribution to the literature on reflected obstacle problems and multi-asset American-type payoffs.

major comments (2)
  1. [§3.1, Remark 3.3; §3.6, Remark 3.19] Theorem 3.2 (V4) and Remark 3.3 / eq. (3.1): under local uniform ellipticity the verification theorem is compatible with μ_u≤0 only when the contact set intersects Δ in zero σ_Δ-measure. The same obstruction reappears for candidate boundaries in Remark 3.19. The paper correctly flags this, but the main free-boundary claims then apply only to configurations that keep the kink in the continuation region (up to null sets). The manuscript should state more prominently, already in the introduction and abstract, that the verification/candidate results exclude positive-length contact along Δ under ellipticity, and should indicate whether any nontrivial model with non-constant coefficients is known to satisfy the diagonal-compatibility condition.
  2. [Assumption 3.1; Assumption 2.1; Theorem 3.13] Assumption 3.1 (Itô–Krylov–Tanaka admissibility) and Assumption 2.1 (reflected SDE) are standing hypotheses, not theorems. The verification and killed-resolvent results therefore remain conditional on Sobolev regularity, signed-measure extension of (L−r)u−c, nonpositive boundary contribution, and UI. For a journal in analysis/probability this is acceptable if clearly labeled, but the paper should either supply at least one non-constant-coefficient example where these are verified, or add a short subsection that isolates the minimal coefficient conditions under which the signed additive functional of Γ and the stopped formula for G are known to exist.
minor comments (4)
  1. [References] The bibliography contains many 2025–26 arXiv items (e.g. [3], [5], [8], [13]–[15], [19], [23]–[25], [28]–[29], [31]–[33], [37], [39], [44], [47], [50]–[51], [54], [56], [59], [61], [66]–[67], [69]–[74], [76]–[77], [79]) that are only loosely related to reflected optimal stopping or free-boundary analysis. Trimming or relocating peripheral self-citations would improve focus.
  2. [§2.4, §3.4] Notation for the killed resolvent alternates between R_C^r and R^C_r; a single consistent form would help. Likewise, the surface measure is written both σ_Δ and σ_Δ(dx).
  3. [Appendix A.5] In the constant-coefficient example (Appendix A.5), the isotropic reduction Γ^Δ(dz)=−((1+α^{2})/2) dy is useful; a one-line comparison with the classical one-dimensional Tanaka formula for |x| would make the factor transparent to non-specialists.
  4. [Throughout] Typographical consistency: “Itô–Krylov–Tanaka” vs “Itô-Krylov-Tanaka”; occasional missing spaces after punctuation in the arXiv source.

Circularity Check

0 steps flagged

No circularity: singular-measure and killed-resolvent claims are direct Tanaka/co-area and stopped-Itô derivations under named hypotheses, not forced by fit or self-citation.

full rationale

The load-bearing claims (Theorems 3.10 and 3.13) are derived from standard semimartingale calculus and potential theory under explicit assumptions, not from definitions that already encode the conclusions. Theorem 3.10 obtains Γ^Δ from Tanaka’s formula for Y=X1−αX2, the identity G=(x1+αx2+|Y|)/2, occupation density, and co-area; the sign and surface factor follow by construction from those identities, which are independent inputs, not a re-labeling of the target. Theorem 3.13 stops the Itô–Tanaka formula for G at τ_D and rearranges under optimality and Assumption 3.12; the killed-versus-unrestricted distinction is an algebraic decomposition (Remark 3.14), not a fitted prediction. Verification (Theorem 3.2) and candidate-boundary verification (Theorem 3.17) are deliberately conditional on Itô–Krylov–Tanaka admissibility, measure-superharmonicity, and the diagonal compatibility obstruction (Remark 3.3 / (3.1)); the paper does not claim those hypotheses follow from viscosity status. The epigraph theorem is a deterministic consequence of assumed monotonicity of H=V−G. Bibliography self-citations (if any) are not load-bearing for these steps; classical references (Tanaka, Revuz–Yor, Peskir–Shiryaev, Crandall–Ishii–Lions) supply the tools. No fitted parameter is renamed a prediction, and no uniqueness theorem of the same author is imported to force the form of Γ or V. Score 0 is therefore the correct honest finding.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The paper is pure theory; load-bearing content is a collection of standing domain assumptions on the reflected diffusion, explicit admissibility hypotheses for the verification theorem, and structural monotonicity for the epigraph result. No numerical free parameters are fitted. No new physical entities are postulated.

axioms (6)
  • domain assumption Existence, uniqueness, strong Markov property and continuous dependence for the normally reflected SDE in the quadrant (Assumption 2.1 A1–A4)
    Taken as standing; not proved in the paper. Invoked for all probabilistic representations.
  • ad hoc to paper Itô–Krylov–Tanaka admissibility of the candidate (signed-measure extension of (L−r)u−c, stopped formula, nonpositive boundary contribution, UI) — Assumption 3.1
    Required for Theorem 3.2; not automatic from viscosity supersolution status (Remark 3.4).
  • ad hoc to paper Global measure-superharmonicity μ_u ≤ 0 as a signed measure (V4) and diagonal compatibility σ_Δ((R²_{++}\O)∩Δ)=0 under local ellipticity
    Forced by the positive singular part of −Γ on the contact set (Remark 3.3); without it verification fails.
  • domain assumption Vertical (and optionally horizontal) monotonicity of the stopping advantage H=V−G (Assumption 3.6)
    Needed for the epigraph theorem; paper states it must be verified separately in concrete models.
  • domain assumption No corner reflection contribution for G (Assumption A.11 / vanishing of local time at (0,0))
    Used to keep Γ = Γ_ac dx + Γ^Δ without an extra corner measure.
  • domain assumption Polynomial growth of V and integrability of discounted payoff/cost terms (A6); Lyapunov condition sufficient but not necessary
    Standing for the value to be finite and for passage to the limit in verification.

pith-pipeline@v1.1.0-grok45 · 35583 in / 3197 out tokens · 30429 ms · 2026-07-14T01:13:49.721939+00:00 · methodology

0 comments
read the original abstract

We study an infinite-horizon optimal stopping problem for a two-dimensional normally reflected diffusion in the quadrant with payoff \(G(x_1,x_2)=x_1\vee \alpha x_2\). The problem combines three features that complicate the usual free-boundary analysis: reflection on the coordinate axes, a genuinely two-dimensional stopping region, and a nonsmooth max-type reward. We formulate the associated reflected obstacle problem, prove a verification theorem under explicit It\^o--Krylov--Tanaka admissibility and measure-superharmonicity assumptions, and derive a conditional epigraph structure for the stopping set. The main technical point is that the stopping-gain object \(\Gamma=c+rG-\mathcal LG\) is a signed measure rather than a function. Its diagonal component is $\Gamma^\Delta(dx) = -\frac{n^\top a(x)n}{2\sqrt{1+\alpha^2}}\sigma_\Delta(dx)$, $n=(1,-\alpha)$, which shows that pointwise stopping-gain sign conditions must be interpreted with care. We also prove that the correct potential representation is the killed-resolvent formula $V(x)=G(x)-R_r^{\mathcal C}\Gamma(x)$, rather than the unrestricted reflected resolvent. A constant-coefficient reflected Brownian example illustrates the diagonal singular term explicitly.

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Works this paper leans on

80 extracted references · 19 linked inside Pith

  1. [1]

    Mathematical Finance7(3), 241–286 (1997)

    Broadie, M., Detemple, J.: The valuation of american options on multiple assets. Mathematical Finance7(3), 241–286 (1997)

    1997

  2. [2]

    Detemple, J.: Optimal exercise for derivative securities. Annu. Rev. Financ. Econ. 6(1), 459–487 (2014)

    2014

  3. [3]

    Results in Applied Mathematics27, 100621 (2025)

    Wang, L.S., Yu, J.: Analysis framework for stochastic predator–prey model with demographic noise. Results in Applied Mathematics27, 100621 (2025)

    2025

  4. [4]

    Chapman and Hall/CRC (2005)

    Detemple, J.: American-style Derivatives: Valuation and Computation. Chapman and Hall/CRC (2005)

    2005

  5. [5]

    arXiv preprint arXiv:2607.03813 (2026)

    Liu, Z., Yu, J., Su, L., Wang, L.S., Du, Y., Liu, J.: Computational oncol- ogy of chemotaxis-driven tumour–immune spatial patterning and stability. arXiv preprint arXiv:2607.03813 (2026)

    Pith/arXiv arXiv 2026

  6. [6]

    Finance and Stochastics3(3), 295–322 (1999)

    Villeneuve, S.: Exercise regions of american options on several assets. Finance and Stochastics3(3), 295–322 (1999)

    1999

  7. [7]

    Karatzas, I., Shreve, S.E., Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance vol. 39. Springer (1998)

    1998

  8. [8]

    Mathematics13(17), 2898 (2025)

    Wang, L.S., Yu, J., Li, S., Liu, Z.: Analysis and mean-field limit of a hybrid pde- abm modeling angiogenesis-regulated resistance evolution. Mathematics13(17), 2898 (2025)

    2025

  9. [9]

    Opera- tions Research52(2), 258–270 (2004)

    Haugh, M.B., Kogan, L.: Pricing american options: A duality approach. Opera- tions Research52(2), 258–270 (2004)

    2004

  10. [10]

    International Journal of Theoretical and Applied Finance 12(03), 319–340 (2009)

    Khaliq, A.Q., Liu, R.: New numerical scheme for pricing american option with regime-switching. International Journal of Theoretical and Applied Finance 12(03), 319–340 (2009)

    2009

  11. [11]

    Mathematical Methods in the Applied Sciences46(9), 10332–10347 (2023)

    Company, R., Egorova, V.N., J´ odar, L.: An etd method for multi-asset american option pricing under jump-diffusion model. Mathematical Methods in the Applied Sciences46(9), 10332–10347 (2023)

    2023

  12. [12]

    In: Capi- tal Market Finance: An Introduction to Primitive Assets, Derivatives, Portfolio Management and Risk, pp

    Poncet, P., Portait, R.: American options and numerical methods. In: Capi- tal Market Finance: An Introduction to Primitive Assets, Derivatives, Portfolio Management and Risk, pp. 501–548. Springer (2022)

    2022

  13. [13]

    arXiv preprint arXiv:2607.03790 (2026)

    Wang, L.S., Yu, J.: Optimal harvesting of size-structured populations with envi- ronmental feedback and fixed recruitment flux. arXiv preprint arXiv:2607.03790 (2026)

    Pith/arXiv arXiv 2026

  14. [14]

    36 arXiv preprint arXiv:2607.02877 (2026)

    Yu, J., Wang, L.S.: Endogenous feedback in size-structured transport equations. 36 arXiv preprint arXiv:2607.02877 (2026)

    Pith/arXiv arXiv 2026

  15. [15]

    arXiv preprint arXiv:2607.01403 (2026)

    Yu, J., Wang, L.S.: Fredholm–residue selection of the unsteady kutta amplitude. arXiv preprint arXiv:2607.01403 (2026)

    Pith/arXiv arXiv 2026

  16. [16]

    Springer (2006)

    Peskir, G., Shiryaev, A.: Optimal Stopping and Free-boundary Problems. Springer (2006)

    2006

  17. [17]

    Springer (2008)

    Shiryaev, A.N.: Optimal Stopping Rules. Springer (2008)

    2008

  18. [18]

    Bensoussan, A., Lions, J.-L.: Applications of Variational Inequalities in Stochastic Control vol. 12. Elsevier (2011)

    2011

  19. [19]

    Bioengineering12(10), 1097 (2025)

    Liu, Z., Wang, L.S., Yu, J., Zhang, J., Martel, E., Li, S.: Bidirectional endothelial feedback drives turing-vascular patterning and drug-resistance niches: a hybrid pde-agent-based study. Bioengineering12(10), 1097 (2025)

    2025

  20. [20]

    Springer (1980)

    Krylov, N.V.: Controlled Diffusion Processes. Springer (1980)

    1980

  21. [21]

    Springer (2013)

    Oksendal, B.: Stochastic Differential Equations: an Introduction with Applica- tions. Springer (2013)

    2013

  22. [22]

    Pham, H.: Continuous-time Stochastic Control and Optimization with Financial Applications vol. 61. Springer (2009)

    2009

  23. [23]

    arXiv preprint arXiv:2607.01347 (2026)

    Yu, J., Wang, L.S.: Bilinear control of age–space structured populations. arXiv preprint arXiv:2607.01347 (2026)

    Pith/arXiv arXiv 2026

  24. [24]

    arXiv preprint arXiv:2606.25288 (2026)

    Wang, L.S.: Vital-rate feedback and threshold harvesting in size-structured populations. arXiv preprint arXiv:2606.25288 (2026)

    Pith/arXiv arXiv 2026

  25. [25]

    arXiv preprint arXiv:2606.12722 (2026)

    Wang, L.S.: Finite-sheeted cauchy operator at rational corners. arXiv preprint arXiv:2606.12722 (2026)

    Pith/arXiv arXiv 2026

  26. [26]

    Bulletin of the American mathematical society 27(1), 1–67 (1992)

    Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bulletin of the American mathematical society 27(1), 1–67 (1992)

    1992

  27. [27]

    Springer (2006)

    Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solu- tions. Springer (2006)

    2006

  28. [28]

    Mathematics13(22), 3583 (2025)

    Liang, Y., Wang, L.S., Yu, J., Liu, Z.: Global well-posedness and stability of nonlocal damage-structured lineage model with feedback and dedifferentiation. Mathematics13(22), 3583 (2025)

    2025

  29. [29]

    arXiv preprint arXiv:2604.15690 (2026) 37

    Yu, J.: Optimization workshop notes for mathematical programming with equi- librium constraints algorithms: Penalty interior-point, implicit-programming, and piecewise sqp. arXiv preprint arXiv:2604.15690 (2026) 37

    Pith/arXiv arXiv 2026

  30. [30]

    Stochastics and Stochastic Reports62(3-4), 285–301 (1998)

    Reikvam, K.: Viscosity solutions of optimal stopping problems. Stochastics and Stochastic Reports62(3-4), 285–301 (1998)

    1998

  31. [31]

    arXiv preprint arXiv:2605.00388 (2026)

    Wang, L.S.: First-order optimality conditions for mathematical programming with equilibrium constraints. arXiv preprint arXiv:2605.00388 (2026)

    Pith/arXiv arXiv 2026

  32. [32]

    arXiv preprint arXiv:2605.00387 (2026)

    Wang, L.S.: Introduction to exact penalization for mathematical programming with equilibrium constraints. arXiv preprint arXiv:2605.00387 (2026)

    Pith/arXiv arXiv 2026

  33. [33]

    arXiv preprint arXiv:2605.30438 (2026)

    Yu, J., Wang, L.S.: Structural dichotomy and mass criticality in indirect chemo- taxis cascades: fourth-order ellipticity versus volterra memory. arXiv preprint arXiv:2605.30438 (2026)

    Pith/arXiv arXiv 2026

  34. [34]

    the Annals of Probability25(2), 702–737 (1997)

    El Karoui, N., Kapoudjian, C., Pardoux, E., Peng, S., Quenez, M.-C.: Reflected solutions of backward sde’s, and related obstacle problems for pde’s. the Annals of Probability25(2), 702–737 (1997)

    1997

  35. [35]

    Stochastic processes and their applications107(2), 173–212 (2003)

    Dayanik, S., Karatzas, I.: On the optimal stopping problem for one-dimensional diffusions. Stochastic processes and their applications107(2), 173–212 (2003)

    2003

  36. [36]

    The Annals of Applied Probability29(1), 505–530 (2019)

    Peskir, G.: Continuity of the optimal stopping boundary for two-dimensional diffusions. The Annals of Applied Probability29(1), 505–530 (2019)

    2019

  37. [37]

    arXiv preprint arXiv:2604.20988 (2026)

    Yu, J.: Optimization workshop notes for mathematical programming with equi- librium constraints (mpecs): Verification of mpec hypotheses. arXiv preprint arXiv:2604.20988 (2026)

    Pith/arXiv arXiv 2026

  38. [38]

    Stochastic Processes and their Applications129(7), 2561–2581 (2019)

    Christensen, S., Crocce, F., Mordecki, E., Salminen, P.: On optimal stopping of multidimensional diffusions. Stochastic Processes and their Applications129(7), 2561–2581 (2019)

    2019

  39. [39]

    arXiv preprint arXiv:2604.20992 (2026)

    Yu, J.: Optimization workshop notes for mathematical programming with equi- librium constraints (mpecs): Second-order optimality conditions. arXiv preprint arXiv:2604.20992 (2026)

    Pith/arXiv arXiv 2026

  40. [40]

    Communications on Pure and Applied Mathematics24(2), 147–225 (1971)

    Stroock, D.W., Varadhan, S.S.: Diffusion processes with boundary conditions. Communications on Pure and Applied Mathematics24(2), 147–225 (1971)

    1971

  41. [41]

    Communications on pure and applied Mathematics37(4), 511–537 (1984)

    Lions, P.-L., Sznitman, A.-S.: Stochastic differential equations with reflecting boundary conditions. Communications on pure and applied Mathematics37(4), 511–537 (1984)

    1984

  42. [42]

    Stochastics: An International Journal of Probability and Stochastic Processes35(1), 31–62 (1991)

    Dupuis, P., Ishii, H.: On lipschitz continuity of the solution mapping to the skorokhod problem, with applications. Stochastics: An International Journal of Probability and Stochastic Processes35(1), 31–62 (1991)

    1991

  43. [43]

    Stochastic Processes: Selected Papers of Hiroshi Tanaka9, 157 38 (1979)

    Tanaka, H.: Stochastic differential equations with reflecting boundary condition in convex regions. Stochastic Processes: Selected Papers of Hiroshi Tanaka9, 157 38 (1979)

    1979

  44. [44]

    Electronic Research Archive34(1), 251–290 (2026)

    Wang, L.S., Yu, J.: Algebraic–spectral thresholds and discrete–continuous stabil- ity transfer in leslie–gower systems. Electronic Research Archive34(1), 251–290 (2026)

    2026

  45. [45]

    The Annals of Probability9(2), 302–308 (1981)

    Harrison, J.M., Reiman, M.I.: Reflected brownian motion on an orthant. The Annals of Probability9(2), 302–308 (1981)

    1981

  46. [46]

    Probability Theory and Related Fields74(3), 455–477 (1987)

    Saisho, Y.: Stochastic differential equations for multi-dimensional domain with reflecting boundary. Probability Theory and Related Fields74(3), 455–477 (1987)

    1987

  47. [47]

    Electronic Research Archive34(6), 4248–4289 (2026)

    Wang, L.S., Yu, J., Liang, Y., Zhang, J.: The breakdown of linear quasi-cycles: Demographic noise and absorbing boundaries in finite predator–prey systems. Electronic Research Archive34(6), 4248–4289 (2026)

    2026

  48. [48]

    Probability Theory and Related Fields96(3), 283–317 (1993)

    Taylor, L.M., Williams, R.J.: Existence and uniqueness of semimartingale reflect- ing brownian motions in an orthant. Probability Theory and Related Fields96(3), 283–317 (1993)

    1993

  49. [49]

    The Annals of Probability, 2971–3037 (2017)

    Burdzy, K., Chen, Z.-Q., Marshall, D., Ramanan, K.: Obliquely reflected brown- ian motion in nonsmooth planar domains. The Annals of Probability, 2971–3037 (2017)

    2017

  50. [50]

    In: 2022 International Conference on Data Analytics, Computing and Artificial Intelligence (ICDACAI), pp

    Gao, Y., Li, L., Yu, J.: Rolling prediction model of closing price based on eemd data noise reduction and hgs-delm. In: 2022 International Conference on Data Analytics, Computing and Artificial Intelligence (ICDACAI), pp. 255–260 (2022). IEEE

    2022

  51. [51]

    Transport Phenomena1(2), 20260061 (2026)

    Wang, L.S., Yu, J., Liang, Y., Zhang, J.: Elliptic criticality versus volterra memory in indirect chemotaxis cascades. Transport Phenomena1(2), 20260061 (2026)

    2026

  52. [52]

    Journal of Theoretical Probability34(4), 2166–2191 (2021)

    Hino, M., Matsuura, K., Yonezawa, M.: Pathwise uniqueness and non-explosion property of skorohod sdes with a class of non-lipschitz coefficients and non-smooth domains. Journal of Theoretical Probability34(4), 2166–2191 (2021)

    2021

  53. [53]

    Stochastics, 1–27 (2026)

    Wei, R., Yang, S., Zhai, J.: Mckean–vlasov stochastic differential equations with oblique reflection on non-smooth time-dependent domains. Stochastics, 1–27 (2026)

    2026

  54. [54]

    Transport Phenomena 1(2), 20260037 (2026)

    Yu, J., Wang, L.S., Ban, S., Liang, Y.: From microscopic damage to macroscopic games: a dimensionality reduction of stem cell homeostasis. Transport Phenomena 1(2), 20260037 (2026)

    2026

  55. [55]

    The Annals of Probability, 115–137 39 (1987)

    Harrison, J.M., Williams, R.J.: Multidimensional reflected brownian motions having exponential stationary distributions. The Annals of Probability, 115–137 39 (1987)

    1987

  56. [56]

    arXiv preprint arXiv:2505.16904 (2025)

    Wang, L.S., Yu, J.: Well-posedness for the rosenzweig-macarthur model with internal stochasticity. arXiv preprint arXiv:2505.16904 (2025)

    Pith/arXiv arXiv 2025

  57. [57]

    Probability Theory and Related Fields75(4), 459–485 (1987)

    Williams, R.J.: Reflected brownian motion with skew symmetric data in a polyhedral domain. Probability Theory and Related Fields75(4), 459–485 (1987)

    1987

  58. [58]

    Probability Theory and Related Fields77(1), 87–97 (1988)

    Reiman, M.I., Williams, R.J.: A boundary property of semimartingale reflecting brownian motions. Probability Theory and Related Fields77(1), 87–97 (1988)

    1988

  59. [59]

    arXiv preprint arXiv:2605.00395 (2026)

    Yu, J.: Controlling the swarm: Sparse actuation and collision avoidance under stochastic delay. arXiv preprint arXiv:2605.00395 (2026)

    Pith/arXiv arXiv 2026

  60. [60]

    Kang, W., Williams, R.J.: An invariance principle for semimartingale reflecting brownian motions in domains with piecewise smooth boundaries (2007)

    2007

  61. [61]

    arXiv preprint arXiv:2605.16780 (2026)

    Yu, J.: A diagnostic framework for implementation risk in bilevel decision prob- lems: The ambiguity premium and the robustness–efficiency frontier. arXiv preprint arXiv:2605.16780 (2026)

    Pith/arXiv arXiv 2026

  62. [62]

    In: Mathematical Proceedings of the Cambridge Philosophical Society, vol

    Hobson, D.G., Rogers, L.C.: Recurrence and transience of reflecting brown- ian motion in the quadrant. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 113, pp. 387–399 (1993). Cambridge University Press

    1993

  63. [63]

    In: Annales de l’Institut Henri Poincar´ e-Probabilit´ es et Statistiques, vol

    Lipshutz, D., Ramanan, K.: Pathwise differentiability of reflected diffusions in convex polyhedral domains. In: Annales de l’Institut Henri Poincar´ e-Probabilit´ es et Statistiques, vol. 55, pp. 1439–1476 (2019)

    2019

  64. [64]

    Electronic Journal of Probability18(34), 1–49 (2013)

    Lamberton, D., Zervos, M.: On the optimal stopping of a one-dimensional diffusion. Electronic Journal of Probability18(34), 1–49 (2013)

    2013

  65. [65]

    Springer (2013)

    Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer (2013)

    2013

  66. [66]

    Mathematical Methods in the Applied Sciences (2026)

    Yu, J., Wang, L.S., Liang, Y.: Rigorous analysis of a nonlocal transport–renewal system for physiologically structured populations. Mathematical Methods in the Applied Sciences (2026)

    2026

  67. [67]

    Plos one 21(2), 0335163 (2026)

    Wang, L.S., Yu, J., Liu, Z.: A damage-structured pde model of stem cell hierar- chies: The dual role of dedifferentiation in tissue homeostasis and aging. Plos one 21(2), 0335163 (2026)

    2026

  68. [68]

    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes vol. 19. Walter de Gruyter (2011)

    2011

  69. [69]

    arXiv preprint arXiv:2604.05882 (2026) 40

    Wang, L.S.: Lecture note for bounded controls in continuous-time and control of several variables. arXiv preprint arXiv:2604.05882 (2026) 40

    Pith/arXiv arXiv 2026

  70. [70]

    arXiv preprint arXiv:2601.20204 (2026)

    Yu, J., Wang, L.S., Liu, Z., Liu, J.: Mode-wise spectral criteria for coupled mass transport in hybrid pde–ode tumor microenvironments. arXiv preprint arXiv:2601.20204 (2026)

    arXiv 2026

  71. [71]

    arXiv preprint arXiv:2605.00386 (2026)

    Wang, L.S.: Introduction to mathematical programming with equilibrium con- straints (mpecs) and bilevel optimization. arXiv preprint arXiv:2605.00386 (2026)

    Pith/arXiv arXiv 2026

  72. [72]

    Mathematics14(11), 2025 (2026)

    Jie, C., Chen, X., Longfei, G., Jiayao, C., Nuo, C., Wang, L.S., Ye, L., Jiguang, Y.: Optimal harvesting for nonlinear size-structured populations with nonlocal environmental feedback. Mathematics14(11), 2025 (2026)

    2025

  73. [73]

    arXiv preprint arXiv:2505.22580 (2025)

    Wang, L.S., Yu, J., Liu, Z.: A hybrid pde-abm model for angiogenesis and tumour microenvironment with application to resistance in cancer treatment. arXiv preprint arXiv:2505.22580 (2025)

    Pith/arXiv arXiv 2025

  74. [74]

    PLoS One21(5), 0350127 (2026)

    Yu, J., Wang, L.S.: Beyond diagonal noise: A better predator-prey modeling framework with cross-covariance. PLoS One21(5), 0350127 (2026)

    2026

  75. [75]

    Citeseer (2004)

    Protter, P.E.: Stochastic Integration and Differential equations. Citeseer (2004)

    2004

  76. [76]

    Transport Phenomena (0) (2026)

    Yu, J., Wang, L.S., Liu, Z., Liu, J.: Pattern suppression and recovery under one- way versus two-way chemotactic coupling in hybrid partial differential equation– ordinary differential equation models. Transport Phenomena (0) (2026)

    2026

  77. [77]

    arXiv preprint arXiv:2601.16337 (2026)

    Yu, J., Wang, L.S., Liu, Z., Liu, J.: Chemotactic feedback controls patterning in hybrid tumor–stroma model. arXiv preprint arXiv:2601.16337 (2026)

    arXiv 2026

  78. [78]

    springer (2014)

    Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus. springer (2014)

    2014

  79. [79]

    Information Technology and Control54(2), 413–438 (2025)

    Wang, Z., Wang, D., Yu, J.: Multi-strategy hybrid improved intelligent algorithm for solving uav-mtsp. Information Technology and Control54(2), 413–438 (2025)

    2025

  80. [80]

    Courier Corporation (2007) 41

    Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. Courier Corporation (2007) 41

    2007