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Exciting games and Monge-Amp\`ere equations

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Pith's one-line read Among all win-martingales on the $d$-dimensional simplex, the unique most-exciting game is the diffusion $dM_s = \sqrt{2(\nabla^2g(M_s))^{-1}/(1-s)}\,dB_s$, where $g$ is the unique smooth solution of the Monge-Amp\`ere equation with…

desk verdict Real multidimensional progress on the most exciting game, with a genuine Monge-Ampère connection; the uniqueness claim is asserted but not proved, and the comparison principle has a gap—both fixable. read the letter →

arxiv 2412.01995 v2 pith:FZ5YTF7C submitted 2024-12-02 math.PR math.AP

Julio Backhoff , Zhizhang Wang , Xin Zhang This is my paper · ORCID
classification math.PRmath.AP MSC 60G4435J9649L2560H10
keywords mostexcitinggamewin-martingalesMonge-AmpèreequationspecificrelativeentropystochasticoptimalcontrolviscositysolutionsAldousmartingaleShannonscalinglimit
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

What is the most suspenseful way to reveal the winner of a contest with $d+1$ players? This paper treats win probabilities as a martingale on the simplex and measures excitement by the time-integrated log-determinant of the volatility, the continuous-time scaling limit of Shannon entropy. It claims that the unique optimal martingale is an explicit diffusion whose volatility is the inverse Hessian of $g$, where $g$ solves the Monge-Amp\`ere equation $g = \log \det(\tfrac12\nabla^2 g)$ on the simplex and blows up on the boundary. Establishing this requires a fine analysis of the degenerate boundary behavior of that PDE, which is the paper's main technical work. If the identification of excitement with this entropy limit is accepted, the result describes exactly how a bookmaker or designer should update odds to keep an audience watching.

What carries the argument

The load-bearing object is the Monge-Amp\`ere equation with infinite boundary condition on the simplex: $g = \log \det(\tfrac12\nabla^2 g)$ for $x\in\Delta$ and $g(x)=\infty$ on $\partial\Delta$. The paper constructs $g$ by approximating the simplex with strictly convex sublevel sets of the explicit barrier $w(x)=-2\sum_i \log x_i -2\log(1-\sum_i x_i)$, applying comparison and Pogorelov-type estimates to obtain uniform $C^2$ bounds, and then sending the approximation to the limit. The crucial boundary analysis rescales coordinates near each face and uses the scaling invariance of the equation to show that $\nabla^2g$ behaves like $\mathrm{diag}(1/x_1^2,\dots,1/x_k^2,1,\dots,1)$; this controls the diffusion coefficient of the Aldous martingale and guarantees both non-exit and convergence to a vertex. The final identification of the optimizer is completed by showing that $\log\det\Sigma_*(t,M_t)$ is a true martingale via the global bound on $\nabla g^{\top}(\nabla^2g)^{-1}\nabla g$.

What would settle it

Starting at any fixed $x\in\Delta$ for $d\ge 2$, simulate the diffusion $dM_s = \sqrt{2(\nabla^2g(M_s))^{-1}/(1-s)}\,dB_s$ with a high-precision numerical solution of the Monge-Amp\`ere equation; if for some $x$ the terminal law is not concentrated on the vertices, meaning positive probability is observed on an open face of the simplex, Theorem 1.2 is false. Equivalently, exhibit any win-martingale with strictly smaller expected cost $\mathbb{E}\int -\log\det\Sigma\,ds$ than the diffusion achieves; the claimed uniqueness would fail.

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Extended reading notes

Core claim

The paper's central discovery is that the Aldous martingale, the diffusion $dM_s = \sqrt{2(\nabla^2g(M_s))^{-1}/(1-s)}\,dB_s$ started anywhere in the simplex, is the unique (in law) optimizer of the cost $\mathbb{E}\int_t^1 -\log\det(\Sigma_s)\,ds$ among all win-martingales. Here $g$ is the unique smooth solution of the Monge-Amp\`ere equation $g = \log \det(\tfrac12\nabla^2 g)$ on the interior of the simplex and $g = \infty$ on its boundary. Along the way, the paper proves new boundary asymptotics: the minimal eigenvalue of $\nabla^2g$ stays bounded as the state approaches a face of the simplex but not a vertex, and the quadratic form $\nabla g^{\top}(\nabla^2g)^{-1}\nabla g$ is bounded globally. These estimates are what let the authors construct the strong solution, show it never exits the simplex before time $1$, and prove that it lands on a vertex almost surely. The value function splits as $v(t,x)=(1-t)g(x)+d(1-t)\log(1-t)$, which reduces the parabolic Hamilton–Jacobi–Bellman equation to the elliptic Monge-Amp\`ere problem.

Load-bearing premise

The load-bearing premise is that Aldous's 'excitement' is quantified by the expected integral of $-\log\det(\Sigma_s)$ over time, coming from a scaling limit of Shannon entropies; that equivalence is imported from the earlier one-dimensional treatment rather than re-derived, so if a different notion of suspense is intended, the interpretation of the optimizer as the most exciting game would not follow even though the mathematical optimality result for this cost still stands.

Editorial extensions

If this is right

  • For $d=1$, the construction reduces to the known explicit diffusion $dM_s = \frac{\sin(\pi M_s)}{\pi\sqrt{1-s}}\,dB_s$, recovering the earlier solution of the most-exciting-game problem.
  • For any finite number of players, there now exists a unique optimal win-martingale, so the question of the most exciting game has a complete answer within this entropy-limit model.
  • The boundary estimates identify how the optimal volatility diverges near the boundary: as some coordinate $x_i\to 0$, the diffusion coefficient grows like $1/x_i$, which is exactly the singular strength that forces an eventual arrival at a vertex.
  • The same analysis shows that the value function is finite, convex, and the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation, so no second optimizer can beat the Aldous martingale.

Reading between the lines

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

  • Beyond the paper: the time-changed Aldous martingale has an invariant sigma-finite measure $e^g\,dx$ on the simplex, which suggests a mirror-Langevin sampler for densities supported on the simplex; the paper notes the moment-measure link but does not pursue sampling applications.
  • Beyond the paper: if the Shannon-type entropy criterion is replaced by another notion of suspense, the same HJB-to-elliptic reduction may survive in modified form, but the singular boundary analysis would have to be reworked for each new cost.
  • Beyond the paper: for $d=2$, a high-precision numerical solution of the Monge-Amp\`ere equation could be used to simulate the Aldous martingale and directly test the prediction that terminal mass concentrates on vertices rather than on open faces of the simplex.
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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

2 major / 4 minor

Summary. The paper studies the continuous-time "most exciting game" problem for d+1 players. A win-martingale is a martingale on the d-dimensional subprobability simplex that terminates at a vertex, and the cost is E∫_t^1 −log det(Σ_s) ds. The authors prove that the value function has the structure (1−t)g(x)+d(1−t)log(1−t), with g solving the Monge-Ampère equation g=log det(1/2 ∇²g) on the simplex with infinite boundary data. They establish existence, uniqueness, regularity and boundary estimates for g (Theorem 1.1), use these to construct the "Aldous martingale" as the strong solution to dM_s = (2(∇²g(M_s))^{-1}/(1−s))^{1/2} dB_s, prove that it is a win-martingale and attains the value function, and assert that it is the unique optimizer in law (Theorem 1.2). The candidate is parameter-free, determined by a PDE with no free constants, and the verification is via the martingale property of log det of the optimal covariance (Lemma 6.4).

Significance. If correct, the paper gives a d-dimensional solution to a natural martingale optimization problem connected with Aldous's question, and it establishes a striking link to Monge-Ampère equations with infinite boundary data on a non-smooth domain. The main mathematical contributions are the boundary estimates for ∇²g on the simplex and the explicit construction of the optimal SDE without fitted parameters. The verification argument is clean and specific. I also note that the manuscript is transparent that the modeling link to a scaling limit of Shannon entropies is taken from [4]; the variational optimality result stands independently of that interpretation, but the "most exciting game" reading is conditional on that identification. However, the claimed uniqueness in law of the optimizer is not established in the paper, and the comparison proof has a gap; these are load-bearing and require repair before the main theorem is fully proven.

major comments (2)
  1. [Section 6, proof of Theorem 1.2, final paragraph] Statement (iii), uniqueness in law of the optimizer, is asserted with the sentence "For the uniqueness of the optimizer we can argue just as in [4]" and is not proved. The one-dimensional argument in [4] uses the explicit solution g(x)=log(π²/sin²(πx)) and the corresponding explicit SDE, so it does not transfer verbatim to the d-dimensional setting, where the coefficient is built from an implicitly defined ∇²g. To justify uniqueness one must show that any optimizer Q satisfies Σ_s = 2(∇²g(M_s))^{-1}/(1−s) for a.e. s and Q-a.s., i.e. one must characterize the equality case in the Legendre inequality −log det Σ + (1/2)Tr((1−s)∇²g(M_s)Σ) ≥ d + d log(1−s) + g(M_s). That equality condition is neither stated nor proved. Please provide a self-contained argument, or clearly state the missing equality-case result and prove it.
  2. [Section 4, Proposition 4.2, Step 3b] The modification of û2 to be +∞ outside a closed small ball around (t0,x0) takes the function outside the class of finite-valued convex functions allowed in Definition 4.1, and the assertion that this modification preserves the super-solution property is not justified. In particular, test functions with local minima at the boundary of the ball are not covered by the condition in Definition 4.1(iii). This modification is essential to force the maximizer of Φα into the interior, so the comparison principle is incomplete as written. Since Corollary 5.1 relies on Proposition 4.2 to identify v with (1−t)g+f, this gap affects the proof of Theorem 1.2 and should be repaired with a rigorous approximation or localization argument.
minor comments (4)
  1. [Section 5.1, proof of Proposition 5.1] The text says "Lemma 5.6 yields" the C^{2,α} estimate after Evans–Krylov, but Lemma 5.6 is a first-order estimate for the scaled functions defined later in Section 5.2; the intended reference appears to be Lemma 5.3 (or a direct Evans–Krylov application to the C² bounds).
  2. [Section 2 vs. Section 6] The time-change in Section 2 is defined as Y_t := M_{1−e^{−t}}, whereas Section 6, before Lemma 6.1, uses Y_t := M_{1−e^{−t/2}}; the two conventions should be reconciled or explicitly distinguished.
  3. [Proposition 3.2] The notation in the convexity proof is dense: the vector of ones is denoted by 1 in the same display as scalar quantities, and the coupling of the Brownian motions is described only in words. A brief notational clarification would improve readability.
  4. [Abstract and Theorem 1.2] The abstract says the winning-probability "is described by" the SDE, while Theorem 1.2 more precisely asserts strong existence, the win-martingale property, and unique optimality in law. The abstract could reflect that the uniqueness part is part of the theorem.

Circularity Check

1 steps flagged · score 4.0 of 10

Main optimality derivation is self-contained; the uniqueness assertion in Theorem 1.2(iii) is deferred to the authors' own one-dimensional paper [4] without the needed d-dimensional equality-case argument.

  1. self citation load bearing [Section 6, proof of Theorem 1.2, final sentence]
    "For the uniqueness of the optimizer we can argue just as in [4]."

    Theorem 1.2(iii) asserts uniqueness in law of the Aldous martingale in arbitrary dimension d. The entire justification is the sentence deferring to [4], which is prior work by one of the present authors and treats only d=1. In d=1 the argument uses the explicit solution g(x)=log(pi^2/sin^2(pi x)); in d dimensions the diffusion coefficient is built from the non-explicit Hessian inverse (nabla^2 g)^{-1}, so uniqueness would require the equality case in the Legendre inequality -log det Sigma + ((1-s)/2) Tr(nabla^2 g Sigma) >= g + d + d log(1-s), i.e. Sigma_s = 2(nabla^2 g(M_s))^{-1}/(1-s) for a.e. s under any optimizer. This equality case is neither stated nor proved.

full rationale

The main derivation of the value function and of the optimality of the Aldous martingale is not circular. The solution g is determined by the parameter-free Monge-Ampere equation (MA); the candidate diffusion (AM) is constructed from nabla^2 g and is then verified against the independently defined variational problem (OPT) through Lemma 6.4 and Corollary 5.1. There are no fitted parameters, no normalization-dependent predictions, and no quantity is defined in terms of the target result. The identification of (OPT) with Aldous's 'most exciting game' is taken from the authors' earlier work [4], but this is a modeling premise rather than a circular derivation; the mathematical optimality result for the stated cost stands independently of that interpretation. The proof of Proposition 4.2 Step 3b modifies u2 to be +infinity outside a small ball and asserts without proof that the super-solution property is preserved; this is an omitted proof that weakens the justification of Corollary 5.1, but it is not a circular step because it does not make the theorem equivalent to its input. The one clear circularity-adjacent issue is the uniqueness assertion of Theorem 1.2(iii), whose proof is reduced to the self-citation [4] without supplying the required d-dimensional equality-case argument. Since the central existence and optimality results are independently established, the overall circularity score is moderate rather than high.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no fitted constants and no new physical entities. The only inputs are the chosen cost functional (inherited from [4]) and standard PDE theories for Monge-Ampère and viscosity solutions.

assumptions (3)
  • domain assumption The cost E∫ −log det(Σ_s) ds is the appropriate scaling limit of Shannon entropies for the 'most exciting game' (taken from Backhoff-Veraguas and Beiglböck [4]).
    This identifies the mathematical objective with Aldous's informal question. The paper cites [4] and does not re-derive the scaling limit.
  • standard math Existence and regularity for the approximating Dirichlet problems follow from Caffarelli-Nirenberg-Spruck [19] for strictly convex domains with smooth boundary.
    Used in Proposition 5.1 to obtain smooth approximants g_n on Ω_n.
  • domain assumption The comparison principle for viscosity solutions of the HJB equation with infinite boundary condition holds, via scaling and doubling variables (Proposition 4.2).
    This is a nonstandard comparison principle; the proof uses an argument from Guan and Jian [43] and a modification of the super-solution outside a ball that is not fully justified in the text.

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Pith. "Pith review of Exciting games and Monge-Amp\`ere equations." pith.science (2026). https://pith.science/paper/FZ5YTF7C

@misc{pith2026241201995,
  author       = {Pith},
  title        = {Pith review of: Exciting games and Monge-Amp\`ere equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FZ5YTF7C}},
  note         = {Machine review of arXiv:2412.01995}
}
abstract

We consider a competition between $d+1$ players, and aim to identify the "most exciting game'' of this kind. This is translated, mathematically, into a stochastic optimization problem over martingales that live on the $d$-dimensional subprobability simplex $\Delta$ and terminate on the vertices of $\Delta$ (so-called win-martingales), with a cost function related to a scaling limit of Shannon entropies. We uncover a surprising connection between this problem and the seemingly unrelated field of Monge-Amp\`{e}re equations: If $g$ solves \begin{equation*} \begin{cases} g(x)=\log \det\left(\frac{1}{2}\nabla^2 g(x)\right), \quad \, \ \ \ \ \, \, \, \, x \in \Delta, \\ g(x)=\infty, \quad \quad \quad \quad \ \ \ \ \ \quad \quad \, \ \ \ \ \ x\in \partial \Delta, \end{cases} \end{equation*} then the winning-probability of the players in the most exciting game is described by $$dM_s=\sqrt{\frac{2 (\nabla^2 g(M_s))^{-1}}{1-s} } \, dB_s.$$ To formalize this, a detailed quantitative analysis of the Monge-Amp\`{e}re equation for $g$ is crucial. This is then leveraged to prove that $M$ is indeed an optimal win-martingale.

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Works this paper leans on

67 extracted references · 67 canonical work pages

  1. [4]

    B ackhoff-Veraguas and M

    J. B ackhoff-Veraguas and M. B eiglb¨ock, The most exciting game , Electronic Communications in Proba- bility, 29 (2024), pp. 1 – 12

    work page 2024

  2. [1]

    A ldous, notes, https://www.stat.berkeley.edu/ aldous/Research/OP/ent-MG.pdf

    D. A ldous, notes, https://www.stat.berkeley.edu/ aldous/Research/OP/ent-MG.pdf

    work page

  3. [2]

    A ldous, What is the max-entropy win-probability martingale? https://www.stat.berkeley.edu/ al- dous/Research/OP/max ent mg.html

    D. A ldous, What is the max-entropy win-probability martingale? https://www.stat.berkeley.edu/ al- dous/Research/OP/max ent mg.html

    work page

  4. [3]

    A vellaneda, C

    M. A vellaneda, C. Friedman, R. Holmes, and D. Samperi, Calibrating volatility surfaces via relative-entropy minimization, Quantitative Finance, 1 (2001), pp. 42–51

    work page 2001

  5. [5]

    B ackhoff-Veraguas, M

    J. B ackhoff-Veraguas, M. B eiglb¨ock, M. H uesmann, and S. K¨allblad, Martingale Benamou–Brenier , The Annals of Probability, 48 (2020), pp. 2258–2289

    work page 2020

  6. [6]

    B ackhoff-Veraguas and E

    J. B ackhoff-Veraguas and E. Belloto, Multidimensional specific relative entropy between contin uous mar- tingales, ArXiv e-prints, (2024)

    work page 2024

  7. [7]

    Geometric Martingale Benamou-Brenier transport and geometric Bass martingales

    J. B ackhoff-Veraguas, G. L oeper, and J. O bloj, Geometric martingale Benamou-Brenier transport and geometric Bass martingales, arXiv preprint arXiv:2406.04016, (2024)

    work page Pith review arXiv 2024

  8. [8]

    B ackhoff-Veraguas and C

    J. B ackhoff-Veraguas and C. Unterberger, On the specific relative entropy between martingale di ffusions on the line, Electronic Communications in Probability, 28 (2023), pp. 1–12

    work page 2023

Show all 67 references
  1. [9]

    B ackhoff-Veraguas and X

    J. B ackhoff-Veraguas and X. Z hang, Specific W asserstein divergence between continuous martin gales, arXiv e-prints, (2024), p. arXiv:2404.19672

    2024 arXiv

  2. [10]

    B eiglb¨ock, A

    M. B eiglb¨ock, A. Cox, and M. Huesmann, Optimal transport and Skorokhod embedding, Invent. Math., 208 (2017), pp. 327–400

    2017

  3. [11]

    B eiglb¨ock, P

    M. B eiglb¨ock, P . Henry-Labord`ere, and F. Penkner, Model-independent bounds for option prices: A mass transport approach, Finance Stoch., 17 (2013), pp. 477–501

    2013

  4. [12]

    B eiglb¨ock and N

    M. B eiglb¨ock and N. Juillet, Shadow couplings, Trans. Amer. Math. Soc., 374 (2021), pp. 4973–5002

    2021

  5. [13]

    B eiglb¨ock, M

    M. B eiglb¨ock, M. N utz, and F. Stebegg, Fine properties of the optimal Skorokhod embedding problem , J. Eur. Math. Soc. (JEMS), 24 (2022), pp. 1389–1429

    2022

  6. [14]

    B eiglb¨ock, M

    M. B eiglb¨ock, M. N utz, and N. Touzi, Complete duality for martingale optimal transport on the li ne, Ann. Probab., 45 (2017), pp. 3038–3074

    2017

  7. [15]

    B eiglb¨ock, G

    M. B eiglb¨ock, G. Pammer, and L. Riess, Change of numeraire for weak martingale transport, arXiv preprint arXiv:2406.07523, (2024). 26 JULIO BACKHOFF, ZHIZHANG W ANG, AND XIN ZHANG

    2024 arXiv

  8. [16]

    B erlyne, Conflict, Arousal, and Curiosity , A Doylestown Foundation paper, McGraw-Hill Book Co., New Y ork, N.Y , 1960

    D. B erlyne, Conflict, Arousal, and Curiosity , A Doylestown Foundation paper, McGraw-Hill Book Co., New Y ork, N.Y , 1960

    1960

  9. [17]

    B ouchard and M

    B. B ouchard and M. N utz, Arbitrage and duality in nondominated discrete-time model s, Ann. Appl. Probab., 25 (2015), pp. 823–859

    2015

  10. [18]

    C affarelli and Y

    L. C affarelli and Y . Li, An extension to a theorem of J¨ orgens, Calabi, and Pogorelov, Communications on Pure and Applied Mathematics, 56 (2003), pp. 549–583

    2003

  11. [19]

    C affarelli, L

    L. C affarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equa- tions i. Monge-Amp´ egre equation, Communications on pure and applied mathematics, 37 (1984) , pp. 369– 402

    1984

  12. [20]

    L. A. C affarelli, The regularity of mappings with a convex potential, Journal of the American Mathematical Society, 5 (1992), pp. 99–104

    1992

  13. [21]

    L. A. C affarelli and X. C abr´e, Fully nonlinear elliptic equations , Colloquium publications / American Mathematical Society, vol. 43, American Mathematical Soci ety, Providence, Rhode Island, 1995

    1995

  14. [22]

    C hen, A.-M

    B. C hen, A.-M. L i, and L. Sheng, Affine techniques on extremal metrics on toric surfaces , Adv. Math., 340 (2018), pp. 459–527. [23] , Extremal metrics on toric surfaces , Adv. Math., 340 (2018), pp. 363–405

    2018

  15. [24]

    Cheng and S.-T

    S.-Y . Cheng and S.-T. Yau, On the existence of a complete K¨ ahler metric on non- compactcomplex manifolds and the regularity of Fe fferman’s equation, Comm. Pure Appl. Math., 33 (1980), pp. 507–544

    1980

  16. [25]

    1980 Beijing Symp

    , The real Monge–Amp` ere equation and a ffine flat structure , Proc. 1980 Beijing Symp. on Di ff. Geom. and Diff. Equations, V ol. I (S.S. Chern and W.T. Wu, eds.), Science Press, Beijing, (1982), pp. 339– 370

    1982

  17. [26]

    C hewi, T

    S. C hewi, T. L e Gouic, C. L u, T. M aunu, P . Rigollet, and A. S tromme, Exponential ergodicity of mirror- Langevin diffusions, Advances in Neural Information Processing Systems, 33 (20 20), pp. 19573–19585

  18. [27]

    C ohen and Y

    A. C ohen and Y . Dolinsky, A scaling limit for utility indi fference prices in the discretised Bachelier model , Finance Stoch., 26 (2022), pp. 335–358

    2022

  19. [28]

    C ordero-Erausquin and B

    D. C ordero-Erausquin and B. K lartag, Moment measures , Journal of Functional Analysis, 268 (2015), pp. 3834–3866

    2015

  20. [29]

    M. G. C randall, H. Ishii, and P .-L. Lions, User’s guide to viscosity solutions of second order partial differ- ential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), pp. 1–67

    1992

  21. [30]

    , User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), pp. 1–67

    1992

  22. [31]

    F. C. Cˆırstea andC. Trombetti, On the Monge-Amp` ere equation with boundary blow-up: existence, unique- ness and asymptotics., Calc. V ar. Partial Differential Equations, 31 (2008), pp. 167–186

    2008

  23. [32]

    J. V . Da vis, B. Kulis, P . Jain, S. Sra, and I. S. Dhillon, Information-theoretic metric learning, in Proceedings of the 24th international conference on Machine learning, 2 007, pp. 209–216

  24. [33]

    D. K. D ey and C. Sriniv asan, Estimation of a covariance matrix under Stein’s loss , The Annals of Statistics, (1985), pp. 1581–1591

    1985

  25. [34]

    E ly, Beeps, American Economic Review, 107 (2017), p

    J. E ly, Beeps, American Economic Review, 107 (2017), p. 31–53

    2017

  26. [35]

    J. E ly, A. F rankel, and E. K amenica, Suspense and surprise , Journal of Political Economy, 123 (2015), pp. 215–260

    2015

  27. [36]

    F athi, Stein kernels and moment maps , The Annals of Probability, 47 (2019), pp

    M. F athi, Stein kernels and moment maps , The Annals of Probability, 47 (2019), pp. 2172–2185

    2019

  28. [37]

    F igalli, The Monge-Amp` ere equation and its applications , Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Z ¨ urich, 2017

    A. F igalli, The Monge-Amp` ere equation and its applications , Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Z ¨ urich, 2017

    2017

  29. [38]

    F igalli and F

    A. F igalli and F. Glaudo, An invitation to optimal transport, Wasserstein distances , and gradient flows , EMS Textbooks in Mathematics, EMS Press, Berlin, 2021

    2021

  30. [39]

    F¨ollmer, Doob decomposition, Dirichlet processes, and entropies on Wiener space, in Dirichlet forms and related topics, vol

    H. F¨ollmer, Doob decomposition, Dirichlet processes, and entropies on Wiener space, in Dirichlet forms and related topics, vol. 394 of Springer Proc. Math. Stat., S pringer, Singapore, [2022] ©2022, pp. 119–141

    2022

  31. [40]

    , Optimal couplings on Wiener space and an extension of Talagr and’s transport inequality , in Sto- chastic analysis, filtering, and stochastic optimization, Springer, Cham, [2022] ©2022, pp. 147–175

    2022

  32. [41]

    G alichon, P

    A. G alichon, P . Henry-Labord`ere, and N. T ouzi, A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options , Ann. Appl. Probab., 24 (2014), pp. 312–336

    2014

  33. [42]

    G antert, Einige grosse Abweichungen der Brownschen Bewegung , vol

    N. G antert, Einige grosse Abweichungen der Brownschen Bewegung , vol. 224 of Bonner Mathematische Schriften [Bonn Mathematical Publications], Universit¨ at Bonn, Mathematisches Institut, Bonn, 1991. Dis- sertation, Rheinische Friedrich-Wilhelms-Universit¨ atBonn, Bonn, 1991

    1991

  34. [43]

    G uan and H.-Y

    B. G uan and H.-Y . Jian, The Monge-Amp` ere equation with infinite boundary value , Pacific J. Math., 216 (2004), pp. 77–94

    2004

  35. [44]

    G. G uo, S. H owison, D. P ossamai, and C. R eisinger, Randomness and early termination: what makes a game exciting?, Forthcoming Probability Theory and Related Fields, 2306. 07133 (2023)

    2023

  36. [45]

    H obson and A

    D. H obson and A. Neuberger, Robust bounds for forward start options , Math. Finance, 22 (2012), pp. 31– 56

    2012

  37. [46]

    H uesmann and D

    M. H uesmann and D. Trevisan, A Benamou–Brenier formulation of martingale optimal trans port, Bernoulli, 25(4A) (2019), pp. 2729–2757. EXCITING GAMES AND MONGE-AMP `ERE EQUA TIONS 27

    2019

  38. [47]

    I shii and P

    H. I shii and P . Lions, Viscosity solutions of fully nonlinear second-order ellip tic partial di fferential equa- tions, Journal of Di fferential Equations, 83 (1990), pp. 26–78

    1990

  39. [48]

    J ames and C

    W. J ames and C. Stein, Estimation with quadratic loss, in Breakthroughs in statistics: Foundations and basic theory, Springer, 1992, pp. 443–460

    1992

  40. [49]

    J ian, Hessian equations with infinite Dirichlet boundary value

    H. J ian, Hessian equations with infinite Dirichlet boundary value. , Indiana Univ. Math. J., 55 (2006), pp. 1045–1062

    2006

  41. [50]

    K aratzas and S

    I. K aratzas and S. Shreve, Brownian motion and stochastic calculus, vol. 113, Springer Science & Business Media, 1991

    1991

  42. [51]

    K hasminskii, Stochastic stability of di fferential equations, vol

    R. K hasminskii, Stochastic stability of di fferential equations, vol. 66 of Stochastic Modelling and Applied Probability, Springer, Heidelberg, second ed., 2012. With contributions by G. N. Milstein and M. B. Nevel- son

    2012

  43. [52]

    K ulis, M

    B. K ulis, M. S ustik, and I. Dhillon, Learning low-rank kernel matrices , in Proceedings of the 23rd inter- national conference on Machine learning, 2006, pp. 505–512

    2006

  44. [53]

    L oeper, Option pricing with linear market impact and nonlinear Blac k–Scholes equations , The Annals of Applied Probability, 28 (2018), pp

    G. L oeper, Option pricing with linear market impact and nonlinear Blac k–Scholes equations , The Annals of Applied Probability, 28 (2018), pp. 2664–2726

    2018

  45. [54]

    Lˆ e, Analysis of Monge-Amp` ere Equations , Graduate studies in mathematics, volume 240, American Mathematical Society, Providence, Rhode Island, 2024

    N. Lˆ e, Analysis of Monge-Amp` ere Equations , Graduate studies in mathematics, volume 240, American Mathematical Society, Providence, Rhode Island, 2024

    2024

  46. [55]

    R. J. M cCann, A convexity principle for interacting gases , Advances in mathematics, 128 (1997), pp. 153– 179

    1997

  47. [56]

    M ohammed, On the existence of solutions to the Monge-Amp` ere equation with infinite boundary values

    A. M ohammed, On the existence of solutions to the Monge-Amp` ere equation with infinite boundary values. , Proc. Amer. Math. Soc., 135 (2007), pp. 141–149

    2007

  48. [57]

    P ham, Continuous-time stochastic control and optimization with financial applications , vol

    H. P ham, Continuous-time stochastic control and optimization with financial applications , vol. 61 of Sto- chastic Modelling and Applied Probability, Springer-V erlag, Berlin, 2009

    2009

  49. [58]

    S antambrogio, Optimal transport for applied mathematicians , vol

    F. S antambrogio, Optimal transport for applied mathematicians , vol. 87 of Progress in Nonlinear Di fferen- tial Equations and their Applications, Birkh¨ auser/Springer, Cham, 2015. Calculus of variations, PDEs, and modeling

    2015

  50. [59]

    , Dealing with moment measures via entropy and optimal transp ort, Journal of Functional Analysis, 271 (2016), pp. 418–436

    2016

  51. [60]

    S hannon, A mathematical theory of communication, Bell Telephone System technical publication

    C. S hannon, A mathematical theory of communication, Bell Telephone System technical publication. Mono- graph ; B-1598, American Telephone and Telegraph Co., New Y ork, 1948

    1948

  52. [61]

    T an and N

    X. T an and N. Touzi, Optimal transportation under controlled stochastic dynamics, The annals of probabil- ity, (2013), pp. 3201–3240

    2013

  53. [62]

    T ouzi, Optimal stochastic control, stochastic target problems, a nd backward SDE , vol

    N. T ouzi, Optimal stochastic control, stochastic target problems, a nd backward SDE , vol. 29 of Fields Institute Monographs, Springer, New Y ork; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2013. With Chapter 13 by Ang` es Tourin

    2013

  54. [63]

    V illani, Topics in optimal transportation, vol

    C. V illani, Topics in optimal transportation, vol. 58 of Graduate Studies in Mathematics, American Math- ematical Society, Providence, RI, 2003

    2003

  55. [64]

    V illani, Optimal Transport

    C. V illani, Optimal Transport. Old and New, vol. 338 of Grundlehren der mathematischen Wissenschafte n, Springer, 2009

    2009

  56. [65]

    W ang and X

    X.-J. W ang and X. Zhu, Ka ¨hler–Ricci solitons on toric manifolds with positive first c hern class, Adv. Math., (2004), pp. 87–103

    2004

  57. [66]

    X iang and X.-P

    N. X iang and X.-P . Yang, The complex Monge-Amp` ere equation with infinite boundary v alue., Nonlinear Anal., 68 (2008), pp. 1075–1081

    2008

  58. [67]

    K. S. Z hang, G. P eyr´e, J. Fadili, and M. Pereyra, W asserstein control of mirror Langevin Monte Carlo, in Conference on learning theory, PMLR, 2020, pp. 3814–3841

    2020

  59. [68]

    Z hong, Optimal dynamic information acquisition, Econometrica, 90 (2022), pp

    W. Z hong, Optimal dynamic information acquisition, Econometrica, 90 (2022), pp. 1537–1582. Department of Mathematics, University of Vienna Email address: julio.backhoff@univie.ac.at School of MathematicalScience, Fudan University, Shanghai, China Email address: zzwang@fudan.e...

    2022

Pith tools

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