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arxiv: 2605.21759 · v1 · pith:KAYLS436new · submitted 2026-05-20 · 💱 q-fin.MF · math.OC· math.PR

An optimal transport foundation for a class of dynamically consistent risk measures

Sven Fuhrmann , Michael Kupper , Max Nendel This is my paper

Pith reviewed 2026-05-22 08:26 UTC · model grok-4.3

classification 💱 q-fin.MF math.OCmath.PR
keywords dynamic risk measuresoptimal transporttime consistencyconvex semigroupsmodel uncertaintystochastic controlWasserstein distancemartingale transport
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The pith

Penalized optimal transport costs identify explicit generators for dynamically consistent risk measures with model uncertainty.

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

The paper builds dynamically consistent risk measures by starting from one-step convex evaluations that penalize deviations from a Markov reference model's transition laws. Time consistency then produces a convex monotone semigroup on payoff functions, which is uniquely determined by its infinitesimal generator. Under a lower bound on the penalties expressed via optimal transport costs, this generator is identified explicitly on smooth test functions. Linear small-time scaling of the transport costs yields a first-order correction driven by a convex Hamiltonian on the gradient, while martingale transport constraints with different scaling produce a second-order correction given by a convex monotone functional on the Hessian. Both cases lead to stochastic control representations of the risk measure, with the control acting on drift or volatility respectively.

Core claim

Imposing time consistency on penalized worst-case expectations over alternative transition laws yields a convex monotone semigroup on bounded continuous functions. When the penalties are bounded from below by suitable optimal transport costs relative to the reference laws, the generator of this semigroup on smooth test functions takes explicit form: a convex Hamiltonian acting on the gradient for optimal transport bounds with linear small-time scaling, and a convex monotone functional acting on the Hessian under martingale transport constraints with a different scaling. Explicit formulas follow from convex conjugates of the transport costs, and the associated risk measures admit stochastic-d

What carries the argument

The risk generator of the convex monotone semigroup, obtained from a lower bound on the family of penalties in terms of optimal transport costs relative to the reference laws.

If this is right

  • Explicit formulas for the generators are obtained via convex conjugates of the underlying transport costs for Wasserstein and martingale Wasserstein penalizations.
  • The dynamic risk measures admit stochastic control representations in which the control acts on the drift in the first-order case and on the volatility in the second-order case.
  • Both regimes are illustrated explicitly for Wasserstein and martingale Wasserstein penalizations.

Where Pith is reading between the lines

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

  • The two scaling regimes suggest a unified way to move between first-order robustification of drifts and second-order robustification of volatilities using the same optimal transport toolbox.
  • The identification technique could extend to non-Markovian reference models provided analogous lower bounds on penalties can be formulated.
  • The stochastic control representations open a direct route to numerical solution of the associated dynamic risk evaluation problems via standard Hamilton-Jacobi-Bellman methods.

Load-bearing premise

The family of penalties admits a lower bound in terms of suitable optimal transport costs relative to the reference laws.

What would settle it

A concrete penalized risk measure whose generator on smooth test functions fails to match the predicted convex Hamiltonian on the gradient or convex monotone functional on the Hessian for the corresponding optimal transport cost.

read the original abstract

We study a class of dynamically consistent risk measures that robustify a time-homogeneous Markovian reference model by allowing for distributional uncertainty in its transition laws. We start from one-step convex risk evaluations in which ambiguity is captured by penalized worst-case expectations over alternative transition laws. Imposing time consistency then yields a convex monotone semigroup on bounded continuous payoff functions, and this semigroup represents the associated dynamic risk measure. The semigroup is uniquely characterized by its risk generator. Under a lower bound on the family of penalties in terms of suitable optimal transport costs relative to the reference laws, we identify the generator on smooth test functions. For optimal transport bounds with linear small-time scaling, this produces a first-order, drift-type correction given by a convex Hamiltonian acting on the gradient. Under martingale transport constraints and a different scaling, however, the leading correction is genuinely of second order and is described by a convex monotone functional acting on the Hessian. We illustrate both regimes for Wasserstein and martingale Wasserstein penalizations and derive explicit formulas via convex conjugates of the underlying transport costs. The associated dynamic risk measures admit stochastic control representations in which the control acts on the drift in the first-order case and on the volatility in the second-order case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The paper develops a framework for dynamically consistent risk measures that incorporate distributional uncertainty into a time-homogeneous Markovian reference model. One-step convex risk evaluations are defined via penalized worst-case expectations over alternative transition laws; time consistency then produces a convex monotone semigroup on bounded continuous functions, which is characterized by its risk generator. Under a lower bound relating the penalty family to optimal transport costs relative to the reference laws, the generator is identified on smooth test functions. Linear small-time scaling of optimal transport bounds yields a first-order drift correction given by a convex Hamiltonian acting on the gradient, while martingale transport constraints with a different scaling produce a second-order correction given by a convex monotone functional acting on the Hessian. Explicit formulas are derived via convex conjugates for the Wasserstein and martingale-Wasserstein cases, and the associated dynamic risk measures are shown to admit stochastic control representations with controls acting on drift or volatility, respectively.

Significance. If the identification results hold, the manuscript supplies a rigorous optimal-transport foundation for a class of dynamic risk measures, linking convex analysis, semigroup theory, and martingale transport in a way that yields concrete, explicit generators and control representations. This could be useful for robust modeling of ambiguity in transition kernels within mathematical finance. The clear separation between first-order and second-order regimes under different scalings and constraints is a substantive contribution, and the stochastic-control interpretations provide a practical bridge to optimization problems.

major comments (1)
  1. [§4, Theorem 4.3] §4, Theorem 4.3 (generator identification): the proof that the lower bound on penalties implies the stated Hamiltonian or Hessian correction appears to rely on a specific convex-conjugate representation; it would be helpful to see an explicit verification that the conjugate of the transport cost indeed produces the claimed monotone functional on the Hessian in the martingale-transport case, including any required growth or convexity conditions on the test functions.
minor comments (3)
  1. [§2.1] Notation: the definition of the one-step penalty functional in §2.1 could explicitly state the domain of the probability measures (e.g., whether they are required to have finite moments of all orders) to avoid ambiguity when passing to the small-time limit.
  2. [Figure 1] Figure 1: the diagram illustrating the semigroup construction would benefit from labeling the arrows with the precise mathematical operations (e.g., “time-consistent extension” or “generator identification”) for quicker navigation.
  3. [References] References: the citation list appears to omit some recent works on martingale optimal transport and dynamic risk measures (e.g., papers on robust hedging under Wasserstein ambiguity); adding 2–3 targeted references would strengthen the positioning.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. We address the major comment below and will incorporate the requested clarification.

read point-by-point responses

  1. Referee: [§4, Theorem 4.3] §4, Theorem 4.3 (generator identification): the proof that the lower bound on penalties implies the stated Hamiltonian or Hessian correction appears to rely on a specific convex-conjugate representation; it would be helpful to see an explicit verification that the conjugate of the transport cost indeed produces the claimed monotone functional on the Hessian in the martingale-transport case, including any required growth or convexity conditions on the test functions.

    Authors: We thank the referee for highlighting this point. The identification result in Theorem 4.3 for the martingale-transport regime does rely on the convex-conjugate representation of the penalty. In the revised manuscript we will expand the proof of Theorem 4.3 (and the supporting Lemma 4.2) to include an explicit verification. Specifically, we will (i) recall the definition of the convex conjugate of the martingale-transport cost functional acting on symmetric matrices, (ii) verify that, for C^{2} test functions whose second derivatives satisfy the natural growth bound implied by the linear scaling of the penalty, the conjugate yields a convex monotone functional on the Hessian, and (iii) confirm that this functional coincides with the expression stated in the theorem. The added steps will be placed immediately after the statement of the lower-bound assumption and will not alter any of the main results. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper begins with one-step convex risk evaluations that penalize worst-case expectations over alternative transition laws. Time consistency is imposed to obtain a convex monotone semigroup on bounded continuous functions, which is then characterized by its generator via standard semigroup theory. Under the explicit structural assumption of a lower bound on the penalty family in terms of optimal transport costs, the generator on smooth test functions is identified by convex analysis, yielding first-order Hamiltonian corrections for linear scaling and second-order Hessian corrections for martingale transport. Explicit formulas are obtained via convex conjugates of the transport costs, and stochastic control representations follow directly. All steps are conditional on the stated lower-bound hypothesis and rely on independent convex-analytic and semigroup tools rather than reducing any claimed generator form to a fitted parameter, self-definition, or load-bearing self-citation chain. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard background results from convex analysis, semigroup theory, and optimal transport; the principal domain assumption is the existence of a time-homogeneous Markovian reference model together with the lower-bound condition on penalties.

axioms (2)
  • domain assumption The reference model is time-homogeneous and Markovian.
    Explicitly stated as the starting point for the robustification procedure.
  • standard math One-step risk evaluations are convex and monotone.
    Invoked when passing from penalized expectations to the dynamic risk measure.

pith-pipeline@v0.9.0 · 5750 in / 1571 out tokens · 52035 ms · 2026-05-22T08:26:15.537377+00:00 · methodology

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Lean theorems connected to this paper

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  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
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    Relation between the paper passage and the cited Recognition theorem.

    Under a lower bound on the family of penalties in terms of suitable optimal transport costs... first-order, drift-type correction given by a convex Hamiltonian acting on the gradient... second order... convex monotone functional acting on the Hessian.

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Works this paper leans on

55 extracted references · 55 canonical work pages · 1 internal anchor

  1. [1]

    Acciaio and I

    B. Acciaio and I. Penner. Dynamic risk measures. InAdvanced Mathematical Methods for Finance, pages 1–34. Springer, Heidelberg, 2011

    work page 2011

  2. [2]

    Artzner, F

    P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath. Coherent measures of risk.Mathematical Finance. An International Journal of Mathematics, Statistics and Financial Economics, 9(3):203–228, 1999

    work page 1999

  3. [3]

    Artzner, F

    P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, and H. Ku. Coherent multiperiod risk adjusted values and Bellman’s principle.Ann. Oper. Res., 152:5–22, 2007

    work page 2007

  4. [4]

    Bartl, S

    D. Bartl, S. Drapeau, J. Ob l´ oj, and J. Wiesel. Sensitivity analysis of Wasserstein distributionally robust optimization problems.Proceedings A, 477(2256):Paper No. 20210176, 19, 2021

    work page 2021

  5. [5]

    Bartl, S

    D. Bartl, S. Eckstein, and M. Kupper. Limits of random walks with distributionally robust transition probabilities.Electron. Commun. Probab., 26:13, 2021. Id/No 28

    work page 2021

  6. [6]

    Bartl, A

    D. Bartl, A. Neufeld, and K. Park. Sensitivity of robust optimization problems under drift and volatil- ity uncertainty.Preprint, arXiv:2311.11248, 2023

    work page arXiv 2023

  7. [7]

    Bartl and J

    D. Bartl and J. Wiesel. Sensitivity of multiperiod optimization problems with respect to the adapted Wasserstein distance.SIAM J. Financ. Math., 14(2):704–720, 2023

    work page 2023

  8. [8]

    Bion-Nadal.Dynamic Risk Measuring: Discrete Time in a Context of Uncertainty and Continuous Time on a Probability Space

    J. Bion-Nadal.Dynamic Risk Measuring: Discrete Time in a Context of Uncertainty and Continuous Time on a Probability Space. ´Ecole polytechnique, 2006

    work page 2006

  9. [9]

    Bion-Nadal

    J. Bion-Nadal. Time consistent dynamic risk processes.Stochastic Process. Appl., 119(2):633–654, 2009

    work page 2009

  10. [10]

    Blanchet and K

    J. Blanchet and K. Murthy. Quantifying distributional model risk via optimal transport.Mathematics of Operations Research, 44(2):565–600, 2019

    work page 2019

  11. [11]

    Blessing, R

    J. Blessing, R. Denk, M. Kupper, and M. Nendel. Convex monotone semigroups and their generators with respect to Γ-convergence.J. Funct. Anal., 288(8):73, 2025. Id/No 110841

    work page 2025

  12. [12]

    Blessing and M

    J. Blessing and M. Kupper. Nonlinear semigroups built on generating families and their Lipschitz sets. Potential Anal., 59(3):857–895, 2023

    work page 2023

  13. [13]

    Blessing and M

    J. Blessing and M. Kupper. Nonlinear semigroups and limit theorems for convex expectations.Ann. Appl. Probab., 35(2):779–821, 2025

    work page 2025

  14. [14]

    Convergence of infinitesimal generators and stability of convex monotone semigroups

    J. Blessing, M. Kupper, and M. Nendel. Convergence of infinitesimal generators and stability of convex monotone semigroups.Preprint, arXiv:2305.18981, 2023

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  15. [15]

    V. I. Bogachev.Measure theory. Vol. I, II. Springer-Verlag, Berlin, 2007

    work page 2007

  16. [16]

    Cheridito, F

    P. Cheridito, F. Delbaen, and M. Kupper. Dynamic monetary risk measures for bounded discrete-time processes.Electron. J. Probab., 11:57–106, 2006. Id/No 3. 30 SVEN FUHRMANN, MICHAEL KUPPER, AND MAX NENDEL

    work page 2006

  17. [17]

    Cheridito and M

    P. Cheridito and M. Kupper. Composition of time-consistent dynamic monetary risk measures in discrete time.Int. J. Theor. Appl. Finance, 14(1):137–162, 2011

    work page 2011

  18. [18]

    Cheridito, H

    P. Cheridito, H. M. Soner, N. Touzi, and N. Victoir. Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs.Communications on Pure and Applied Mathematics, 60(7):1081–1110, 2007

    work page 2007

  19. [19]

    P. R. Chernoff. Note on product formulas for operator semigroups.J. Functional Analysis, 2:238–242, 1968

    work page 1968

  20. [20]

    P. R. Chernoff.Product formulas, nonlinear semigroups, and addition of unbounded operators, volume

    work page

  21. [21]

    American Mathematical Soc., 1974

    work page 1974

  22. [22]

    Coquet, Y

    F. Coquet, Y. Hu, J. M´ emin, and S. Peng. Filtration-Consistent Nonlinear Expectations and Related g-Expectations.Probability Theory and Related Fields, 123(1):1–27, 2002

    work page 2002

  23. [23]

    F. Delbaen. The structure of m-stable sets and in particular of the set of risk neutral measures. InIn Memoriam Paul-Andr´ e Meyer: S´ eminaire de Probabilit´ es XXXIX, volume 1874 ofLecture Notes in Math., pages 215–258. Springer, Berlin, 2006

    work page 2006

  24. [24]

    Delbaen, S

    F. Delbaen, S. Peng, and E. Rosazza Gianin. Representation of the penalty term of dynamic concave utilities.Finance and Stochastics, 14(3):449–472, 2010

    work page 2010

  25. [25]

    Duffie and L

    D. Duffie and L. G. Epstein. Stochastic differential utility.Econometrica, 60(2):353–394, 1992

    work page 1992

  26. [26]

    El Karoui, S

    N. El Karoui, S. Peng, and M. C. Quenez. Backward stochastic differential equations in finance. Mathematical Finance. An International Journal of Mathematics, Statistics and Financial Economics, 7(1):1–71, 1997

    work page 1997

  27. [27]

    W. H. Fleming and H. M. Soner.Controlled Markov processes and viscosity solutions, volume 25 of Stoch. Model. Appl. Probab.New York, NY: Springer, 2nd ed. edition, 2006

    work page 2006

  28. [28]

    F¨ ollmer

    H. F¨ ollmer. Optimal couplings on Wiener space and an extension of Talagrand’s transport inequality. InStochastic analysis, filtering, and stochastic optimization. A commemorative volume to honor Mark H. A. Davis’s contributions, pages 147–175. Cham: Springer, 2022

    work page 2022

  29. [29]

    F¨ ollmer and I

    H. F¨ ollmer and I. Penner. Convex risk measures and the dynamics of their penalty functions.Statistics & Decisions. International Journal for Statistical Theory and Related Fields, 24(1):61–96, 2006

    work page 2006

  30. [30]

    F¨ ollmer and A

    H. F¨ ollmer and A. Schied. Convex measures of risk and trading constraints.Finance and Stochastics, 6(4):429–447, 2002

    work page 2002

  31. [31]

    F¨ ollmer and A

    H. F¨ ollmer and A. Schied.Stochastic finance. An introduction in discrete time.De Gruyter Textb. Berlin: de Gruyter, 4th revised edition edition, 2016

    work page 2016

  32. [32]

    Frittelli and E

    M. Frittelli and E. R. Rosazza Gianin. Putting order in risk measures.Journal of Banking & Finance, 26(7):1473–1486, 2002

    work page 2002

  33. [33]

    Fuhrmann, M

    S. Fuhrmann, M. Kupper, and M. Nendel. Wasserstein perturbations of Markovian transition semi- groups.Ann. Inst. Henri Poincar´ e Probab. Stat., 59(2):904–932, 2023

    work page 2023

  34. [34]

    Gao and A

    R. Gao and A. J. Kleywegt. Distributionally Robust Stochastic Optimization with Wasserstein Dis- tance, 2016

    work page 2016

  35. [35]

    Hu and S

    M. Hu and S. Peng.$G$-L´ evy processes under sublinear expectations.Probability, Uncertainty and Quantitative Risk, 6(1):1–22, 2021

    work page 2021

  36. [36]

    Karatzas and S

    I. Karatzas and S. E. Shreve.Brownian motion and stochastic calculus., volume 113 ofGrad. Texts Math.New York etc.: Springer-Verlag, 2nd ed. edition, 1991

    work page 1991

  37. [37]

    Kazi-Tani, D

    N. Kazi-Tani, D. Possama¨ ı, and C. Zhou. Second order BSDEs with jumps: Existence and probabilistic representation for fully-nonlinear PIDEs.Electronic Journal of Probability, 20:no. 65, 31, 2015

    work page 2015

  38. [38]

    F. K¨ uhn. Viscosity solutions to Hamilton-Jacobi-Bellman equations associated with sublinear L´ evy(- type) processes.ALEA. Latin American Journal of Probability and Mathematical Statistics, 16(1):531– 559, 2019

    work page 2019

  39. [39]

    Kupper, M

    M. Kupper, M. Nendel, and A. Sgarabottolo. Risk measures based on weak optimal transport.Quant. Finance, 25(2):163–180, 2025

    work page 2025

  40. [40]

    H. Lam. Robust sensitivity analysis for stochastic systems.Math. Oper. Res., 41(4):1248–1275, 2016

    work page 2016

  41. [41]

    A. Lazrak. Generalized stochastic differential utility and preference for information.Ann. Appl. Probab., 14(4):2149–2175, 2004

    work page 2004

  42. [42]

    Mohajerin Esfahani and D

    P. Mohajerin Esfahani and D. Kuhn. Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations.Mathematical Program- ming, 171(1-2, Ser. A):115–166, 2018

    work page 2018

  43. [43]

    M. Nendel. Chernoff-Mehler Approximation for L´ evy Processes with Drift.Preprint, arXiv:2511.19414, 2025

    work page arXiv 2025

  44. [44]

    Nendel, A

    M. Nendel, A. Neufeld, K. Park, and A. Sgarabottolo. Scaling limits of multi-period distributionally robust optimization problems.Preprint, arXiv:2511.20126, 2025. OPTIMAL TRANSPORT AND DYNAMIC RISK MEASURES 31

    work page arXiv 2025

  45. [45]

    Nendel and A

    M. Nendel and A. Sgarabottolo. A parametric approach to the estimation of convex risk functionals based on Wasserstein distance.Appl. Math. Optim., 93(1):44, 2026. Id/No 8

    work page 2026

  46. [46]

    Nutz and H

    M. Nutz and H. M. Soner. Superhedging and dynamic risk measures under volatility uncertainty. SIAM Journal on Control and Optimization, 50(4):2065–2089, 2012

    work page 2065

  47. [47]

    S. Peng. Nonlinear Expectations, Nonlinear Evaluations and Risk Measures. InStochastic Methods in Finance, volume 1856 ofLecture Notes in Math., pages 165–253. Springer, Berlin, 2004

    work page 2004

  48. [48]

    Peng.G-expectation,G-Brownian motion and related stochastic calculus of Itˆ o type

    S. Peng.G-expectation,G-Brownian motion and related stochastic calculus of Itˆ o type. InStochastic Analysis and Applications, volume 2 ofAbel Symp., pages 541–567. Springer, Berlin, 2007

    work page 2007

  49. [49]

    Pichler and R

    A. Pichler and R. Schlotter. Quantification of risk in classical models of finance.Quantitative Finance, 22(1):31–45, 2022

    work page 2022

  50. [50]

    F. Riedel. Dynamic coherent risk measures.Stochastic Processes and their Applications, 112(2):185– 200, 2004

    work page 2004

  51. [51]

    E. R. Rosazza Gianin. Risk measures via g-expectations.Insurance: Mathematics and Economics, 39(1):19–34, 2006

    work page 2006

  52. [52]

    Schroder and C

    M. Schroder and C. Skiadas. Optimal consumption and portfolio selection with stochastic differential utility.Journal of Economic Theory, 89(1):68–126, 1999

    work page 1999

  53. [53]

    H. M. Soner, N. Touzi, and J. Zhang. Wellposedness of second order backward SDEs.Probability Theory and Related Fields, 153(1-2):149–190, 2012

    work page 2012

  54. [54]

    Villani.Optimal Transport: Old and New, volume 338 ofGrundlehren Der Mathematischen Wis- senschaften [Fundamental Principles of Mathematical Sciences]

    C. Villani.Optimal Transport: Old and New, volume 338 ofGrundlehren Der Mathematischen Wis- senschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2008

    work page 2008

  55. [55]

    Y. Xu. Multidimensional Dynamic Risk Measure via Conditionalg-Expectation.Mathematical Fi- nance. An International Journal of Mathematics, Statistics and Financial Economics, 26(3):638–673, 2016. Department of Mathematics and Statistics, University of Konstanz Email address:sven.fuhrmann@uni-konstanz.de Department of Mathematics and Statistics, University...

    work page 2016