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Path-Space Model Risk via Signature-Induced Optimal Transport

T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Path-space model risk for affine signature payoffs and half-space events reduces to a baseline-only computation governed by the single effective budget κ(ℓ,δ), and the same scalar drives budget-aware sparse surrogates for general path-depen

desk verdict Core affine Sig-OT bounds are real and clean; the reserve-risk and option applications oversell coverage because the surrogate-to-target event gap is unquantified, and the tightness ratios are partly built from the calibration loop. read the letter →

arxiv 2607.20343 v2 pith:TKQYQMEL submitted 2026-07-22 q-fin.RM math.OCq-fin.MF

Tomoyuki Ichiba , Qijin Shi This is my paper
classification q-fin.RMmath.OCq-fin.MF MSC 91G7049Q2260L10
keywords modelriskoptimaltransportpathsignaturesdistributionallyrobustoptimizationworst-caseprobabilitysignaturesurrogatespath-dependentambiguityset
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

The paper tries to establish that distributional model risk on path laws—the gap between what a baseline model predicts and what nearby alternative models could produce—can be made tractable by measuring ambiguity in signature coordinates. It defines an ambiguity set in which an alternative law is plausible if a single coupling to the baseline keeps the expected change of each selected signature coordinate below a budget δ_I. For payoffs that are affine functions of those coordinates, and for threshold events defined by such affine scores, the worst-case expectation and worst-case probability over the ambiguity set admit explicit upper bounds that depend only on the baseline law and on the scalar κ(ℓ,δ). The same κ drives a budget-aware sparse regression to build surrogates for general path-dependent claims. If the framework works as intended, robust pricing and reserve-risk capital calculations can be stress-tested without solving an infinite-dimensional optimization.

What carries the argument

Signature-induced optimal transport (Sig-OT): a coordinate-wise transport geometry on selected time-augmented path-signature coordinates φ_I(x), with coordinate budgets δ_I certified by one common coupling between alternative and baseline path laws. The workhorse is the effective budget κ(ℓ,δ)=Σ|ℓ_I|δ_I: it is the exact robust premium for affine scores in the relaxed feature space, and it determines the threshold shift u_κ for half-space probability bounds. The ambient feature-space relaxation—replacing realizability-constrained signature feature laws by all laws on R^W with the same coordinate transport budgets—is what makes the closed-form formulas possible.

What would settle it

Construct a target reserve event F(R)≥u and a stress law inside the calibrated Sig-OT ball such that the true stress probability exceeds the reported Sig-OT robust probability for the affine surrogate at the same u; if such a law exists, the surrogate step does not deliver the stated protection. Concretely, use a jump-heavy insurance surplus process with heavy-tailed claims inside the ball and compare its ruin probability at a deep capital level with the corresponding reported Sig-OT robust probability.

Watch

Extended reading notes

Core claim

The central discovery is the effective-budget identity: for an affine signature score s_{ℓ,β}(x)=β+Σ_I ℓ_I φ_I(x), the robust expectation over the Sig-OT ambiguity set is at most the baseline expectation plus κ(ℓ,δ)=Σ|ℓ_I|δ_I, and the worst-case probability of the half-space event {s_ℓ ≥ b} is at most the baseline probability of the shifted event {s_ℓ ≥ b − u_κ}, where u_κ solves a baseline-only moment equation. These bounds are obtained by relaxing the path-level problem to an ambient feature space, so they are valid upper bounds even though the relaxation can be strict; an explicit step-2 example in the appendix exhibits the strictness. The same effective budget is then used as a lasso pen

Load-bearing premise

The load-bearing premise is that the sparse affine signature score fitted to a target path functional is a faithful enough stand-in for that functional: the paper proves robust bounds for the surrogate score, and Remark 5.1 concedes that sparse affine signatures cannot approximate discontinuous barrier-style functionals from finite samples, so if the surrogate gap is large in the tail, the reported robust probabilities bound a different event than the stated reserve-risk even

Editorial extensions

If this is right

  • Robust expectation and worst-case probability for any affine signature functional can be computed from baseline samples plus the scalar κ; no search over alternative path laws is needed.
  • The same ambiguity set can be reused across different payoffs and events, since the geometry lives in signature coordinates rather than in a payoff-specific cost.
  • Sparse affine surrogates with budget-weighted lasso inherit an explicit robust correction, making the framework applicable to smooth path-dependent payoffs such as Asian and lookback options.
  • The probability bound has a threshold-shift interpretation: robustifying a tail event under model ambiguity is equivalent to lowering the threshold by u_κ and evaluating the baseline law.
  • Because the relaxed bound can be strict, the reported robust values are guaranteed upper bounds, not exact worst-cases, for path-level problems.

Reading between the lines

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

  • The paper leaves open how tight the surrogate route is for discontinuous payoffs; a testable extension is to compare the Sig-OT robust bound for a barrier option against the exact worst-case over the same ambiguity set, since the paper's Remark 5.1 concedes sparse affine signatures cannot approximate such payoffs from finite samples.
  • One could replace the selected coordinate set W with learned or data-dependent coordinates—chosen, for example, by their sensitivity to the target event—and the effective-budget formula would still apply, suggesting a way to tighten the bounds without changing the framework.
  • The common-coupling requirement is what certifies all coordinate budgets simultaneously; if one instead used coordinate-wise decoupled transport, as the paper's proxy calibration does, coverage of stress laws is no longer guaranteed. A natural diagnostic is to quantify when the decoupled proxy over- or under-states the certified budget.
  • The framework suggests a model-risk accounting metric: report the scalar κ per risk functional, making model risk comparable across desks or portfolios; this is an extension the authors do not explicitly pursue.
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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

3 major / 3 minor

Summary. The paper proposes a signature-induced optimal transport formulation for model risk on path space. Coordinate budgets on selected time-augmented signature coordinates define an ambiguity set U_δ(P0) through a single common coupling. For affine signature scores s_{ℓ,β}, Proposition 3.3 gives a robust expectation bound E_{P0}[s]+κ(ℓ,δ), and Theorem 3.5 gives a worst-case half-space probability bound via a shifted threshold, after relaxing to an ambient feature space. Section 4 introduces a budget-weighted lasso for fitting affine surrogates to more general path functionals. Sections 5.1–5.3 calibrate budgets empirically and present numerical experiments for sparse affine risk factors, path-dependent option pricing, and insurance reserve ruin probabilities, including a comparison with the Skorokhod J1 benchmark.

Significance. The affine signature results are a genuine contribution: the dual derivation is explicit, the effective-budget premium is exact for the relaxed feature-space problem, and the shifted-threshold probability formula is cleanly derived. The paper is also honest about the realizability gap in Appendix B and provides extensive numerical diagnostics. However, the practical reach claimed in the abstract and Section 4 depends on the surrogate step, and that step lacks a theorem or error certificate. The numerical experiments are useful as calibration checks, but they do not yet establish that the reported robust values protect the stated path-dependent target events. If the surrogate gap can be closed or the claims appropriately narrowed, the paper is publishable.

major comments (3)
  1. [§5.3.2, Tables 6–7; §4 (surrogate workflow)] The reported ‘Sig-OT’ probabilities in Table 6 are upper bounds for the surrogate event {s_ℓ ≥ b−u*}, not for the target reserve event {F(R) ≥ u}. No theorem or numerical error bound controls the gap between sup_Q Q(s_ℓ ≥ b) and sup_Q Q(F(R) ≥ u). High in-sample FitR² (0.996, 0.983) does not imply tail-region dominance. The table itself shows the failure: for the early-shortfall target at u=200, Sig-OT/CL < 1.8×10^-5, i.e., the robust surrogate probability is far below the stated CL risk, while at u=100 it is only 0.314×CL. Remark 5.1 concedes the problem for discontinuous barrier payoffs, but the same unbounded gap applies to the continuous reserve functionals in Tables 4 and 6. The paper must either provide a one-sided error certificate for the surrogate or explicitly relabel all such numbers as bounds on surrogate events.
  2. [§5.1, Eq. (29); §5.2–5.3] The budget vector δ_joint is obtained by solving the empirical OT problem with the same stress law P1 and the same affine direction ℓ that is later used to compute κ and the robust bound. Hence P1 ∈ U_δ(P0) by construction. The tables therefore demonstrate internal consistency of the calibration, but they do not provide evidence that the method protects against model uncertainty outside the fitted stress law. The tightness ratios are conditional on the calibrated δ and cannot be interpreted as coverage probabilities for a pre-specified ambiguity set. Please add an out-of-sample protocol (e.g., calibrate δ on one stress model and evaluate on another, or preselect δ from the Appendix C bounds) or state clearly that all reported stress tests are in-sample calibration checks.
  3. [§4; Remark 5.1] The surrogate step relies on an unquantified approximation. Proposition 2.5 gives uniform approximation of continuous functionals on compact path classes by linear signature functionals, but Section 4 fits a lasso on a finite baseline sample and post-processes with a monotone link. No sample-complexity bound, no sup-norm control, and no tail-region control is given. Since the affine robust formulas are then applied to the surrogate, the absence of a statement relating sup_Q Q(s_ℓ ≥ b) to sup_Q Q(F ≥ b) is a load-bearing gap, not a presentation issue. The paper should either add such a statement, use an outer approximation of the target event by a certified family of affine events, or restrict the methodological claim to affine signature scores.
minor comments (3)
  1. [Abstract and §3.2] The abstract says the bounds 'depend only on the baseline model'; more precisely they depend on δ, ℓ, W, the normalizers a_I, and the baseline distribution. Suggest rewording to avoid overstatement.
  2. [§3.3, Eq. (24)] In the sentence following (24), 'the same scalar gives a shifted-threshold baseline bound' should note that the equality with P0(s_ℓ ≥ b−u*) holds under the no-atom condition; the theorem does state this, but the introduction and abstract omit it.
  3. [§5.3.2, Table 6] The FitR² values are reported without indicating whether they are in-sample or out-of-sample. Please clarify, and consider reporting a split-sample or cross-validated R² since the lasso penalty is selected on the same baseline data.

Circularity Check

1 steps flagged · score 4.0 of 10

Core Sig-OT bounds are internally derived; empirical coverage checks are by-construction because δ is fit from the same stress law.

  1. fitted input called prediction [Section 5.1 (Eq. 29); Sections 5.2.1–5.2.2 (Tables 2–3); same pattern in §5.3.1]
    "Given a benchmark or stress law P1, any coupling π∈Π(P1,P0) induces the coordinate budgets δπ_I := ∫|φI(x)−φI(y)|π(dx,dy), ∀I∈W, and hence, certifies P1∈Uδπ(P0). ... For each stress model, we first calibrate the joint budget δ̂joint_I ... so that the stress law is covered by the Sig-OT ambiguity set. ... the relaxed Sig-OT robust premium induced by δ̂joint_I is then κ̂joint=Σ|ℓ_I|δ̂joint_I."

    The robust bound is applied to an ambiguity set Uδ that was constructed to contain the stress law used as the comparison. For the coupling π defining δ̂joint, the affine stress shift equals ∫Σℓ_I(φ_I(y)−φ_I(x))dπ ≤ Σ|ℓ_I|∫|φ_I(y)−φ_I(x)|dπ = κ̂joint. Hence Stress/Base shift ≤ κ̂joint and Robust/Stress ≥ 1 are algebraic consequences of the calibration, not empirical findings. The 'tightness ratios' in Tables 2–4 therefore restate the coverage constraint rather than validate the model-risk prediction.

full rationale

The theoretical chain is self-contained: multi-budget OT duality (Theorem 2.6) is applied to coordinate costs |φ_I(x)−φ_I(y)|, giving the feature-space relaxation; Proposition 3.3 and Theorem 3.5 then derive the affine expectation premium E_P0[s]+κ(ℓ,δ) and the shifted-threshold half-space bound from the definitions of ℓ and δ. No load-bearing self-citation occurs: universal approximation and OT duality are cited to external literature, and the formulas do not assume the stress law. The circular aspect is confined to the numerical demonstrations: δ in Eq. (29) is calibrated from the same stress model that the robust values are then compared against, so the reported dominance and tightness ratios are coverage checks by construction rather than independent predictions. This does not undermine the core theorems, but it means Tables 2–4 should be read as implementation consistency checks, not out-of-sample validations. Remark 5.1 explicitly concedes that sparse affine surrogates cannot approximate discontinuous barrier-style payoffs from finite samples; the missing bound between the surrogate event and the target reserve event is a validity gap rather than a circularity, but it is a real limitation of the §5.3.2 reported probabilities.

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

The central theoretical result assumes the affine coefficients ℓ and budgets δ are given, and the numerical workflow estimates them from baseline and stress samples. The most consequential unproved premise is that a sparse affine signature score event tracks the target path-dependent event closely enough for tail-risk statements; the paper itself flags this as an open challenge.

free parameters (5)
  • coordinate budgets δ_I = calibrated from baseline/stress samples via score-weighted Sinkhorn OT (Eq. 29)
    Definition 3.1 treats δ as input; in experiments δ is estimated from the same stress law the bound then claims to cover, so the robust values inherit finite-sample calibration error.
  • affine score coefficients ℓ_I = hand-picked in §5.2; lasso-fitted in §5.3
    The effective budget κ depends directly on ℓ; fitting on baseline data means the reported robust values are not out-of-sample predictions for the stress laws.
  • feature normalizers a_I = baseline MAD scale with floor 10^-8
    Every signature coordinate is divided by a baseline-fitted robust scale; changing these normalizers rescales δ and κ and changes the ambiguity geometry.
  • word set W and truncation level N = k=2 or 4; W listed per example
    The ambiguity geometry is defined only on selected coordinates and truncation level; results and tightness depend on this choice, with ablations in Appendix E.3.
  • lasso tuning α and postlink g = α=1 (options), α=3 (reserve); monotone 1-Lipschitz spline
    Manual hyperparameters control the tradeoff between surrogate fit and effective budget; the postlink is a post-fitting refinement.
assumptions (5)
  • standard math Time-augmented signatures uniquely determine paths and linear signatures approximate continuous path functionals (Propositions 2.4–2.5)
    Invoked in the introduction and §3.2 to justify representing payoffs and events by affine signature scores.
  • standard math Multi-budget OT-DRO strong duality (Theorem 2.6)
    Proved in Appendix A for upper semicontinuous f and δ∈(0,∞)^m; it underlies both main robust bounds.
  • domain assumption Selected signature coordinates have finite first moments and a finite-cost common coupling exists
    Needed for Definition 3.1 and Proposition 3.3; stated around §3.1 and supported by model-based bounds in Appendix C.
  • ad hoc to paper The ambient feature-space relaxation gap is acceptable for the applications
    Appendix B shows strictness for a step-2 example, but no quantitative control of the gap is given for the examples in §5.
  • ad hoc to paper Sparse affine signature surrogate events approximate the target path-dependent events in the tail region
    Used in §5.3.2 to report Sig-OT robust probabilities against Cramér–Lundberg ruin probabilities; no error bound is provided, and Remark 5.1 concedes failure for discontinuous payoffs.

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Pith. "Pith review of Path-Space Model Risk via Signature-Induced Optimal Transport." pith.science (2026). https://pith.science/paper/TKQYQMEL

@misc{pith2026260720343,
  author       = {Pith},
  title        = {Pith review of: Path-Space Model Risk via Signature-Induced Optimal Transport},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TKQYQMEL}},
  note         = {Machine review of arXiv:2607.20343}
}
read the original abstract

We propose a signature-induced, optimal transport framework for path-space model risk, in which ambiguity between stochastic path laws is factorized through optimal transport costs on signature coordinates under a common coupling. Via an ambient feature-space relaxation, we derive for affine signature scores and affine half-space events explicit robust expectation and probability bounds that depend only on the baseline model. In both cases, the correction is governed by the same effective budget, which depends only on the affine score and the budget vector of the ambiguity set. Moreover, the same quantity leads to a budget-aware sparse signature surrogate method for more general, possibly implicit, or data-driven, path functionals. We illustrate the resulting methodology through controlled stress tests and benchmark model-misspecification problems in finance and insurance.

Figures

Figures reproduced from arXiv: 2607.20343 by the authors.

Figure 1
Figure 1. Raw parametric coverage scale at k = 4. Each panel plots the decoupled empirical radius ∆4(P0, Pθ) = ∥δbdec 4 (P0, Pθ)∥1 against the swept model parameter. The six panels cover GBM drift, GBM volatility, Merton jump intensity, GBM-to-Heston volatility-of-volatility, within-Heston volatility-of-volatility, and CEV/local￾volatility elasticity. materially enlarges the separation of volatility and stochastic-volatility … view at source ↗
Figure 2
Figure 2. Normalized parametric coverage profiles across truncation levels. Each panel plots [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗
Figure 3
Figure 3. Heterogeneous coverage of the Sig-OT ambiguity ball. The entries are pairwise empirical radii [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗

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Works this paper leans on

40 extracted references · 12 canonical work pages

  1. [1]

    (2010).Ruin probabilities(2nd ed., Vol

    Asmussen, S., & Albrecher, H. (2010).Ruin probabilities(2nd ed., Vol. 14). World Scientific. doi: 10.1142/7431

    work page doi:10.1142/7431 2010

  2. [2]

    Bartl, D., Drapeau, S., & Tangpi, L. (2020). Computational aspects of robust optimized certainty equivalents and option pricing.Mathematical Finance,30(1), 287–309. doi: 10.1111/mafi.12203

    work page doi:10.1111/mafi.12203 2020

  3. [3]

    Bayer, C., dos Reis, G., Horvath, B., & Oberhauser, H. (Eds.). (2026).Signature methods in finance: An introduction with computational applications. Springer. Beiglb¨ ock, M., Henry-Labord` ere, P., & Penkner, F. (2013). Model-independent bounds for option prices: A mass transport approach.Finance and Stochastics,17(3), 477–501. doi: 10.1007/s00780-013-0205-8

    work page doi:10.1007/s00780-013-0205-8 2026

  4. [4]

    Blanchet, J., & Murthy, K. (2019). Quantifying distributional model risk via optimal transport.Operations Research,67(3), 650–665

    2019

  5. [5]

    Boedihardjo, H., Geng, X., Lyons, T., & Yang, D. (2016). The signature of a rough path: Uniqueness.Advances in Mathematics,293, 720–737. doi: 10.1016/j.aim.2016.02.011

    doi:10.1016/j.aim.2016.02.011 2016

  6. [6]

    Bonnier, P., Liu, C., & Oberhauser, H. (2023). Adapted topologies and higher rank signatures.The Annals of Applied Probability,33(3), 2136–2175. doi: 10.1214/22-AAP1862

    work page doi:10.1214/22-aap1862 2023

  7. [7]

    Brown, H., Hobson, D., & Rogers, L. C. G. (2001). Robust hedging of barrier options.Mathematical Finance, 11(3), 285–314. doi: 10.1111/1467-9965.00116

    arXiv 2001

  8. [8]

    Carassus, L., & Wiesel, J. (2025). Strategies with minimal norm are optimal for expected utility maximisation under high model ambiguity.Finance and Stochastics,29(2), 519–551. doi: 10.1007/s00780-025-00558-4

    work page doi:10.1007/s00780-025-00558-4 2025

Show all 40 references
  1. [9]

    Cass, T., Messadene, R., & Turner, W. F. (2023). Signature asymptotics, empirical processes, and optimal transport.Electronic Journal of Probability,28, 1–19. doi: 10.1214/23-EJP1048 28 (a) Time–price path (b) Time–price–realized-variation path Figure 3: Heterogeneous coverage...

    2023 doi

  2. [10]

    Chevyrev, I., & Kormilitzin, A. (2026). A primer on the signature method in machine learning. In C. Bayer, G. dos Reis, B. Horvath, & H. Oberhauser (Eds.),Signature methods in finance: An introduction with computational applications(pp. 3–64). Springer. Retrieved from https://...

    2026 doi

  3. [11]

    Chevyrev, I., & Oberhauser, H. (2022). Signature moments to characterize laws of stochastic processes.Journal of Machine Learning Research,23(176), 1–42

    2022

  4. [12]

    Cont, R. (2006). Model uncertainty and its impact on the pricing of derivative instruments.Mathematical Finance,16(3), 519–547. doi: 10.1111/j.1467-9965.2006.00281.x

    2006

  5. [13]

    Cox, A. M. G., & Kallblad, S. (2017). Model-independent bounds for asian options: A dynamic programming approach.SIAM Journal on Control and Optimization,55(6), 3409–3436. doi: 10.1137/16M1087527

    2017 doi

  6. [14]

    Cuchiero, C., Gazzani, G., M¨ oller, J., & Svaluto-Ferro, S. (2025). Joint calibration to SPX and VIX options with signature-based models.Mathematical Finance,35(1), 161–213. doi: 10.1111/mafi.12442

    2025 doi

  7. [15]

    Cuchiero, C., Gazzani, G., & Svaluto-Ferro, S. (2023). Signature-based models: Theory and calibration.SIAM Journal on Financial Mathematics,14(3), 910–957

    2023

  8. [16]

    Cuchiero, C., Primavera, F., & Svaluto-Ferro, S. (2025). Universal approximation theorems for continuous functions of c` adl` ag paths and L´ evy-type signature models.Finance and Stochastics,29(2), 289–342. doi: 10.1007/s00780-025-00557-5

    2025 doi

  9. [17]

    Cuchiero, C., Schmocker, P., & Teichmann, J. (2026). Global universal approximation of functional input maps on weighted spaces.Constructive Approximation. Retrieved from https://doi.org/10.1007/ s00365-025-09726-3doi: 10.1007/s00365-025-09726-3

    2026 doi

  10. [18]

    Delage, E., & Ye, Y. (2010). Distributionally robust optimization under moment uncertainty with application to data-driven problems.Operations Research,58(3), 595–612

    2010

  11. [19]

    Denis, L., & Martini, C. (2006). A theoretical framework for the pricing of contingent claims in the presence of model uncertainty.The Annals of Applied Probability,16(2), 827–852. doi: 10.1214/105051606000000169

    2006 doi

  12. [20]

    C., & Landau, B

    Dowson, D. C., & Landau, B. V. (1982). The Fr´ echet distance between multivariate normal distributions. Journal of Multivariate Analysis,12(3), 450–455. doi: 10.1016/0047-259X(82)90077-X

    1982 doi

  13. [21]

    (2014).A course on rough paths: with an introduction to regularity structures.Springer

    Friz, P., & Hairer, M. (2014).A course on rough paths: with an introduction to regularity structures.Springer. doi: 10.1007/978-3-319-08332-2

    2014 doi

  14. [22]

    (2010).Multidimensional stochastic processes as rough paths: Theory and applications (Vol

    Friz, P., & Victoir, N. (2010).Multidimensional stochastic processes as rough paths: Theory and applications (Vol. 120). Cambridge University Press. 29

    2010

  15. [23]

    Galichon, A., Henry-Labord` ere, P., & Touzi, N. (2014). A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options.The Annals of Applied Probability,24(1), 312–336. doi: 10.1214/13-AAP925

    2014 doi

  16. [24]

    Gao, R., & Kleywegt, A. J. (2023). Distributionally robust stochastic optimization with Wasserstein distance. Mathematics of Operations Research,48(2), 603–655

    2023

  17. [25]

    Glasserman, P., & Xu, X. (2014). Robust risk measurement and model risk.Quantitative Finance,14(1), 29–58

    2014

  18. [26]

    Goh, J., & Sim, M. (2010). Distributionally robust optimization and its tractable approximations.Operations Research,58(4), 902–917

    2010

  19. [27]

    Guo, I., & Loeper, G. (2021). Path dependent optimal transport and model calibration on exotic derivatives. The Annals of Applied Probability,31(3), 1232–1263. doi: 10.1214/20-AAP1617

    2021 doi

  20. [28]

    Guo, X., Wang, B., Zhang, R., & Zhao, C. (2025). On consistency of signature using lasso.Operations Research, 73(5), 2530–2549. doi: 10.1287/opre.2024.1133

    2025

  21. [29]

    M., & Lyons, T

    Hambly, B. M., & Lyons, T. J. (2010). Uniqueness for the signature of a path of bounded variation and the reduced path group.Annals of Mathematics,171(1), 109–167

    2010

  22. [30]

    Hobson, D. G. (1998). Robust hedging of the lookback option.Finance and Stochastics,2(4), 329–347

    1998

  23. [31]

    Kuhn, D., Shafiee, S., & Wiesemann, W. (2025). Distributionally robust optimization.Acta Numerica,34, 579–804. doi: 10.1017/S0962492924000084

    2025 doi

  24. [32]

    Liao, S., Ni, H., Sabate-Vidales, M., Szpruch, L., Wiese, M., & Xiao, B. (2024). Sig-Wasserstein gans for conditional time series generation.Mathematical Finance,34(2), 622–670. doi: 10.1111/mafi.12423

    2024 doi

  25. [33]

    Lyons, T. J. (1995). Uncertain volatility and the risk-free synthesis of derivatives.Applied Mathematical Finance, 2(2), 117–133. doi: 10.1080/13504869500000007 Mohajerin Esfahani, P., & Kuhn, D. (2018). Data-driven distributionally robust optimization using the Wasserstein me...

    1995 doi

  26. [34]

    S., Xiao, B., Wiese, M., & Liao, S

    Ni, H., Szpruch, L., Vidales, M. S., Xiao, B., Wiese, M., & Liao, S. (2021). Sig-Wasserstein GANs for time series generation. InICAIF’21: 2nd ACM international conference on AI in finance, virtual event, november 3–5, 2021(pp. 28:1–28:8). ACM. Retrieved from https://doi.org/10...

    2021

  27. [35]

    Nutz, M., & Soner, H. M. (2012). Superhedging and dynamic risk measures under volatility uncertainty.SIAM Journal on Control and Optimization,50(4), 2065–2089. doi: 10.1137/100814925

    2012 doi

  28. [36]

    Rahimian, H., & Mehrotra, S. (2022). Frameworks and results in distributionally robust optimization.Open Journal of Mathematical Optimization,3, 1–85

    2022

  29. [37]

    Reizenstein, J., & Graham, B. (2020). Algorithm 1004: The iisignature library: Efficient calculation of iterated-integral signatures and log signatures.ACM Transactions on Mathematical Software (TOMS)

    2020

  30. [38]

    Rockafellar, R. T. (1970).Convex analysis(Vol. 28). Princeton, NJ: Princeton University Press

    1970

  31. [39]

    M., & Kuhn, D

    Rychener, Y., Esteban-P´ erez, A., Morales, J. M., & Kuhn, D. (2024).Wasserstein distributionally robust optimization with heterogeneous data sources.arXiv preprint. Retrieved from https://arxiv.org/abs/ 2407.13582

    2024 arXiv

  32. [40]

    M., Touzi, N., & Zhang, J

    Soner, H. M., Touzi, N., & Zhang, J. (2012). Wellposedness of second order backward SDEs.Probability Theory and Related Fields,153(1–2), 149–190. doi: 10.1007/s00440-011-0342-y 30

    2012 doi

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