Beyond Monte Carlo: Harnessing Diffusion Models to Simulate Financial Market Dynamics

Andrew Lesniewski; Giulio Trigila

REVIEW 4 major objections 6 minor 26 references

Beyond Monte Carlo: Harnessing Diffusion Models to Simulate Financial Market Dynamics

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that a quadrature-trained diffusion model generates synthetic equity returns that pass the two-sample Cramér–von Mises test, match observed quantiles in the tails, and yield better-conditioned covariance estimates than…

desk verdict The analytic quadrature reduction is a real contribution; the empirical market-fidelity claims are not yet supported. read the letter →

arxiv 2412.00036 v3 pith:NOO4HEO4 submitted 2024-11-21 q-fin.CP cs.AIcs.CEq-fin.PM

Andrew Lesniewski , Giulio Trigila This is my paper · ORCID

classification q-fin.CPcs.AIcs.CEq-fin.PM

keywords diffusionmodelsscorematchingsyntheticfinancialdataCramér–vonMisestestcovarianceregularizationGauss–Hermitequadraturereverse-timeSDEequityreturns

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

This paper claims that a diffusion model can generate synthetic daily equity returns that are statistically indistinguishable from observed returns as judged by the two-sample Cramér–von Mises test on an equally weighted portfolio, with Q–Q plots showing agreement across quantiles including the tails. It also claims that covariance matrices estimated from the synthetic scenarios are better conditioned than the historical sample covariance, making them usable as regularized covariance estimates for portfolio problems. The practical advance is training without Monte Carlo: the denoising score-matching objective is evaluated by Gauss–Hermite quadrature, which reduces the high-dimensional integrals to one- and two-dimensional Gaussian integrals. The method is demonstrated on 33 U.S. equities over five one-year windows and one five-year window, including the COVID crash period, with Cramér–von Mises p-values between 0.64 and 1.00.

What carries the argument

The load-bearing mechanism is the denoising score-matching objective combined with a single-hidden-layer noise conditional score network. Because each activation depends on the input only through one linear functional $\langle w_j | x \rangle$, the $d$-dimensional Gaussian integrals in the objective factor: after an orthogonal rotation, each integral collapses to a one- or two-dimensional Gaussian integral with $\det \Gamma = 1$, evaluated by Gauss–Hermite quadrature. The forward process is the variance-preserving SDE $dX_t = -\frac{1}{2} \beta(t) X_t dt + \sqrt{\beta(t)} dW_t$, which after the time change $\tau(t)=\int_0^t \beta(s)\,ds$ becomes an Ornstein–Uhlenbeck process with explicit Gaussian transition density; the reverse-time SDE driven by the fitted score generates the synthetic sample. The efficiency claim rests on replacing Monte Carlo sampling of the training objective with these closed-form quadrature reductions, plus Simpson's rule over time.

What would settle it

Run a genuine multivariate two-sample test, for instance the projection-averaging test cited in the paper as [12], on the historical and synthetic return vectors for any of the six experiments; if the test rejects at the 5% level, the central claim that the synthetic data come from the same distribution as the observed data is falsified. A simpler diagnostic is to compute the Cramér–von Mises p-value for many random portfolio weight vectors: if the p-values are not roughly uniform across directions, the single equally weighted portfolio's high p-value does not indicate multivariate agreement.

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Extended reading notes

Core claim

The central discovery is that a variance-preserving denoising SDE, trained with a score network whose weights are fit by a fully deterministic numerical-integration scheme, produces synthetic return vectors whose distribution aligns with the observed data in the tested windows. The authors take the empirical distribution of historical returns as the initial condition, add noise through the VP SDE until the data become white noise, and then simulate the reverse-time SDE with the trained score; the t=0 values of those reverse paths are the synthetic scenarios. For validation, they form an equally weighted portfolio of the 33 assets and run the two-sample Cramér–von Mises test on historical versus synthetic portfolio returns, obtaining p-values from 0.64 to 1.00 across six experiments, and they report Q–Q plots consistent in the tails. The covariance matrix of the synthetic scenarios has a condition number roughly half that of the historical sample covariance in every experiment, and the condition number falls further as more synthetic scenarios are generated.

Load-bearing premise

The load-bearing premise is that one univariate two-sample Cramér–von Mises test on a single equally weighted portfolio, with that portfolio's choice justified by an unpublished companion paper and the synthetic data compared to the same data used in training, is enough to certify that the full multivariate distribution of asset returns has been reproduced.

Editorial extensions

If this is right

Synthetic scenarios can be produced in unlimited quantity, and the covariance matrix estimated from them has a condition number roughly half that of the historical sample covariance in every reported experiment; the condition number drops further as more synthetic points are generated.
Because the score is fit by quadrature rather than Monte Carlo, the training step avoids the simulation cost that becomes impractical for portfolios with dimension greater than 100.
The generated data reproduce the empirical distribution of returns including tail quantiles, addressing a known weakness of earlier market generators based on restricted Boltzmann machines and generative adversarial networks.
The synthetic data are suitable as inputs to portfolio allocation and risk quantification, where the better-conditioned covariance matrix is the relevant object.
All six experiments cover different market regimes, including the COVID crash, low volatility, rate hikes, and election uncertainty, and the model produces high p-values in each, so the claim is not tied to one volatility state.

Reading between the lines

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

Because each synthetic path starts from a randomly drawn training point and is then noised and denoised, the generated distribution is a smoothed perturbation of the empirical training distribution; this suggests the reported p-values measure in-sample fidelity, and testing on a holdout window the model never saw is a natural next check.
The better-conditioned covariance matrices likely behave like an implicit shrinkage estimator, since the denoising process compresses extreme eigen-directions; comparing against standard shrinkage estimators on the same data would show whether the improvement is a new mechanism or a known regularization effect.
The quadrature speedup is tied to the single-hidden-layer architecture: with deeper score networks the integrals no longer collapse to one- and two-dimensional Gaussian integrals, so the efficiency claim and the architecture choice stand or fall together.
The same recipe, treating an empirical measure as the initial condition and training a reverse SDE by quadrature, could apply to any high-dimensional underdetermined covariance estimation problem, not only equity returns.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

One thing to know: the genuinely new piece here is the reduction of the denoising score matching objective to Gauss-Hermite quadrature for a single-hidden-layer score network. That is a real, checkable contribution and the best reason to take the paper seriously. The empirical demonstration, however, does not support the abstract's language about passing the CvM test for portfolios of assets; it supports only an in-sample univariate check on one equal-weighted portfolio.

The math in Section 5 checks out. For the VP SDE, the transition kernel is Gaussian with diagonal covariance, so after an orthogonal rotation the d-dimensional integrals factor into 1D or 2D Gaussian integrals. That reduction is correct. The statement "det(Γ)=1" is wrong as written—Γ is a projection onto a codimension-1 or -2 subspace, so its determinant is 0—but what the argument needs is that the leftover Gaussian integral over the orthogonal complement equals 1, which is true. So it's a minor slip, not a fatal flaw.

The empirics are a proof of concept, nothing more. The CvM test is applied to a single univariate portfolio projection; the proper multivariate test is explicitly deferred to an unpublished companion paper. The comparison is in-sample: each synthetic scenario starts from a randomly chosen training return and runs forward and backward through the SDE. That procedure can pass an in-sample two-sample test by returning slightly perturbed training points without learning the population distribution. The reported p-values—1.00, 0.64, 1.00, 0.83, 0.98, 0.99—are all high, which would be unusual if the test were well calibrated for independent samples; no bootstrap or competing-generator baseline is given, so we have no sense of the test's power here. The condition number comparison is against the raw sample covariance only; shrinkage baselines like Ledoit-Wolf are missing. Q-Q plots are visual and suggestive but not quantitative. No code or data are released, which makes reproducibility hard to assess.

Who gets value: people working on fast training of score networks for low- or moderate-dimensional financial data, and possibly the quadrature trick carries over to other settings. The paper is not yet a demonstration that diffusion models capture the full multivariate distribution of returns.

I would send this to a serious referee, but with the expectation of major revision. The analytic contribution deserves scrutiny, and the empirical claims are fixable with out-of-sample evaluation, baselines, and code or detailed reproducibility instructions.

Referee Report

4 major / 6 minor

Summary. The paper proposes a diffusion-model method for generating synthetic financial return data. The forward process is a variance-preserving linear SDE, and the score function is parameterized by a single-hidden-layer network. The main technical contribution is an analytical reduction of the denoising score-matching objective to one- and two-dimensional Gauss-Hermite quadratures, avoiding Monte Carlo integration. The method is applied to daily returns of 33 U.S. equities over six windows. Validation relies on a univariate two-sample Cramér-von Mises test applied to the equally weighted portfolio, Q-Q plots, and the condition numbers of synthetic versus historical covariance matrices. The authors claim high fidelity of the synthetic data and suitability of the synthetic covariance as a regularized estimator.

Significance. The quadrature-based training procedure is a useful methodological contribution that could make diffusion-model training more efficient and reproducible for financial applications. If the empirical claims held, the method would provide a practical tool for scenario generation and covariance regularization. However, the current empirical evidence does not yet establish that the full multivariate return distribution is reproduced: the only statistical test is univariate, applied to a single projection, and the evaluation is in-sample with no competing baseline. The technical derivation in Section 5.2 contains a correctable error in the determinant justification. These issues are addressable, and the core idea is promising.

major comments (4)

[6.2, 7] The only distributional test reported is the two-sample Cramér-von Mises test applied to the univariate equally weighted portfolio, with the choice of that portfolio itself justified by the authors' unpublished companion paper [16]. Passing a single univariate projection does not control the multivariate distribution, so the abstract's wording 'pass the two-sample Cramer - von Mises test for portfolios of assets' overstates the evidence. The authors should either implement a genuine multivariate test, such as the projection-averaging test in [12] that they cite, or clearly restrict the claims to the univariate projection and present the multivariate test as future work.
[6.2, 7] The validation is in-sample: each window is used both for training the score network and for the CvM comparison, and the synthetic samples are produced by forward/reverse SDEs whose forward pass is initialized at randomly selected training points. While the theoretical VP SDE drives the forward process toward white noise, the practical discretization and the initialization from the training set make the effective dependence on the training data unclear; an out-of-sample or holdout evaluation is needed. In addition, the reported p-values (1.00, 0.64, 1.00, 0.83, 0.98, 0.99) all lie near 1, which is more consistent with a low-power or miscalibrated test than with strong evidence of distributional equality, and no competing generator is benchmarked. Please provide a power analysis and a baseline comparison.
[5.2, Eq. (87)] The statement 'as det(Γ)=1' following Eq. (87), and the analogous statement in the derivation of I2, is mathematically incorrect: Γ = I - |E⟩⟨E| (and its two-dimensional analog) has determinant zero. The quadrature formula itself is correct, because the Gaussian integral over the orthogonal complement equals (2π)^{(d-1)/2} (resp. (2π)^{(d-2)/2}) and cancels against the normalization; please replace the determinant justification with this argument.
[7, condition numbers] The claim that the synthetic covariance matrices are suitable as regularized versions of the sample covariance is not supported by the evidence provided. Lower condition numbers are reported, but there is no comparison against standard regularized estimators (e.g., Ledoit-Wolf shrinkage or factor models), and no downstream application is tested. The paper should either add such comparisons or soften the claim to a descriptive observation.

minor comments (6)

[4.2.1] The heading 'Forward-time VE SDE' appears to be a typo; the subsection is about the sub-variance-preserving SDE and should read 'Forward-time sub-VP SDE'.
[7] The number of time steps K used in the Euler-Maruyama scheme is not specified, and training details such as learning rate, number of epochs, and batch size are not reported; these are needed for reproducibility.
[7] The pCvM values are reported to only two decimals, and values of exactly 1.00 appear; please describe how the p-values are computed (exact distribution or asymptotic approximation) and report more significant digits or confidence intervals.
[5.2, Eq. (87)] The notation 'dy dd−1x' is nonstandard; the coordinate decomposition should be written explicitly as x = yE + u with u in the orthogonal complement of E.
[8] The statement that the model can generate an arbitrary number of synthetic data points without compromising tail behavior or the CvM p-value exceeds the evidence, which only tests up to 16,384 scenarios.
[3.1.2] The use of vector-valued α1(t) and the exponential of its integral in Eq. (22) relies on the Hadamard-product convention, but this is not stated explicitly in the solution formula; please add a note to avoid confusion.

Circularity Check

1 steps flagged · score 4.0 of 10

The empirical validation relies on an unpublished same-author companion paper to justify the CvM portfolio choice; the core derivation is otherwise self-contained.

self citation load bearing [Section 6.2 (Generating synthetic market scenarios) and Section 7 (Experiments)]
"In a forthcoming paper [16], we propose a practical methodology that we believe is well-suited for situations like the one addressed in this study. ... While no choice of the portfolio weights g is a priori preferred, numerical simulations [16] suggest that an equally weighted portfolio g = (1 /d, . . . ,1/d) often lies near the maximizer (i.e. the worst case scenario) of TCvM ."

The paper's central empirical evidence is the set of pCvM values in Section 7, computed on the equally weighted portfolio. The only stated justification for choosing that portfolio as a near-worst-case projection is numerical simulations reported in [16], an unpublished companion paper by the same two authors. The proper multivariate test is also deferred to [16]. Thus the evidential force of the reported high p-values depends on a self-citation that is not independently checkable, and a single univariate projection cannot by itself control the multivariate distribution; the equal-weight choice is load-bearing for the validation claim.

full rationale

The core derivation—denoising score matching, reverse-time SDEs, Gauss-Hermite evaluation of the DSM objective, and Euler-Maruyama sampling—is mathematically self-contained and relies on external, checkable results [1,2,7,22,24]. I find no self-definitional or fitted-input-as-prediction circularity in that derivation. The synthetic-data validation is in-sample in the sense that the same dataset is used for training and for the CvM comparison, which weakens the empirical claim, but that is an evidentiary limitation rather than a statement that reduces by construction; the paper does not call this a held-out prediction. The one genuine circular element is the reliance on the unpublished same-author companion paper [16] to justify both the portfolio selection for the CvM test and the deferred multivariate test. Because the central methodology has independent content and the self-citation is limited to the validation protocol rather than the mathematical derivation, a score of 4 is appropriate.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the chosen noise schedule, the adequacy of a small one-hidden-layer network for the score, the i.i.d. assumption on returns, and the use of a univariate projection test as a proxy for multivariate goodness of fit. No independent evidence is provided for these choices.

free parameters (7)

Noise schedule hyperparameters b_i and a = b_i=0.1, a=0 (all i) in experiments
These control the rate at which data are noised into white noise via β_i(t)=b_i(1-t)^(-(1+a)); chosen by hand without sensitivity analysis (Section 7).
Hidden layer size h = 16 (32 for full dataset)
Number of neurons in the single-hidden-layer score network; chosen without ablation (Section 7).
Weight function λ0(t) = 1
Set to constant in experiments; no justification or sensitivity analysis (Section 7).
Gauss-Hermite order D and Simpson subintervals = D=4, 8 subintervals
Numerical integration settings chosen for accuracy; no error analysis given (Section 7).
Number of time steps K in Euler-Maruyama = not reported
The discretization of the forward and reverse SDEs is load-bearing for sampling accuracy, but K is never specified (Sections 6.1 and 7).
Training hyperparameters (learning rate, epochs, batch size) = not reported (batch size 32 mentioned once)
Adam optimizer is named but learning rate and iteration count are omitted, so the fitted network weights are not reproducible (Section 7).
Portfolio weights g for CvM test = equally weighted (1/d,...,1/d)
The evaluation metric depends on this choice; it is justified by an unpublished companion paper, not by data (Section 6.2).

assumptions (5)

domain assumption The forward VP SDE with β(t)=b(1-t)^(-(1+a)) drives the empirical distribution to standard normal white noise as t→1 (Section 4.1)
This is a standard construction in score-based generative modeling, but it assumes the chosen schedule is adequate for financial returns with heavy tails.
standard math The reverse-time SDE from Anderson's theory applies with the estimated score function (Section 3.2)
Assumes sufficient regularity for time reversal of diffusions, which holds for linear SDEs with the true score; with a fitted score it is approximate.
ad hoc to paper A single-hidden-layer neural network with h=16 neurons can approximate the score function well enough in d=33 dimensions (Section 5.1)
No approximation-error analysis or capacity argument is provided; the paper itself notes this is one of many possible configurations.
domain assumption Daily returns in each window are treated as i.i.d. draws from an unknown stationary distribution P0 (Section 1)
Real equity returns are non-stationary and autocorrelated; the method ignores time dependence and regime shifts within a window.
ad hoc to paper Passing the univariate two-sample Cramér-von Mises test on the equally weighted portfolio is a sufficient proxy for multivariate distributional equality (Section 6.2)
The paper acknowledges multivariate tests are challenging and defers a proper test to the authors' unpublished companion paper [16].

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Cite this review

Pith. "Pith review of Beyond Monte Carlo: Harnessing Diffusion Models to Simulate Financial Market Dynamics." pith.science (2026). https://pith.science/paper/NOO4HEO4

@misc{pith2026241200036,
  author       = {Pith},
  title        = {Pith review of: Beyond Monte Carlo: Harnessing Diffusion Models to Simulate Financial Market Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NOO4HEO4}},
  note         = {Machine review of arXiv:2412.00036}
}

read the original abstract

We propose a highly efficient and accurate methodology for generating synthetic financial market data using a diffusion model approach. The synthetic data produced by our methodology align closely with observed market data in several key aspects: (i) they pass the two-sample Cramer - von Mises test for portfolios of assets, and (ii) Q - Q plots demonstrate consistency across quantiles, including in the tails, between observed and generated market data. Moreover, the covariance matrices derived from a large set of synthetic market data exhibit significantly lower condition numbers compared to the estimated covariance matrices of the observed data. This property makes them suitable for use as regularized versions of the latter. For model training, we develop an efficient and fast algorithm based on numerical integration rather than Monte Carlo simulations. The methodology is tested on a large set of equity data.

Figures

Figures reproduced from arXiv: 2412.00036 by the authors.

**Figure 2.** Q-Q plot of the historical versus synthetic returns in Experiment 1. Experiment 2. We use days 250 through 506 of the data set. This period was marked by historically [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

**Figure 3.** Histograms of historical (left) and synthetic (right) returns in Experiment 2 [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: Q-Q plot of the historical versus synthetic returns in Experiment 2. Experiment 3. We use days 500 through 756 of the data set. This period was marked by rising inflation, elevated market volatility, and interest rate hikes implemented by the Federal Reserve. pCvM = 1.…

**Figure 5.** Figure 5: Histograms of historical (left) and synthetic (right) returns in Experiment 3 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]

**Figure 6.** Figure 6: Q-Q plot of the historical versus synthetic returns in Experiment 3. Experiment 4. We use days 750 through 1006 of the data set. During this period, the Federal Reserve concluded its rate-hiking program. pCvM = 0.83, κhist = 95.82, κsynth = 51.05. (101) [PITH_FULL_IMA…

**Figure 7.** Figure 7: Histograms of historical (left) and synthetic (right) returns in Experiment 4 [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]

**Figure 8.** Figure 8: Q-Q plot of the historical versus synthetic returns in Experiment 4. Experiment 5. We use days 1000 through 1256 of the data set. This period was characterized by occasional spikes in market volatility driven by the presidential election. pCvM = 0.98, κhist = 438.51, κ…

**Figure 9.** Figure 9: Histograms of historical (left) and synthetic (right) returns in Experiment 5 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]

**Figure 10.** Figure 10: Q-Q plot of the historical versus synthetic returns in Experiment 5. Experiment 6. In our final experiment we use days 1 through 1248 of the data set, which covers the entire five year period7 . In order to accommodate a larger data set, for this experiment we increas…

**Figure 11.** Figure 11: Histograms of historical (left) and synthetic (right) returns in Experiment 6 [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]

**Figure 12.** Figure 12: Q-Q plot of the historical versus synthetic returns in Experiment 6. It is worth noting that the period used for our experiments was characterized by rapidly shifting economic regimes, including market volatility from the pandemic, falling interest rates, subsequent i…

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [16]

Lesniewski, A., and Trigila, G.: Projected Cramer - von Mises test, in preparation

work page

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Kim, I., Balakrishnan, S., and Wasserman, L.: Robust Multivariate Nonparametric Tests via Projection-Averaging, Annals of Statistics, 48, 3417 - 3441 (2020)

work page 2020

[3] [1]

Anderson, B. D. O.: Reverse-time diffusion equation models, Stochastic Processes and their Applications, 12, 313 - 326 (1982)

work page 1982

[4] [2]

W.: On the distribution of the two-sample Cramer-von Mises criterion,The Annals of Mathematical Statistics, 1148 - 1159 (1962)

Anderson, T. W.: On the distribution of the two-sample Cramer-von Mises criterion,The Annals of Mathematical Statistics, 1148 - 1159 (1962)

work page 1962

[5] [3]

Johansson, K., Ogut, N.G, Pelger, M., Schmelzer, T., and Stephen Boyd, S.: A Simple Method for Predicting Covariance Matrices of Financial Returns, F oundations and Trends in Economet- rics, 12, 324 – 407 (2023)

work page 2023

[6] [4]

Fan, J., Li, R., Zhang, CH, and Hui, Z.: Statistical F oundations of Data Science , CRC Press (2020)

work page 2020

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G., and Pardoux, E.: Time Reversal of Diffusions, The Annals of Probability , 14, 1188 - 1205 (1986)

Haussmann, U. G., and Pardoux, E.: Time Reversal of Diffusions, The Annals of Probability , 14, 1188 - 1205 (1986)

work page 1986

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Simulating financial market dynamics 27

Ho, J., Jain, A., and Abbeel, P.: Denoising diffusion probabilistic models, Advances in Neural Information Processing Systems, 33, 6840 - 6851 (2020). Simulating financial market dynamics 27

work page 2020