StochLOB: Heavy-Tailed Score Matching for Robust Limit Order Book Trend Prediction

# StochLOB: Heavy-Tailed Score Matching for Robust Limit Order Book Trend Prediction
Source: [https://arxiv.org/html/2610.21081](https://arxiv.org/html/2610.21081)
\\workshoptitle

AI for Stochastic Dynamics

Arshia Abolghasemi Seyed Reza Tavakoli11footnotemark:1Parsa Naderi Nazanin MirzaeiBehrad Jahani Saba Yousefzadeh Tahereh Javaheri Mohammad Hosein RohbanGholamali Aminian Ali HabibniaDataism Laboratory for Quantitative Finance Sharif University of Technology Amirkabir University of Technology Boston University The Alan Turing Institute Virginia Tech††thanks:Equal contribution\.††thanks:Corresponding author: Ali Habibnia,[habibnia@vt\.edu](mailto:[email protected])\.

###### Abstract

Score\-based diffusion methods usually perturb data with Brownian Gaussian noise, corresponding to a light\-tailed forward stochastic differential equation \(SDE\)\. For market time series this SDE can be a poor abstraction: limit order book \(LOB\) order flow is heavy\-tailed, bursty, and occasionally jump\-like\. We studyStochLOB, a training\-only score\-matching framework that changes the forward diffusion corruption law while leaving the deployed predictor deterministic\. Instead of Brownian\-only Gaussian corruption, the noising process uses two non\-Gaussian SDE\-inspired alternatives: jump\-Lévynoise through a compound\-Poisson variance mixture, and clippedα\\alpha\-stable noise through a positive\-stable Gaussian scale mixture\. These corruptions inject finite jumps and heavy\-tailed residuals into training without requiring reverse\-time diffusion sampling at inference\. The resulting score target has a useful interpretation: its radial factorht​\(r\)h\_\{t\}\(r\)is the posterior expected corruption precision, so large residuals are weighted according to the latent corruption scale that most plausibly generated them\. On chronologically held\-out Coinbase LOB data, the proposed variants remain competitive on uncorrupted BTC and altcoin test horizons and improve several realized tail\-window stress diagnostics\. An exploratory Feishu A\-share transfer study further tests whether the probability outputs remain usable under transaction costs and trading\-rule constraints\. Code and configuration files are available at[https://github\.com/ArshiAbolghasemi/Penny](https://github.com/ArshiAbolghasemi/Penny)\.

## 1Introduction

Score\-based diffusion methods are commonly built around a Brownian forward stochastic differential equation \(SDE\), whose finite\-time corruption is Gaussian and light\-tailed\. This modeling choice is convenient, but it also defines the residual geometry seen by a denoising score objective: large coordinate movements become exponentially unlikely, and the learned restoring force is tied to a Brownian quadratic penalty\. For data generated by stochastic systems with bursts, jumps, or persistent heavy tails, the Brownian\-only SDE can therefore be a poor training corruption law\.

Limit order book \(LOB\) prediction makes this mismatch concrete\. Order\-flow imbalance \(OFI\), depth changes, and short\-horizon returns can remain quiet for many bins and then move abruptly after an order sweep, a liquidity withdrawal, or a short burst of directional trading\. These large residuals are not rare under the market process, but they are rare under Gaussian diffusion corruption\. This is difficult to reconcile with the heavy tails and jump\-like behavior observed in financial time series and LOB order flow\([Mandelbrot, 1963](https://arxiv.org/html/2610.21081#bib.bib1);[Cont, 2001](https://arxiv.org/html/2610.21081#bib.bib3);[Cont et al\., 2014](https://arxiv.org/html/2610.21081#bib.bib7)\)\.

The question we study is whether the forward diffusion SDE used for score\-based regularization should be modified so its noise law better matches the stochastic input process\. We introduceStochLOB, a training\-only framework for robust mid\-price trend prediction\. The deployed model remains a standard classifier: the corruption sampler and score head are used only while learning the shared representation, and test\-time inference is a single deterministic forward pass\. This isolates the effect of changing the stochastic diffusion corruption from the cost of reverse\-time diffusion sampling at deployment\.

We consider two non\-Gaussian SDE\-inspired corruption families\. The jump\-Lévyvariant adds a compound\-Poisson jump component to a Brownian background, implemented as a Gaussian scale mixture that exposes the representation to mostly ordinary perturbations with occasional finite jumps\. Theα\\alpha\-stable variant replaces the fixed Brownian variance with a clipped positive\-stable scale mixture, producing broad heavy\-tailed perturbations while retaining numerical stability\. Both variants use the same classifier architecture and differ only in the stochastic training law and auxiliary loss\. The score target is also interpretable: the radial score factorht​\(r\)h\_\{t\}\(r\)is a posterior precision under the latent corruption scale\. Thus the denoising objective does not apply the same restoring force to every residual\. Ordinary residuals receive high posterior precision, while extreme residuals can be explained by larger latent corruption scales\.

Our empirical analysis follows the same logic\. First, distribution\-fit and tail diagnostics on Coinbase OFI show that the Gaussian assumption is a poor description of the observed inputs\. Second, on Coinbase BTC and four additional crypto symbols, the proposed variants remain competitive on standard held\-out mid\-price trend prediction and win selected horizons, showing that the stochastic auxiliary objective does not require a large standard\-test accuracy tradeoff\. Third, realized stress\-window diagnostics evaluate the regimes that motivate the method: high\-stress, high\-variance, and jump\-residual windows in which Gaussian corruption is least plausible\. Figure[1](https://arxiv.org/html/2610.21081#S1.F1)gives the headline stress diagnostic atk=100k=100\. We also include an exploratory Feishu A\-share transfer study in which model probabilities are converted into constrained portfolio signals, checking whether the forecasts remain useful after transaction costs, lot\-size rules, and T\+1 selling constraints\. This transfer check is not meant to define a complete trading system; it tests whether the learned probabilities survive a realistic downstream decision rule\.

Contributions\.We make three contributions\. First, we formulate robust supervised representation learning as a mismatch between Brownian diffusion\-SDE corruption and heavy\-tailed, jump\-like stochastic inputs, using LOB trend prediction as the concrete test bed\. Second, we modify the training\-time diffusion corruption with jump\-Lévyand clippedα\\alpha\-stable noise while keeping the LOB predictor deterministic and free of inference\-time sampling\. Third, we provide a posterior\-precision interpretation of the resulting score functionhth\_\{t\}, with proofs and future\-work mathematics in Appendix[G](https://arxiv.org/html/2610.21081#A7)\. Appendices[A](https://arxiv.org/html/2610.21081#A1)–[F](https://arxiv.org/html/2610.21081#A6)provide the data details, score\-table construction, implementation and training algorithms, additional crypto results, and the Feishu portfolio analysis\.

Figure 1:Main stress result on chronologically held\-out Coinbase BTC windows at horizonk=100k=100\(31,84231\{,\}842test windows\)\. Columns sort windows by realized stress, realized variance, and variance\-controlled jump residual, all computed from one\-step increments of the level\-averaged feature stream\. Decile 10 is the most turbulent\. The top row reports macro\-F1 by decile; the bottom row reports paired differences against the same StochLOB trunk trained with cross\-entropy only\. Shaded bands are95%95\\%circular block\-bootstrap confidence intervals\.
## 2Related Work

LOB trend prediction\.LOB forecasting has moved from linear models toward architectures that exploit both level structure and temporal order\. DeepLOB combines convolutional, Inception, and recurrent blocks for mid\-price movement prediction\([Zhang et al\., 2019](https://arxiv.org/html/2610.21081#bib.bib10)\)\. TABL and BiN\-TABL add bilinear mappings, temporal attention, and input normalization for financial time series\([Tran et al\., 2019](https://arxiv.org/html/2610.21081#bib.bib32);[Tran et al\., 2021](https://arxiv.org/html/2610.21081#bib.bib33)\)\. Later work studies stronger attention and benchmarking designs, including TLOB, LiT, LOBench, Axial\-LOB, and order\-flow attention networks\([Berti and Kasneci, 2025](https://arxiv.org/html/2610.21081#bib.bib28);[Xiao et al\., 2025](https://arxiv.org/html/2610.21081#bib.bib29);[Zhong et al\., 2025](https://arxiv.org/html/2610.21081#bib.bib30);[Kisiel and Gorse, 2022](https://arxiv.org/html/2610.21081#bib.bib31);[Bandealinaeini et al\., 2025](https://arxiv.org/html/2610.21081#bib.bib9)\)\. Our experiments keep the standard three\-class mid\-price trend protocol and compare against linear, recurrent, convolutional\-attention, transformer, and single\-asset order\-flow attention baselines trained on the same chronological windows\.

Heavy tails and jumps\.Heavy tails are a classical stylized fact of financial returns\([Mandelbrot, 1963](https://arxiv.org/html/2610.21081#bib.bib1);[Cont, 2001](https://arxiv.org/html/2610.21081#bib.bib3)\); LOB\-specific studies show non\-Gaussian behavior in liquidity, impact, and order flow\([Bouchaud et al\., 2002](https://arxiv.org/html/2610.21081#bib.bib4);[Mike and Farmer, 2008](https://arxiv.org/html/2610.21081#bib.bib5);[Gould et al\., 2013](https://arxiv.org/html/2610.21081#bib.bib6);[Cont et al\., 2014](https://arxiv.org/html/2610.21081#bib.bib7)\)\. Jump\-diffusion models provide a finite\-activity language for abrupt price moves\([Merton, 1976](https://arxiv.org/html/2610.21081#bib.bib2)\), while stable processes model persistent heavy\-tailed fluctuations\([Samorodnitsky and Taqqu, 1994](https://arxiv.org/html/2610.21081#bib.bib18)\)\. We use these objects as training\-time corruption laws, not as complete market simulators\.

Diffusion as representation learning\.Score\-based and denoising diffusion models are typically introduced as generative models\([Ho et al\., 2020](https://arxiv.org/html/2610.21081#bib.bib11);[Song et al\., 2021](https://arxiv.org/html/2610.21081#bib.bib14)\), but the denoising task can also shape supervised representations\. Joint diffusion training uses shared networks for denoising and prediction\([Deja et al\., 2023](https://arxiv.org/html/2610.21081#bib.bib12)\), and related work studies diffusion features as representation learners\([Yang and Wang, 2023](https://arxiv.org/html/2610.21081#bib.bib13)\)\. Non\-Gaussian diffusion replaces Brownian noise with jump or heavy\-tailed processes\([Yoon et al\., 2023](https://arxiv.org/html/2610.21081#bib.bib20);[Shariatian et al\., 2025](https://arxiv.org/html/2610.21081#bib.bib23);[Pandey et al\., 2025](https://arxiv.org/html/2610.21081#bib.bib22)\)\. LOB diffusion work models event streams, counterfactual books, or volume snapshots\([Zheng et al\., 2024](https://arxiv.org/html/2610.21081#bib.bib24);[Wang and Ventre, 2026](https://arxiv.org/html/2610.21081#bib.bib26);[Wang and Ventre, 2025](https://arxiv.org/html/2610.21081#bib.bib25)\);StochLOBinstead uses non\-Gaussian corruption as a training\-time auxiliary regularizer for deterministic LOB classification\.

## 3Background

Limit order book \(LOB\) data record the outstanding limit orders waiting at an exchange\. At any time, the book has a bid side, where traders are willing to buy, and an ask side, where traders are willing to sell\. Each side is ordered by price into levels: the first bid and ask levels are the immediately executable quotes, while deeper levels describe nearby liquidity that would be reached after the top quotes are consumed\. A raw book snapshot is therefore a structured row rather than a flat tabular observation: for each displayed level it contains a bid price, bid volume, ask price, and ask volume\. With twenty displayed levels, one snapshot contains eighty raw book fields before trade\-flow and derived features are added\.

Across time, consecutive snapshots show how queues build, vanish, or move across levels\. These changes are the main signal in LOB prediction because they reveal buying and selling pressure before it appears as a mid\-price move\.

Following the multi\-level OFI representation of[Kolm et al\. \(2023\)](https://arxiv.org/html/2610.21081#bib.bib8), which builds on the order\-flow imbalance construction of[Cont et al\. \(2014\)](https://arxiv.org/html/2610.21081#bib.bib7), we use OFI as the primary LOB representation\. OFI summarizes whether liquidity is added, cancelled, or consumed on the bid and ask sides before the mid\-price moves\. New bid depth or removed ask depth pushes signed flow positive; removed bid depth or new ask depth pushes it negative\. Aggregating these signed changes converts irregular book updates into a regular time\-series signal for neural sequence models\. Appendix[A\.1](https://arxiv.org/html/2610.21081#A1.SS1)gives the full LOB notation, Coinbase feature construction, and exact OFI aggregation formula in Eq\. \([10](https://arxiv.org/html/2610.21081#A1.E10)\)\. Appendix[A\.4](https://arxiv.org/html/2610.21081#A1.SS4)reports the train\-only heavy\-tail diagnostics, including Figures[4](https://arxiv.org/html/2610.21081#A1.F4)and[5](https://arxiv.org/html/2610.21081#A1.F5)\.

The representation\-learning question is whether the auxiliary corruption task should match the market residuals seen by OFI windows\. Many bins contain small queue adjustments, but active periods can contain abrupt order sweeps, liquidity withdrawals, or bursts of one\-sided trading\. A pooled Hill/Pareto tail\-index estimate on Coinbase OFI magnitudes givesα^=1\.62\\hat\{\\alpha\}=1\.62, well below Gaussian light\-tail behavior \(Appendix[A\.4](https://arxiv.org/html/2610.21081#A1.SS4)\)\. A Gaussian denoising task emphasizes a fixed quadratic geometry around the clean input; the non\-Gaussian score tasks below instead expose the shared classifier trunk to larger residuals during training\. The reverse diffusion chain is never run at inference, so the method is stochastic representation learning rather than a generative market simulator\.

## 4Methodology

StochLOBis designed to change the training signal while leaving the deployed predictor unchanged\. Each minibatch is seen in two views\. The clean windowx0x\_\{0\}is passed through the classifier path and optimized with the ordinary three\-class trend loss\. A second, stochastically corrupted viewxtx\_\{t\}is passed through a score head attached to the same temporal trunk\. Only the shared representation is common between these objectives; at test time the noisy view, corruption sampler, score table, and score head are all disabled\.

### 4\.1Problem and Labels

Given a normalized LOB feature windowx0∈ℝT×Fx\_\{0\}\\in\\mathbb\{R\}^\{T\\times F\}, the classifier returns probabilities over\{down,stationary,up\}\\\{\\mathrm\{down\},\\mathrm\{stationary\},\\mathrm\{up\}\\\}\. For Coinbase,T=60T=60andF=31F=31: each window covers 10 minutes sampled every 10 seconds\. Following the DeepLOB smoothed\-mid protocol\([Zhang et al\., 2019](https://arxiv.org/html/2610.21081#bib.bib10)\), the return target at horizonkkis

rk​\(τ\)=1k​∑i=1kpτ\+i−1k​∑i=1kpτ−i1k​∑i=1kpτ−i,r\_\{k\}\(\\tau\)=\\frac\{\\frac\{1\}\{k\}\\sum\_\{i=1\}^\{k\}p\_\{\\tau\+i\}\-\\frac\{1\}\{k\}\\sum\_\{i=1\}^\{k\}p\_\{\\tau\-i\}\}\{\\frac\{1\}\{k\}\\sum\_\{i=1\}^\{k\}p\_\{\\tau\-i\}\},\(1\)and labels are assigned by train\-calibrated thresholds

yτ=\{up,rk​\(τ\)\>αlab,down,rk​\(τ\)<−αlab,stationary,\|rk​\(τ\)\|≤αlab\.y\_\{\\tau\}=\\begin\{cases\}\\mathrm\{up\},&r\_\{k\}\(\\tau\)\>\\alpha\_\{\\mathrm\{lab\}\},\\\\ \\mathrm\{down\},&r\_\{k\}\(\\tau\)<\-\\alpha\_\{\\mathrm\{lab\}\},\\\\ \\mathrm\{stationary\},&\|r\_\{k\}\(\\tau\)\|\\leq\\alpha\_\{\\mathrm\{lab\}\}\.\\end\{cases\}\(2\)The standard supervised loss isℒcls=CE⁡\(fθ​\(x0\),y\)\\mathcal\{L\}\_\{\\mathrm\{cls\}\}=\\mathrm\{CE\}\(f\_\{\\theta\}\(x\_\{0\}\),y\)\.

The chronological construction is important\. For a window centered atτ\\tau, the input uses only observations up toτ\\tau, while the label uses the smoothed future mid\-price in Eq\. \([1](https://arxiv.org/html/2610.21081#S4.E1)\)\. Windows are therefore assigned to train, validation, or test only when the full look\-back context and the full future label horizon remain inside the same split\. This avoids leakage from overlapping windows near split boundaries\.

### 4\.2Generalized Scale\-Mixture Score Matching

StochLOBcorrupts an unperturbed input through a Gaussian scale mixture:

xt=at​x0\+u,u=Wt​ξ,ξ∼𝒩⁡\(0,Id\)\.x\_\{t\}=a\_\{t\}x\_\{0\}\+u,\\qquad u=\\sqrt\{W\_\{t\}\}\\xi,\\qquad\\xi\\sim\\mathcal\{N\}\(0,I\_\{d\}\)\.\(3\)Hereata\_\{t\}follows a standard variance\-preserving or cosine schedule, while the varianceWtW\_\{t\}is random\. Gaussian diffusion is the special caseWt=σt2W\_\{t\}=\\sigma\_\{t\}^\{2\}\. Non\-Gaussian corruption changes only the scale lawpt​\(W\)p\_\{t\}\(W\), so the same classifier can be trained under light\-tailed, jump\-like, or heavy\-tailed perturbations\.

We choose Gaussian scale mixtures because they keep the score target tractable while allowing non\-Gaussian residuals\. Conditional onWt=wW\_\{t\}=w, the corruption is Gaussian and isotropic\. Marginally, however, the variance is random, so the residual distribution can place much more probability on large moves\. This gives a controlled way to expose the classifier representation to market\-shaped perturbations without changing the input schema or the inference\-time architecture\.

The additive\-noise density is

qt\(u\)=𝔼Wt\[\(2πWt\)−d/2exp\(−∥u∥22​Wt\)\]\.q\_\{t\}\(u\)=\\mathbb\{E\}\_\{W\_\{t\}\}\\left\[\(2\\pi W\_\{t\}\)^\{\-d/2\}\\exp\\left\(\-\\frac\{\\lVert u\\rVert^\{2\}\}\{2W\_\{t\}\}\\right\)\\right\]\.\(4\)Its score is radial:

∇u​log​qt​\(u\)=−u​ht​\(∥u∥\),\\nabla\_\{u\}\\log q\_\{t\}\(u\)=\-u\\,h\_\{t\}\(\\lVert u\\rVert\),\(5\)where

ht​\(r\)=𝔼W\[W−1W−d/2exp\(−r2/2W\)\]𝔼W\[W−d/2exp\(−r2/2W\)\]\.h\_\{t\}\(r\)=\\frac\{\\mathbb\{E\}\_\{W\}\[W^\{\-1\}W^\{\-d/2\}\\exp\(\-r^\{2\}/2W\)\]\}\{\\mathbb\{E\}\_\{W\}\[W^\{\-d/2\}\\exp\(\-r^\{2\}/2W\)\]\}\.\(6\)The functionht​\(r\)h\_\{t\}\(r\)can be interpreted as the posterior expected corruption precision,𝔼⁡\[Wt−1∣∥u∥=r\]\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\\right\], as shown in more detail in Appendix[G](https://arxiv.org/html/2610.21081#A7)\(Posterior\-Precision Interpretation ofhth\_\{t\}\)\. Following[Baule \(2025\)](https://arxiv.org/html/2610.21081#bib.bib17),StochLOBtabulates this one\-dimensional function by Monte Carlo over residual radii and diffusion steps, then interpolates it during training\. Appendix[B\.1](https://arxiv.org/html/2610.21081#A2.SS1)establishes that this tabulation procedure is a self\-normalized importance\-sampling estimator\([Geweke, 1989](https://arxiv.org/html/2610.21081#bib.bib34)\)of a Tweedie\-type posterior expectation\([Efron, 2011](https://arxiv.org/html/2610.21081#bib.bib43)\)\. For each fixedr\>0r\>0, the estimator is strongly consistent, satisfies an explicit non\-asymptotic error bound, and is asymptotically normal with a closed\-form variance under the Gaussian, jump\-Poisson, and clipped alpha\-stable corruption laws used in this paper\. This posterior\-score view is a Gaussian scale\-mixture instance of classical empirical\-Bayes score identities\([Andrews and Mallows, 1974](https://arxiv.org/html/2610.21081#bib.bib40);[Efron, 2011](https://arxiv.org/html/2610.21081#bib.bib43)\)\.

The score loss is

ℒscore=𝔼t,Wt,ξ​\[wt​‖sθ​\(xt,t\)\+u​ht​\(∥u∥\)‖mean2\]\.\\mathcal\{L\}\_\{\\mathrm\{score\}\}=\\mathbb\{E\}\_\{t,W\_\{t\},\\xi\}\\left\[w\_\{t\}\\left\\\|s\_\{\\theta\}\(x\_\{t\},t\)\+u\\,h\_\{t\}\(\\lVert u\\rVert\)\\right\\\|^\{2\}\_\{\\mathrm\{mean\}\}\\right\]\.\(7\)The scalarwtw\_\{t\}is a deterministic per\-step denoising\-score\-matching weight:wt=𝔼⁡\[Wt\]w\_\{t\}=\\mathbb\{E\}\[W\_\{t\}\]for finite\-variance Gaussian and jump corruptions, andwt=1w\_\{t\}=1for the reported clippedα\\alpha\-stable runs\. The classifier and score head share the temporal trunk, but only the classifier path is used at test time\. The complete training objective is

ℒ=ℒcls\+λdiff​ℒscore\.\\mathcal\{L\}=\\mathcal\{L\}\_\{\\mathrm\{cls\}\}\+\\lambda\_\{\\mathrm\{diff\}\}\\mathcal\{L\}\_\{\\mathrm\{score\}\}\.

### 4\.3Two Market\-Shaped Corruptions

StochLOB\(Lévy\)\.The jump variant uses a Brownian background plus a compound\-Poisson gamma variance:

Wt=σt2\+∑j=1NtSj,Nt∼Poisson⁡\(Λt\),Sj∼Gamma⁡\(κ,θ\)\.W\_\{t\}=\\sigma\_\{t\}^\{2\}\+\\sum\_\{j=1\}^\{N\_\{t\}\}S\_\{j\},\\qquad N\_\{t\}\\sim\\mathrm\{Poisson\}\(\\Lambda\_\{t\}\),\\quad S\_\{j\}\\sim\\mathrm\{Gamma\}\(\\kappa,\\theta\)\.\(8\)This exposes the representation to mostly ordinary perturbations with occasional large shocks, matching the intuition of order sweeps or liquidity withdrawals\.

StochLOB\(α\\alpha\-stable\)\.The stable variant uses

Wt=σt2​A,A∼𝒮α/2\+,𝔼⁡\[exp⁡\(−s​A\)\]=exp⁡\(−sα/2\)\.W\_\{t\}=\\sigma\_\{t\}^\{2\}A,\\qquad A\\sim\\mathcal\{S\}^\{\+\}\_\{\\alpha/2\},\\qquad\\mathbb\{E\}\[\\exp\(\-sA\)\]=\\exp\(\-s^\{\\alpha/2\}\)\.\(9\)This creates polynomial\-tail perturbations in the ideal case\. The implementation clipsAAat a high quantile for numerical stability, giving a truncated\-stable corruption law that remains much heavier\-tailed than a Gaussian\. Appendix[A](https://arxiv.org/html/2610.21081#A1)\(Data Details, Alpha Selection, and Tail Diagnostics\), especially Appendices[A\.2](https://arxiv.org/html/2610.21081#A1.SS2)and[A\.3](https://arxiv.org/html/2610.21081#A1.SS3), explains howα\\alphaandαlab\\alpha\_\{\\mathrm\{lab\}\}are selected without test leakage\.

### 4\.4Architecture

All proposed variants use the same classifier family: optional BiN normalization, a two\-layer bidirectional GRU, one temporal self\-attention block with timestep conditioning, attention pooling, and a three\-class classifier head\. The score head is a training\-only branch attached to the shared representation\. Figure[2](https://arxiv.org/html/2610.21081#S4.F2)shows the complete training workflow and the inference path without stochastic corruption\. Using a shared architecture isolates the effect of the corruption law: changes in test behavior are attributable to the auxiliary stochastic objective rather than to a larger inference model\. Training, inference, and score\-table pseudocode are given in Appendix[B](https://arxiv.org/html/2610.21081#A2)\(GSM Score Table\) and Appendix[D](https://arxiv.org/html/2610.21081#A4)\(Training and Inference Pseudocode\)\.

Figure 2:StochLOB workflow\. The classifier pools the unperturbed sequence representation for three trend logits\. The training\-only score decoder receives the noisyT×FT\\times Fwindow, timestep embedding, and per\-timestep trunk context, and predicts aT×FT\\times Fscore field\. At inference, the corruption sampler and score decoder are disabled\.

## 5Datasets

We use point\-in\-time market data from crypto and equity order books\. Appendix[A](https://arxiv.org/html/2610.21081#A1)gives the exact timestamps, feature construction, normalization, split counts, label calibration, and tail diagnostics\.

Coinbase crypto\.The main benchmark is Coinbase order\-book and trade data for BTC, ETH, BNB, XRP, and USDT\. The main text focuses on BTC as the deepest symbol; ETH, BNB, XRP, and USDT results are in Appendix[E](https://arxiv.org/html/2610.21081#A5)\.

Feishu A\-share universe\.Feishu is a broader Chinese A\-share transfer setting\. Each asset\-day combines intraday OFI summaries with daily OHLCV and technical variables\. The transfer study evaluates the resulting directional probabilities through a common constrained portfolio rule, described in Appendix[F](https://arxiv.org/html/2610.21081#A6)\.

## 6Experiments

### 6\.1Experiment Setup

All Coinbase experiments use chronological 70/15/15 train, validation, and test splits, so every model is evaluated on later market periods than those used for training\. Checkpoints are selected by validation macro\-F1 and then frozen for all reported test metrics\. The main tables use OFI windows withT=60T=60,F=31F=31, and horizonsk∈\{10,20,50,100\}k\\in\\\{10,20,50,100\\\}\. Optimizer, corruption, seed\-sweep, and Feishu transfer details are given in Appendix[C](https://arxiv.org/html/2610.21081#A3)\.

### 6\.2Mid\-Price Trend Prediction Results

Table[1](https://arxiv.org/html/2610.21081#S6.T1)reports standard held\-out BTC mid\-price trend prediction results with seed\-sweep variability\. StochLOB \(jump\-Poisson\) has the highest mean macro\-F1 atk=10k=10, StochLOB \(alpha\-stable\) atk=20k=20, and StochLOB variants are competitive at the remaining horizons\. The standard\-test ranking is mixed, so the proposed variants are not presented as across\-the\-board leaderboard winners\. The primary claim is that non\-Gaussian score corruption remains competitive on BTC and improves stability in realized tail\-window regimes\.

Table 1:BTC OFI trend prediction on chronologically held\-out windows\. Accuracy \(Acc\) and macro\-F1 are reported as mean±\\pmstd over five seeds \(13,21,42,84,16813,21,42,84,168\)\. Bold marks the best mean result in each horizon/metric column; underlining marks the second best\.
### 6\.3Stress Stability

Standard held\-out trend prediction does not fully test behavior under difficult market states\. Figure[1](https://arxiv.org/html/2610.21081#S1.F1)shows the BTC stress comparison, and Table[2](https://arxiv.org/html/2610.21081#S6.T2)reports a compact summary of extreme\-window results across stress, high\-variance, and jump\-residual regimes\. For each normalized OFI window, we average features across levels and compute one\-step incrementsΔ​x\\Delta x\. The stress score ismax⁡\|Δ​x\|/std⁡\(Δ​x\)\\max\|\\Delta x\|/\\mathrm\{std\}\(\\Delta x\), the high\-variance score isvar⁡\(Δ​x\)\\mathrm\{var\}\(\\Delta x\), and the jump\-residual score is the residual from regressinglog⁡max​\|Δ​x\|\\log\\max\|\\Delta x\|onlog⁡var⁡\(Δ​x\)\\log\\mathrm\{var\}\(\\Delta x\)within the same horizon\. Extreme windows are the top decile of the corresponding score\. The full pre\-specified regime\-by\-horizon grid is in Appendix Table[9](https://arxiv.org/html/2610.21081#A5.T9), with implementation details in Appendix[C](https://arxiv.org/html/2610.21081#A3)\. These comparisons support the paper’s empirical claim: StochLOB variants are not uniformly best on standard held\-out data, but heavy\-tailed score corruption gives measurable stability gains in several tail\-region test windows\.

Table 2:Compact summary of extreme\-window macro\-F1 comparisons on held\-out BTC regimes\. Each row reports the best heavy\-tailed StochLOB variant in that regime and horizon, the matched Gaussian StochLOB baseline, and the strongest supervised comparator in the same row\.Best F1 in each row is shown in bold\.Gains are absolute macro\-F1 differences; Appendix Table[9](https://arxiv.org/html/2610.21081#A5.T9)reports the full regime\-by\-horizon grid\.
### 6\.4Feishu Portfolio Transfer

Table[3](https://arxiv.org/html/2610.21081#S6.T3)and Figure[3](https://arxiv.org/html/2610.21081#S6.F3)report the Feishu out\-of\-sample portfolio transfer check\. The classifier is trained for mid\-price direction; the fixed downstream portfolio rule tests whether its probabilities remain usable after frictions and trading constraints\. Details are in Appendices[F\.1](https://arxiv.org/html/2610.21081#A6.SS1),[F\.2](https://arxiv.org/html/2610.21081#A6.SS2), and[F\.3](https://arxiv.org/html/2610.21081#A6.SS3)\. Among the active models shown, StochLOB \(jump\-Poisson\) reaches 38\.78% CAGR and Sharpe 1\.04, with the lowest maximum drawdown and volatility\.

Table 3:Feishu out\-of\-sample constrained portfolio metrics\. Bold denotes the best result in each column across the shown active models\. Other evaluated models are omitted because they mostly predict the flat class and therefore generate little or no trading activity\.Figure 3:Exploratory Feishu portfolio\-management transfer result\. Cumulative returns are shown for the constrained portfolio rule using StochLOB variants, supervised baselines, and the passive equal\-weight reference\.

## 7Discussion

The central message ofStochLOBis not that one architecture should dominate every standard benchmark cell\. It is that the auxiliary corruption used for representation learning should match the regime in which robustness is needed\. Crypto OFI inputs are heavy\-tailed, ordinary Gaussian corruption is a limited proxy for market stress, and non\-Gaussian score matching gives the shared representation a geometry that is more appropriate for large\-residual states\. The posterior\-precision view ofhth\_\{t\}makes this mechanism explicit: as the residual magnitude changes, the score adapts to the corruption scale that is plausible under the chosen noise law\.

There are limits\. The compound\-Poisson law is still isotropic after scaling; real book shocks are signed, level\-correlated, and state dependent\. The alpha\-stable law is clipped for numerical stability, so the implementation is a truncated\-stable training distribution rather than a pure infinite\-variance process\. Feishu also shows that robust probabilities are not automatically portfolio\-ready; the alpha\-stable model may need threshold or utility\-aware calibration before its conservative signal becomes economically useful\.

Future work should turn this into a sharper coverage criterion\. The present corruptions improve radial coverage by making large residual magnitudes plausible during training, but they do not guarantee angular coverage: real book stress is signed, cross\-level, and concentrated along a small number of market directions\. Appendix[G](https://arxiv.org/html/2610.21081#A7), especially Appendix[G\.1](https://arxiv.org/html/2610.21081#A7.SS1), formalizes this with a second\-order change\-of\-measure argument in which robustness transfer depends on both radial tail compatibility and an angular Rényi term that can dominate in high\-dimensional LOB windows\. This motivates four next steps: estimate state\-dependent directional stress spectra and replace scalarht​\(r\)h\_\{t\}\(r\)with matrix\-valued posterior precision; replay signed empirical jumps from real order\-book residuals; separate the score loss and temporal readout with multi\-seed ablations; and select portfolio thresholds on validation data under transaction costs and risk constraints\.

## 8Conclusion

This work reframes LOB diffusion regularization around the noise geometry observed in market data\.StochLOBkeeps inference as a deterministic classifier, but trains the shared representation with jump\-like or heavy\-tailed score\-matching corruption\. The resulting models remain competitive on standard Coinbase trend prediction, improve several realized stress\-window diagnostics, and produce probabilities that can be tested under a constrained Feishu portfolio rule\. Appendix[G](https://arxiv.org/html/2610.21081#A7)\(Posterior\-Precision Interpretation ofhth\_\{t\}\) proves thathth\_\{t\}is a posterior expected corruption precision; Appendices[D](https://arxiv.org/html/2610.21081#A4)\(Training and Inference Pseudocode\) and[C](https://arxiv.org/html/2610.21081#A3)\(Experimental Setup and Implementation Details\) provide the operational details\.

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- Zhanget al\.\(2019\)Z\. Zhang, S\. Zohren, and S\. RobertsDeepLOB: deep convolutional neural networks for limit order books\.IEEE Transactions on Signal Processing67\(11\),pp\. 3001–3012\.External Links:[Document](https://dx.doi.org/10.1109/TSP.2019.2907260),[Link](https://doi.org/10.1109/TSP.2019.2907260)Cited by:[§2](https://arxiv.org/html/2610.21081#S2.p1.1),[§4\.1](https://arxiv.org/html/2610.21081#S4.SS1.p1.1)\.
- Zhenget al\.\(2024\)Z\. Zheng, G\. Li, D\. Ouyang, D\. Liang, and J\. ShaoLimit order book event stream prediction with diffusion model\.arXiv preprint arXiv:2412\.09631\.External Links:[Document](https://dx.doi.org/10.48550/arXiv.2412.09631),[Link](https://doi.org/10.48550/arXiv.2412.09631)Cited by:[§2](https://arxiv.org/html/2610.21081#S2.p3.1)\.
- Zhonget al\.\(2025\)M\. Zhong, Y\. Lin, and P\. YangRepresentation learning of limit order book: a comprehensive study and benchmarking\.arXiv preprint arXiv:2505\.02139\.External Links:[Document](https://dx.doi.org/10.48550/arXiv.2505.02139),[Link](https://doi.org/10.48550/arXiv.2505.02139)Cited by:[§2](https://arxiv.org/html/2610.21081#S2.p1.1)\.

## Appendix AData Details, Alpha Selection, and Tail Diagnostics

### A\.1Coinbase Crypto Protocol and LOB Representation

The Coinbase dataset covers BTC, ETH, BNB, XRP, and USDT\. The raw data were crawled from Coinbase as paired order\-book and trade CSV files and then normalized into the shared parquet schema used by the experiments\. The feed is sampled every 10 seconds from 2026\-06\-18 23:42:10 UTC to approximately 2026\-07\-13 23:10:40 UTC, giving 26 calendar dates and roughly 213k snapshots per symbol\. Each input window hasT=60T=60steps, so it covers 10 minutes of recent book and trade\-flow state\. Prediction horizonsk=\{10,20,50,100\}k=\\\{10,20,50,100\\\}correspond to 100, 200, 500, and 1000 seconds\.

At sampling timeτ\\tau, the exchange exposes the topLLbid and ask levels\. Bid levelffis theff\-th highest buy price and ask levelffis theff\-th lowest sell price, so

bτ1≥bτ2≥⋯≥bτL,aτ1≤aτ2≤⋯≤aτL\.b\_\{\\tau\}^\{1\}\\geq b\_\{\\tau\}^\{2\}\\geq\\cdots\\geq b\_\{\\tau\}^\{L\},\\qquad a\_\{\\tau\}^\{1\}\\leq a\_\{\\tau\}^\{2\}\\leq\\cdots\\leq a\_\{\\tau\}^\{L\}\.With aggregate resting volumesvτ,bfv\_\{\\tau,b\}^\{f\}andvτ,afv\_\{\\tau,a\}^\{f\}, a depth\-LLsnapshot is

ℬτ=\{\(bτf,vτ,bf\),\(aτf,vτ,af\)\}f=1L\.\\mathcal\{B\}\_\{\\tau\}=\\left\\\{\(b\_\{\\tau\}^\{f\},v\_\{\\tau,b\}^\{f\}\),\(a\_\{\\tau\}^\{f\},v\_\{\\tau,a\}^\{f\}\)\\right\\\}\_\{f=1\}^\{L\}\.The level\-1 spread isaτ1−bτ1a\_\{\\tau\}^\{1\}\-b\_\{\\tau\}^\{1\}, and the mid\-price is\(bτ1\+aτ1\)/2\(b\_\{\\tau\}^\{1\}\+a\_\{\\tau\}^\{1\}\)/2\. The Coinbase raw snapshot columns are\{bτf,vτ,bf,aτf,vτ,af\}f=120\\\{b\_\{\\tau\}^\{f\},v\_\{\\tau,b\}^\{f\},a\_\{\\tau\}^\{f\},v\_\{\\tau,a\}^\{f\}\\\}\_\{f=1\}^\{20\}, sampled into 10\-second bins before OFI and auxiliary feature construction\.

OFI mode uses 20 bid/ask order\-flow coordinates plus the 11 auxiliary features in Table[5](https://arxiv.org/html/2610.21081#A1.T5)\. Feature extraction is per calendar day for order\-flow differences\. Normalization is a causal trailing rollingzz\-score over the chronological series, using a 2000\-bin window\. Thus a feature at binttuses only statistics from the past and current bin, never future test data\.

Following[Cont et al\. \[2014\]](https://arxiv.org/html/2610.21081#bib.bib7)and[Kolm et al\. \[2023\]](https://arxiv.org/html/2610.21081#bib.bib8), cumulative OFI for levelffover an interval\(t−h,t\)\(t\-h,t\)is

OFIt−h,tf=∑n=N⁡\(t−h\)\+1N⁡\(t\)\(bOFnf−aOFnf\)\.\\mathrm\{OFI\}\_\{t\-h,t\}^\{f\}=\\sum\_\{n=N\(t\-h\)\+1\}^\{N\(t\)\}\\left\(\\mathrm\{bOF\}\_\{n\}^\{f\}\-\\mathrm\{aOF\}\_\{n\}^\{f\}\\right\)\.\(10\)
All Coinbase splits are chronological: 70% train, 15% validation, and 15% test\. A window is assigned to a split only when its look\-back context and future label horizon remain inside the same raw interval\. This prevents overlapping windows from leaking across train/test boundaries\. Table[4](https://arxiv.org/html/2610.21081#A1.T4)reports the crawl coverage and raw test\-start boundary for each crypto symbol\.

For BTC, the row\-index split boundaries are 2026\-06\-18 23:42:10 UTC for the first training row, 2026\-07\-06 11:40:10 UTC for the first validation row, and 2026\-07\-10 05:24:10 UTC for the first test row; the final raw timestamp is 2026\-07\-13 23:10:40 UTC\. Because the input look\-back and label horizon must remain inside the test interval, the first BTC test label center is 2026\-07\-10 05:34:00 UTC\. The last label centers are 2026\-07\-13 23:09:00, 23:07:20, 23:02:10, and 22:53:50 UTC fork=10,20,50,100k=10,20,50,100, respectively; in all four cases the final future label timestamp is 2026\-07\-13 23:10:40 UTC\. Table[6](https://arxiv.org/html/2610.21081#A1.T6)gives the resulting BTC split sizes and train\-calibrated label thresholds by horizon\.

Table 4:Coinbase crawl coverage and chronological test\-start timestamps computed from the normalized local parquet files\. Test start denotes the raw snapshot index at the 85% chronological split boundary before applying horizon\-specific look\-back and future\-label validity constraints\.Table 5:The 11 auxiliary features appended to the 20 OFI levels\. Repeated activity proxies are intentional substitutes for unavailable intra\-bin quote\-update counts in the resampled exchange feed\.Table 6:BTC split sizes and train\-calibrated label thresholds\. Other crypto symbols use the same protocol with symbol\-specific thresholds\.
### A\.2Howαlab\\alpha\_\{\\mathrm\{lab\}\}Is Selected

The label thresholdαlab\\alpha\_\{\\mathrm\{lab\}\}controls the dead band between “stationary” and a directional move\. For crypto, it is calibrated on the training split only\. We first compute the smoothed\-mid trend ratiork​\(τ\)r\_\{k\}\(\\tau\)from Eq\. \([1](https://arxiv.org/html/2610.21081#S4.E1)\) over the training interval, then set

αlab=quantile1/3\(\|rk\(τ\)\|:τ∈train\)\.\\alpha\_\{\\mathrm\{lab\}\}=\\mathrm\{quantile\}\_\{1/3\}\\left\(\\left\|r\_\{k\}\(\\tau\)\\right\|:\\tau\\in\\mathrm\{train\}\\right\)\.This gives an approximately balanced three\-class split while preserving the chronological validation and test intervals\. If a config setslabel\_alphato a positive number, that fixed value is used instead; the Coinbase experiments reported here uselabel\_alpha: \-1, meaning train\-only calibration\.

For Feishu mid\-price experiments, the default label mode is quantile rather than a symmetric fixed band\. The training\-split return distribution is cut at the1/31/3and2/32/3quantiles:

ℓ=quantile1/3\(rt:t∈train\),u=quantile2/3\(rt:t∈train\)\.\\ell=\\mathrm\{quantile\}\_\{1/3\}\(r\_\{t\}:\\,t\\in\\mathrm\{train\}\),\\qquad u=\\mathrm\{quantile\}\_\{2/3\}\(r\_\{t\}:\\,t\\in\\mathrm\{train\}\)\.Labels are down forrt<ℓr\_\{t\}<\\ell, stationary forℓ≤rt≤u\\ell\\leq r\_\{t\}\\leq u, and up forrt\>ur\_\{t\}\>u\. The same\(ℓ,u\)\(\\ell,u\)thresholds are then frozen for validation and out\-of\-sample portfolio evaluation\. A fixed±α\\pm\\alphaband is also available, but the Feishu mid\-price experiments reported here uselabel\_mode: "quantile"andclass\_quantiles: \[0\.3333, 0\.6667\]\.

### A\.3How the Alpha\-Stable Tail Index Is Selected

The alpha\-stable parameterα\\alphais separate fromαlab\\alpha\_\{\\mathrm\{lab\}\}\. It controls the tail thickness of the training corruption: smaller values show the score head larger perturbations more often, whileα=2\\alpha=2recovers Gaussian\-type behavior\. We select this quantity from training data, not from validation or test performance\.

For the Coinbase configs,α\\alphais based on a Hill/Pareto tail estimate of normalized OFI magnitudes\. The pooled diagnostic givesα^=1\.62\\hat\{\\alpha\}=1\.62, which motivated the heavy\-tailed corruption family\. The final per\-symbol configs use clipped symbol\-specific estimates in the practical range\[1\.2,1\.9\]\[1\.2,1\.9\], shown in Table[7](https://arxiv.org/html/2610.21081#A1.T7)\. BTC therefore uses a slightly heavier\-than\-Gaussian setting near the observed OFI tail, while BNB is clipped at the upper end because its estimate is closer to light\-tailed behavior under this diagnostic\.

Table 7:Coinbase alpha\-stable corruption indices from train\-only OFI tail calibration\.For Feishu, we do not reuse the crypto per\-symbol calibration because the task is a broad A\-share transfer experiment with daily asset windows and a different market microstructure\. The Feishu alpha\-stable configs therefore useα=1\.5\\alpha=1\.5as a conservative heavy\-tailed transfer default\. This value is not tuned on the out\-of\-sample portfolio period\.

### A\.4Heavy\-Tail Evidence

The distribution\-fit diagnostics rank candidate laws by Kolmogorov\-Smirnov distance on subsampled series\. Figure[4](https://arxiv.org/html/2610.21081#A1.F4)shows the BTC mid\-price\-increment and feature\-PC1 distribution fits\. Across all crypto symbols, the normal reference is not the leading fit\. For mid\-price increments, it ranks 11th \(BTC\), 13th \(BNB\), 12th \(ETH\), 20th \(XRP\), and 16th \(USDT\) among roughly 83 candidates\. For the first principal component of the full feature tensor, the normal ranks are 31st, 65th, 25th, 40th, and 26th\. Feishu shows the same direction: normal ranks 24th for mid\-price increments and 63rd for the first feature principal component\. Figure[5](https://arxiv.org/html/2610.21081#A1.F5)gives the pooled OFI tail evidence used to motivate the alpha\-stable family\.

\(a\)BTC mid\-price increments\(b\)BTC feature PC1
Figure 4:Heavy\-tail distribution\-fit diagnostics\. The dashed normal reference is not the leading fit: it ranks 11th for BTC increments and 31st for the first principal component of the feature tensor\.![Refer to caption](https://arxiv.org/html/2610.21081v1/crypto-tail-fit.png)Figure 5:Log\-log complementary CDF of pooled absolute crypto OFI, normalized by symbol median\. The top 5% Hill/Pareto fit givesα^=1\.62\\hat\{\\alpha\}=1\.62, supporting heavy\-tailed OFI corruption\.
### A\.5Feishu Data Details

The Feishu dataset is the larger Chinese A\-share protocol used for the portfolio transfer experiment\. The feature tensor has 259 coordinates per asset\-day\. The first 240 coordinates are intraday OFI features: 24 trading\-day slots times 10 book levels\. The slot grid uses the Chinese A\-share continuous trading sessions: 09:40, 09:50, …, 11:20 in the morning and 13:00, 13:10, …, 15:00 in the afternoon\. OFI is computed at the tick level within each asset\-day, aligned to the 10\-minute slot marks, and zero\-filled for off\-grid 5\-minute bars\. The OFI block is normalized with a causal 5\-day rollingzz\-score and clipped to\[−5,5\]\[\-5,5\]\.

The remaining 19 coordinates are daily OHLCV and technical variables: 1\-, 5\-, 10\-, and 20\-day returns; 5\- and 20\-day volatility; Amihud illiquidity; volumezz\-score; 14\-day RSI; distances from 5\- and 20\-day moving averages; open\-to\-close return; high\-low range; close\-VWAP distance; and raw open, close, volume, low, and high\. These daily features are cross\-sectionally standardized across assets on the same day and clipped to the same numeric range as the OFI block\. Table[8](https://arxiv.org/html/2610.21081#A1.T8)summarizes the Feishu universe, feature count, splits, and evaluation period\.

Table 8:Feishu data and evaluation protocol\.

## Appendix BGSM Score Table

Foru=W​ξu=\\sqrt\{W\}\\xi,ξ∼𝒩⁡\(0,Id\)\\xi\\sim\\mathcal\{N\}\(0,I\_\{d\}\), andW\>0W\>0, the marginal density is

ϕt\(u\)=𝔼W\[\(2πW\)−d/2exp\(−∥u∥2/2W\)\]\.\\phi\_\{t\}\(u\)=\\mathbb\{E\}\_\{W\}\[\(2\\pi W\)^\{\-d/2\}\\exp\(\-\\lVert u\\rVert^\{2\}/2W\)\]\.Differentiating under the integral gives

∇uϕt\(u\)=−u𝔼W\[W−1\(2πW\)−d/2exp\(−∥u∥2/2W\)\]\.\\nabla\_\{u\}\\phi\_\{t\}\(u\)=\-u\\,\\mathbb\{E\}\_\{W\}\[W^\{\-1\}\(2\\pi W\)^\{\-d/2\}\\exp\(\-\\lVert u\\rVert^\{2\}/2W\)\]\.Dividing byϕt​\(u\)\\phi\_\{t\}\(u\)yields Eq\. \([5](https://arxiv.org/html/2610.21081#S4.E5)\)\. The table is therefore one\-dimensional in the radius even thoughd=T⋅F=1860d=T\\cdot F=1860for the crypto dataset\. Appendix[G](https://arxiv.org/html/2610.21081#A7)gives the complete posterior\-precision proof and its Gaussian, jump, and alpha\-stable corollaries\.

Algorithm 1BuildScoreTable1:diffusion horizon

TmaxT\_\{\\max\}; corruption sampler

pt​\(W\)p\_\{t\}\(W\); dimension

dd; grid size

nrn\_\{r\}; Monte Carlo size

MM
2:per\-timestep radius grids

RtR\_\{t\}and radial precision values

HtH\_\{t\}
3:for

t=1,…,Tmaxt=1,\\dots,T\_\{\\max\}do

4:draw

W1,…,WM∼pt​\(W\)W\_\{1\},\\dots,W\_\{M\}\\sim p\_\{t\}\(W\)
5:draw

C1,…,CM∼χd2C\_\{1\},\\dots,C\_\{M\}\\sim\\chi^\{2\}\_\{d\}
6:estimate typical radii

rj=Wj​Cjr\_\{j\}=\\sqrt\{W\_\{j\}C\_\{j\}\}
7:set

rmax=1\.02​quantile0\.9999​\(\{rj\}\)r\_\{\\max\}=1\.02\\,\\mathrm\{quantile\}\_\{0\.9999\}\(\\\{r\_\{j\}\\\}\)
8:create

Rt=\{r1⋆,…,rnr⋆\}R\_\{t\}=\\\{r\_\{1\}^\{\\star\},\\dots,r\_\{n\_\{r\}\}^\{\\star\}\\\}on

\[0,rmax\]\[0,r\_\{\\max\}\]
9:foreach

r⋆∈Rtr^\{\\star\}\\in R\_\{t\}do

10:compute log weights

gj=−d2​log⁡Wj−\(r⋆\)2/\(2​Wj\)g\_\{j\}=\-\\tfrac\{d\}\{2\}\\log W\_\{j\}\-\(r^\{\\star\}\)^\{2\}/\(2W\_\{j\}\)
11:set

m=maxj⁡gjm=\\max\_\{j\}g\_\{j\}
12:set

Ht​\(r⋆\)=∑jegj−m​Wj−1/∑jegj−mH\_\{t\}\(r^\{\\star\}\)=\\sum\_\{j\}e^\{g\_\{j\}\-m\}W\_\{j\}^\{\-1\}/\\sum\_\{j\}e^\{g\_\{j\}\-m\}
13:endfor

14:endfor

15:return

\{\(Rt,Ht\)\}t=1Tmax\\\{\(R\_\{t\},H\_\{t\}\)\\\}\_\{t=1\}^\{T\_\{\\max\}\}

### B\.1Monte Carlo Justification

Algorithm[1](https://arxiv.org/html/2610.21081#alg1)estimatesht​\(r\)h\_\{t\}\(r\)by replacing the two expectations in Eq\. \([6](https://arxiv.org/html/2610.21081#S4.E6)\) with Monte Carlo averages\. The argument below is pointwise in the tabulated radius: reusing one batch of draws across the radius grid creates correlation between grid entries, but it does not change the marginal guarantee at any fixed grid point\.

Fix a timestepttand a radiusr\>0r\>0\. Herer=∥u∥r=\\lVert u\\rVert, so negative values are outside the domain\. We state the Monte Carlo guarantee for positive radii only\. The table may includer=0r=0as a numerical interpolation endpoint, but Proposition[1](https://arxiv.org/html/2610.21081#Thmproposition1)does not claim a finite\-sample bound there for the clipped alpha\-stable law: the bounds in Lemma[1](https://arxiv.org/html/2610.21081#Thmlemma1)diverge asr↓0r\\downarrow 0\. IfWtW\_\{t\}is bounded away from zero, as in the Gaussian and jump\-Poisson corruptions used here, the same ratio\-estimator argument extends tor=0r=0with bounds based on that lower bound\. For clipped alpha\-stable corruption, the zero endpoint would require separate negative\-moment/concentration conditions\. This is fine because under continuous Gaussian\-mixture corruption, sampled training residuals satisfyℙ⁡\(∥u∥=0\)=0\\mathbb\{P\}\(\\lVert u\\rVert=0\)=0: conditional on any realizationWt=w\>0W\_\{t\}=w\>0, the residualu\|Wt=w∼𝒩⁡\(0,w​Id\)u\\mid W\_\{t\}=w\\sim\\mathcal\{N\}\(0,wI\_\{d\}\)is absolutely continuous, so

ℙ⁡\(∥u∥=0∣Wt=w\)=ℙ⁡\(u=0∣Wt=w\)=0\.\\mathbb\{P\}\(\\lVert u\\rVert=0\\mid W\_\{t\}=w\)=\\mathbb\{P\}\(u=0\\mid W\_\{t\}=w\)=0\.Taking expectation overWtW\_\{t\}givesℙ⁡\(∥u∥=0\)=0\\mathbb\{P\}\(\\lVert u\\rVert=0\)=0\. Define

g1\(w,r\)=w−1−d/2exp\(−r22​w\),g2\(w,r\)=w−d/2exp\(−r22​w\),g\_\{1\}\(w,r\)=w^\{\-1\-d/2\}\\exp\\\!\\left\(\-\\frac\{r^\{2\}\}\{2w\}\\right\),\\qquad g\_\{2\}\(w,r\)=w^\{\-d/2\}\\exp\\\!\\left\(\-\\frac\{r^\{2\}\}\{2w\}\\right\),and write

Nt​\(r\)=𝔼pt​\[g1​\(W,r\)\],Dt​\(r\)=𝔼pt​\[g2​\(W,r\)\]\.N\_\{t\}\(r\)=\\mathbb\{E\}\_\{p\_\{t\}\}\[g\_\{1\}\(W,r\)\],\\qquad D\_\{t\}\(r\)=\\mathbb\{E\}\_\{p\_\{t\}\}\[g\_\{2\}\(W,r\)\]\.Thenht​\(r\)=Nt​\(r\)/Dt​\(r\)h\_\{t\}\(r\)=N\_\{t\}\(r\)/D\_\{t\}\(r\)\. The empirical table entry is

h^t​\(r\)=M−1​∑i=1Mg1​\(Wi,r\)M−1​∑i=1Mg2​\(Wi,r\),Wi∼i\.i\.d\.pt​\(W\)\.\\widehat\{h\}\_\{t\}\(r\)=\\frac\{M^\{\-1\}\\sum\_\{i=1\}^\{M\}g\_\{1\}\(W\_\{i\},r\)\}\{M^\{\-1\}\\sum\_\{i=1\}^\{M\}g\_\{2\}\(W\_\{i\},r\)\},\\qquad W\_\{i\}\\stackrel\{\{\\scriptstyle\\mathrm\{i\.i\.d\.\}\}\}\{\{\\sim\}\}p\_\{t\}\(W\)\.Sinceg1​\(w,r\)=w−1​g2​\(w,r\)g\_\{1\}\(w,r\)=w^\{\-1\}g\_\{2\}\(w,r\), this is equivalently

h^t​\(r\)=∑i=1Mω^i​Wi−1,ω^i=g2​\(Wi,r\)∑j=1Mg2​\(Wj,r\)\.\\widehat\{h\}\_\{t\}\(r\)=\\sum\_\{i=1\}^\{M\}\\widehat\{\\omega\}\_\{i\}W\_\{i\}^\{\-1\},\\qquad\\widehat\{\\omega\}\_\{i\}=\\frac\{g\_\{2\}\(W\_\{i\},r\)\}\{\\sum\_\{j=1\}^\{M\}g\_\{2\}\(W\_\{j\},r\)\}\.The table value is therefore a self\-normalized importance\-sampling estimator \([Geweke, 1989](https://arxiv.org/html/2610.21081#bib.bib34); see also[Owen, 2013](https://arxiv.org/html/2610.21081#bib.bib35), Ch\. 9for a general treatment\)\. The proposal is the corruption lawpt​\(W\)p\_\{t\}\(W\), the unnormalized target is

πt​\(d​w∣r\)=g2​\(w,r\)​pt​\(d​w\)Dt​\(r\),\\pi\_\{t\}\(dw\\mid r\)=\\frac\{g\_\{2\}\(w,r\)p\_\{t\}\(dw\)\}\{D\_\{t\}\(r\)\},and the estimated quantity is

𝔼πt\(⋅∣r\)\[W−1\]=𝔼pt​\[W−1​g2​\(W,r\)\]𝔼pt​\[g2​\(W,r\)\]=ht\(r\)\.\\mathbb\{E\}\_\{\\pi\_\{t\}\(\\cdot\\mid r\)\}\[W^\{\-1\}\]=\\frac\{\\mathbb\{E\}\_\{p\_\{t\}\}\[W^\{\-1\}g\_\{2\}\(W,r\)\]\}\{\\mathbb\{E\}\_\{p\_\{t\}\}\[g\_\{2\}\(W,r\)\]\}=h\_\{t\}\(r\)\.By Proposition[2](https://arxiv.org/html/2610.21081#Thmproposition2), this is the Tweedie\-type posterior expected corruption precision𝔼⁡\[Wt−1∣∥u∥=r\]\\mathbb\{E\}\[W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\]\.

###### Lemma 1\(Pointwise boundedness\)\.

For everyr\>0r\>0,

supw\>0g1\(w,r\)=\(d\+2e​r2\)1\+d/2=:B1\(r\),supw\>0g2\(w,r\)=\(de​r2\)d/2=:B2\(r\)\.\\sup\_\{w\>0\}g\_\{1\}\(w,r\)=\\left\(\\frac\{d\+2\}\{er^\{2\}\}\\right\)^\{1\+d/2\}=:B\_\{1\}\(r\),\\qquad\\sup\_\{w\>0\}g\_\{2\}\(w,r\)=\\left\(\\frac\{d\}\{er^\{2\}\}\\right\)^\{d/2\}=:B\_\{2\}\(r\)\.Both bounds are finite for fixedr\>0r\>0\.

###### Proof\.

Both functions have the formfa\(w\)=w−aexp\(−r2/2w\)f\_\{a\}\(w\)=w^\{\-a\}\\exp\(\-r^\{2\}/2w\), witha=d/2a=d/2forg2g\_\{2\}anda=1\+d/2a=1\+d/2forg1g\_\{1\}\. Since

dd​w​log⁡fa​\(w\)=−aw\+r22​w2,\\frac\{d\}\{dw\}\\log f\_\{a\}\(w\)=\-\\frac\{a\}\{w\}\+\\frac\{r^\{2\}\}\{2w^\{2\}\},the unique critical point isw⋆=r2/\(2​a\)w^\{\\star\}=r^\{2\}/\(2a\)\. Alsofa​\(w\)→0f\_\{a\}\(w\)\\to 0asw→0\+w\\to 0^\{\+\}and asw→∞w\\to\\infty, so this critical point is the global maximum\. Evaluatingfa​\(w⋆\)f\_\{a\}\(w^\{\\star\}\)gives

fa​\(w⋆\)=\(2​ae​r2\)a\.f\_\{a\}\(w^\{\\star\}\)=\\left\(\\frac\{2a\}\{er^\{2\}\}\\right\)^\{a\}\.Substitutinga=d/2a=d/2anda=1\+d/2a=1\+d/2gives the two displayed bounds\. ∎

This boundedness is the only fact needed to cover the corruption laws used in the paper\. For Gaussian corruption,ptp\_\{t\}is the point massδσt2\\delta\_\{\\sigma\_\{t\}^\{2\}\}\. For the jump\-Poisson law,

Wt=σt2\+∑j=1NtSj≥σt2\>0a\.s\.W\_\{t\}=\\sigma\_\{t\}^\{2\}\+\\sum\_\{j=1\}^\{N\_\{t\}\}S\_\{j\}\\geq\\sigma\_\{t\}^\{2\}\>0\\quad\\text\{a\.s\.\}For the clipped alpha\-stable law,

Wt=σt2​min⁡\(A,Q0\.999​\(A\)\)\>0a\.s\.W\_\{t\}=\\sigma\_\{t\}^\{2\}\\min\(A,Q\_\{0\.999\}\(A\)\)\>0\\quad\\text\{a\.s\.\}The clipped law may still have support arbitrarily close to zero, but for any fixedr\>0r\>0Lemma[1](https://arxiv.org/html/2610.21081#Thmlemma1)makesg1g\_\{1\}andg2g\_\{2\}bounded uniformly overw\>0w\>0\. ThusNt​\(r\)N\_\{t\}\(r\)andDt​\(r\)D\_\{t\}\(r\)are finite, andDt​\(r\)\>0D\_\{t\}\(r\)\>0\.

###### Proposition 1\(Statistical accuracy of one score\-table entry\)\.

Fixttandr\>0r\>0, and letptp\_\{t\}be the Gaussian, jump\-Poisson, or clipped alpha\-stable corruption law used byStochLOB\. Then:

h^t​\(r\)→ht​\(r\)almost surely as​M→∞\.\\widehat\{h\}\_\{t\}\(r\)\\to h\_\{t\}\(r\)\\qquad\\text\{almost surely as \}M\\to\\infty\.This is the usual consistency property of self\-normalized importance\-sampling estimators\[[Geweke, 1989](https://arxiv.org/html/2610.21081#bib.bib34)\]\. Moreover, for anyδ∈\(0,1\)\\delta\\in\(0,1\), if

M≥2​B2​\(r\)2​log⁡\(4/δ\)Dt​\(r\)2,M\\geq\\frac\{2B\_\{2\}\(r\)^\{2\}\\log\(4/\\delta\)\}\{D\_\{t\}\(r\)^\{2\}\},then with probability at least1−δ1\-\\delta,

\|h^t​\(r\)−ht​\(r\)\|≤2​\{B1​\(r\)\+ht​\(r\)​B2​\(r\)\}Dt​\(r\)​log⁡\(4/δ\)2​M\.\\left\|\\widehat\{h\}\_\{t\}\(r\)\-h\_\{t\}\(r\)\\right\|\\leq\\frac\{2\\\{B\_\{1\}\(r\)\+h\_\{t\}\(r\)B\_\{2\}\(r\)\\\}\}\{D\_\{t\}\(r\)\}\\sqrt\{\\frac\{\\log\(4/\\delta\)\}\{2M\}\}\.This gives a finite\-sample guarantee of the kind developed for importance\-sampling estimators more generally\[[Agapiou et al\., 2017](https://arxiv.org/html/2610.21081#bib.bib36)\]\. Finally,

M​\(h^t​\(r\)−ht​\(r\)\)→𝑑𝒩⁡\(0,σt2​\(r\)\),\\sqrt\{M\}\\bigl\(\\widehat\{h\}\_\{t\}\(r\)\-h\_\{t\}\(r\)\\bigr\)\\xrightarrow\{d\}\\mathcal\{N\}\\\!\\left\(0,\\sigma\_\{t\}^\{2\}\(r\)\\right\),This is the corresponding asymptotic normality result for this estimator class \([Geweke, 1989](https://arxiv.org/html/2610.21081#bib.bib34);[Owen, 2013](https://arxiv.org/html/2610.21081#bib.bib35), Ch\. 9\), where

σt2​\(r\)=1Dt​\(r\)2​Varpt​\[g2​\(W,r\)​\(W−1−ht​\(r\)\)\]\.\\sigma\_\{t\}^\{2\}\(r\)=\\frac\{1\}\{D\_\{t\}\(r\)^\{2\}\}\\operatorname\{Var\}\_\{p\_\{t\}\}\\\!\\left\[g\_\{2\}\(W,r\)\\bigl\(W^\{\-1\}\-h\_\{t\}\(r\)\\bigr\)\\right\]\.For deterministic Gaussian corruption this variance is zero, recovering the exact valueht​\(r\)=1/σt2h\_\{t\}\(r\)=1/\\sigma\_\{t\}^\{2\}\.

###### Proof\.

Lemma[1](https://arxiv.org/html/2610.21081#Thmlemma1)gives𝔼pt​\|g1​\(W,r\)\|<∞\\mathbb\{E\}\_\{p\_\{t\}\}\|g\_\{1\}\(W,r\)\|<\\inftyand𝔼pt​\|g2​\(W,r\)\|<∞\\mathbb\{E\}\_\{p\_\{t\}\}\|g\_\{2\}\(W,r\)\|<\\infty\. The strong law of large numbers therefore gives

M−1​∑i=1Mg1​\(Wi,r\)→Nt​\(r\),M−1​∑i=1Mg2​\(Wi,r\)→Dt​\(r\)M^\{\-1\}\\sum\_\{i=1\}^\{M\}g\_\{1\}\(W\_\{i\},r\)\\to N\_\{t\}\(r\),\\qquad M^\{\-1\}\\sum\_\{i=1\}^\{M\}g\_\{2\}\(W\_\{i\},r\)\\to D\_\{t\}\(r\)almost surely\. SinceDt​\(r\)\>0D\_\{t\}\(r\)\>0, the continuous mapping theorem givesh^t​\(r\)→Nt​\(r\)/Dt​\(r\)=ht​\(r\)\\widehat\{h\}\_\{t\}\(r\)\\to N\_\{t\}\(r\)/D\_\{t\}\(r\)=h\_\{t\}\(r\)almost surely\.

For the finite\-sample statement, write

N^t=M−1​∑ig1​\(Wi,r\),D^t=M−1​∑ig2​\(Wi,r\)\.\\widehat\{N\}\_\{t\}=M^\{\-1\}\\sum\_\{i\}g\_\{1\}\(W\_\{i\},r\),\\qquad\\widehat\{D\}\_\{t\}=M^\{\-1\}\\sum\_\{i\}g\_\{2\}\(W\_\{i\},r\)\.Algebra gives

\|h^t​\(r\)−ht​\(r\)\|≤\|N^t−Nt​\(r\)\|\+ht​\(r\)​\|D^t−Dt​\(r\)\|D^t\.\\left\|\\widehat\{h\}\_\{t\}\(r\)\-h\_\{t\}\(r\)\\right\|\\leq\\frac\{\|\\widehat\{N\}\_\{t\}\-N\_\{t\}\(r\)\|\+h\_\{t\}\(r\)\|\\widehat\{D\}\_\{t\}\-D\_\{t\}\(r\)\|\}\{\\widehat\{D\}\_\{t\}\}\.By Lemma[1](https://arxiv.org/html/2610.21081#Thmlemma1),g1​\(Wi,r\)∈\[0,B1​\(r\)\]g\_\{1\}\(W\_\{i\},r\)\\in\[0,B\_\{1\}\(r\)\]andg2​\(Wi,r\)∈\[0,B2​\(r\)\]g\_\{2\}\(W\_\{i\},r\)\\in\[0,B\_\{2\}\(r\)\]\. Hoeffding’s inequality\[[Hoeffding, 1963](https://arxiv.org/html/2610.21081#bib.bib38)\]and a union bound imply that, with probability at least1−δ1\-\\delta,

\|N^t−Nt​\(r\)\|≤B1​\(r\)​log⁡\(4/δ\)2​M,\|D^t−Dt​\(r\)\|≤B2​\(r\)​log⁡\(4/δ\)2​M\.\|\\widehat\{N\}\_\{t\}\-N\_\{t\}\(r\)\|\\leq B\_\{1\}\(r\)\\sqrt\{\\frac\{\\log\(4/\\delta\)\}\{2M\}\},\\qquad\|\\widehat\{D\}\_\{t\}\-D\_\{t\}\(r\)\|\\leq B\_\{2\}\(r\)\\sqrt\{\\frac\{\\log\(4/\\delta\)\}\{2M\}\}\.The stated lower bound onMMensuresD^t≥Dt​\(r\)/2\\widehat\{D\}\_\{t\}\\geq D\_\{t\}\(r\)/2on this event\. Substitution gives the displayed non\-asymptotic error bound\.

For asymptotic normality, apply the multivariate central limit theorem to

Yi=\(g1​\(Wi,r\),g2​\(Wi,r\)\)\.Y\_\{i\}=\(g\_\{1\}\(W\_\{i\},r\),g\_\{2\}\(W\_\{i\},r\)\)\.Its covariance is finite by Lemma[1](https://arxiv.org/html/2610.21081#Thmlemma1)\. The delta method\[[van der Vaart, 1998](https://arxiv.org/html/2610.21081#bib.bib39)\]applied toφ⁡\(x,y\)=x/y\\varphi\(x,y\)=x/y, whose gradient at\(Nt​\(r\),Dt​\(r\)\)\(N\_\{t\}\(r\),D\_\{t\}\(r\)\)is

∇φ=\(1Dt​\(r\),−ht​\(r\)Dt​\(r\)\),\\nabla\\varphi=\\left\(\\frac\{1\}\{D\_\{t\}\(r\)\},\-\\frac\{h\_\{t\}\(r\)\}\{D\_\{t\}\(r\)\}\\right\),yields an asymptotic variance

1Dt​\(r\)2​Varpt​\[g1​\(W,r\)−ht​\(r\)​g2​\(W,r\)\]\.\\frac\{1\}\{D\_\{t\}\(r\)^\{2\}\}\\operatorname\{Var\}\_\{p\_\{t\}\}\\\!\\left\[g\_\{1\}\(W,r\)\-h\_\{t\}\(r\)g\_\{2\}\(W,r\)\\right\]\.Usingg1​\(W,r\)=W−1​g2​\(W,r\)g\_\{1\}\(W,r\)=W^\{\-1\}g\_\{2\}\(W,r\)gives the stated closed form\. ∎

The proposition shows that the Monte Carlo part of the table has the usualM−1/2M^\{\-1/2\}pointwise error scale, with a variance determined by how much posterior\-weighted precisionW−1W^\{\-1\}varies under the corruption law\. The log\-sum\-exp computation in Algorithm[1](https://arxiv.org/html/2610.21081#alg1)implements the same ratio estimator in a numerically stable form\. Linear interpolation and clamping are deterministic post\-processing steps applied after the grid values have been estimated; they do not change the pointwise Monte Carlo claim above, although they add the ordinary discretization error of a finite radius grid\.

StochLOB \(Lévy\) usesnr=512,M=20,000n\_\{r\}=512,M=20\{,\}000\. StochLOB \(α\\alpha\-stable\) usesnr=256,M=8192n\_\{r\}=256,M=8192\. Training radii are linearly interpolated and clamped beyond the largest tabulated radius\. The Gaussian limitW≡σt2W\\equiv\\sigma\_\{t\}^\{2\}recoversht​\(r\)=1/σt2h\_\{t\}\(r\)=1/\\sigma\_\{t\}^\{2\}, so no table is needed for a purely Gaussian baseline\.

## Appendix CExperimental Setup and Implementation Details

### C\.1Shared Architecture

BothStochLOBvariants use optional BiN normalization, a two\-layer bidirectional GRU with 64 hidden units per direction, one DiT\-style temporal self\-attention block with four heads and adaLN\-Zero timestep conditioning, attention pooling for trend classification, and a small convolutional score head\. The corruption law and loss differ; the inference path without stochastic corruption is the same\. Figure[2](https://arxiv.org/html/2610.21081#S4.F2)shows the workflow in the main text\.

### C\.2Optimization

Optimization is fixed across the proposed models: AdamW\[[Loshchilov and Hutter, 2019](https://arxiv.org/html/2610.21081#bib.bib27)\], learning rate10−410^\{\-4\}, default moments\(β1,β2\)=\(0\.9,0\.999\)\(\\beta\_\{1\},\\beta\_\{2\}\)=\(0\.9,0\.999\),ϵ=10−8\\epsilon=10^\{\-8\}, weight decay3×10−43\\times 10^\{\-4\}, and a per\-update schedule with 500 linear warmup steps followed by cosine decay over the planned training steps\. We also use gradient clipping at 1\.0, label smoothing 0\.1, and dropout 0\.2\. Coinbase runs use batch size 64 and at most 80 epochs with patience 15\. Feishu runs use batch size 256 and at most 30 epochs\. Checkpoints are selected by validation macro\-F1 on unperturbed windows, not by the score loss, because the final deployed model is a classifier\. The diffusion auxiliary weight isλdiff=1\.0\\lambda\_\{\\mathrm\{diff\}\}=1\.0\. The jump\-Poisson setting uses a variance\-preserving schedule withTmax=1000T\_\{\\max\}=1000and gamma jump shape/scale 1\. The alpha\-stable setting clips the positive stable scale at the 0\.999 quantile\.

All reported Coinbase configuration files setdevice: "cuda"and use single\-GPU workers\. The reported training runs and checkpoint artifact repair/validation were carried out on a single NVIDIA GeForce RTX 4090 worker with 24 GB VRAM\. The StochLOB classifier trunk has about 0\.53M trainable parameters\. Logged per\-sample forward costs for the supervised baselines range from effectively zero for linear models to about 0\.122 GFLOPs for Axial\-LOB; DeepLOB and TLOB are logged at about 0\.093 and 0\.061 GFLOPs/sample, respectively\. The score\-table Monte Carlo preprocessing is performed once per corruption law and schedule and is reused by the training runs\.

### C\.3Crypto Evaluation

For each symbol and horizon, all baselines use the same chronological train/validation/test windows, the same feature normalization, and the same train\-calibrated label thresholds\. Standard held\-out metrics are accuracy and macro\-F1 on the held\-out test interval\. The main paper reports BTC; the appendix reports ETH, BNB, XRP, and USDT\. Seed\-sweep results are reported as mean and sample standard deviation over seeds13,21,42,84,16813,21,42,84,168\.

The realized stress evaluation freezes the trained checkpoint and partitions held\-out BTC windows by statistics computed from the observed test inputs\. The stress score is the maximum one\-step level\-averaged feature increment divided by realized volatility; the variance score is the realized variance of those increments; and the jump\-residual score is the residual from regressing the largest increment on realized variance in log\-log space\. This keeps the data split, trained weights, class thresholds, and decision rule fixed while testing the regimes where stochastic score\-matching corruption should matter\.

### C\.4Feishu Evaluation

Feishu uses a two\-year in\-sample period and one\-year out\-of\-sample release period\. The in\-sample period is split chronologically per asset into train/validation windows\. The out\-of\-sample period is never used for label thresholds, normalization decisions, checkpoint selection, or trading threshold selection\. Model probabilities are exported day by day and consumed by the common portfolio engine described in Appendix[F](https://arxiv.org/html/2610.21081#A6)\.

## Appendix DTraining and Inference Pseudocode

Algorithms[1](https://arxiv.org/html/2610.21081#alg1),[2](https://arxiv.org/html/2610.21081#alg2), and[3](https://arxiv.org/html/2610.21081#alg3)give the operational recipe\. The score table is constructed once per corruption law and schedule; training then uses table interpolation for each sampled residual norm\.

Algorithm 2TrainNonGaussianLOB1:training windows

𝒟\\mathcal\{D\}; classifier

fθf\_\{\\theta\}; score head

sθs\_\{\\theta\}; corruption sampler

pt​\(W\)p\_\{t\}\(W\); score table

hth\_\{t\}; loss weight

λdiff\\lambda\_\{\\mathrm\{diff\}\}
2:checkpoint with best validation macro\-F1

3:forepoch

=1,…,E=1,\\dots,Edo

4:forminibatch

\(x0,y\)∼𝒟\(x\_\{0\},y\)\\sim\\mathcal\{D\}do

5:compute unperturbed logits

z0=fθ​\(x0\)z\_\{0\}=f\_\{\\theta\}\(x\_\{0\}\)
6:compute

ℒcls=CE⁡\(z0,y\)\\mathcal\{L\}\_\{\\mathrm\{cls\}\}=\\mathrm\{CE\}\(z\_\{0\},y\)
7:sample diffusion step

t∼Uniform⁡\{1,…,Tmax\}t\\sim\\mathrm\{Uniform\}\\\{1,\\dots,T\_\{\\max\}\\\}
8:sample variance

W∼pt​\(W\)W\\sim p\_\{t\}\(W\)and noise

ξ∼𝒩⁡\(0,Id\)\\xi\\sim\\mathcal\{N\}\(0,I\_\{d\}\)
9:form

u=W​ξu=\\sqrt\{W\}\\xiand

xt=at​x0\+ux\_\{t\}=a\_\{t\}x\_\{0\}\+u
10:interpolate

ht​\(∥u∥\)h\_\{t\}\(\\lVert u\\rVert\)from the score table

11:compute target score

g⋆=−u​ht​\(∥u∥\)g^\{\\star\}=\-u\\,h\_\{t\}\(\\lVert u\\rVert\)
12:compute

ℒscore=∥sθ​\(xt,t\)−g⋆∥mean2\\mathcal\{L\}\_\{\\mathrm\{score\}\}=\\lVert s\_\{\\theta\}\(x\_\{t\},t\)\-g^\{\\star\}\\rVert^\{2\}\_\{\\mathrm\{mean\}\}
13:update

θ\\thetausing

ℒcls\+λdiff​ℒscore\\mathcal\{L\}\_\{\\mathrm\{cls\}\}\+\\lambda\_\{\\mathrm\{diff\}\}\\mathcal\{L\}\_\{\\mathrm\{score\}\}
14:endfor

15:evaluate validation macro\-F1 on unperturbed windows and save the best checkpoint

16:endfor

Algorithm 3InferStochLOB1:normalized LOB window

x0x\_\{0\}; trained classifier path

fθf\_\{\\theta\}
2:class probabilities for down, stationary, and up

3:pass

x0x\_\{0\}through the same causal normalization used in training

4:encode the unperturbed window through BiGRU, temporal attention, and pooling

5:compute trend logits

z=fθ​\(x0\)z=f\_\{\\theta\}\(x\_\{0\}\)
6:return

p⁡\(y∣x0\)=softmax⁡\(z\)p\(y\\mid x\_\{0\}\)=\\mathrm\{softmax\}\(z\)
7:The corruption sampler, score table, score head, and reverse diffusion process are not used at inference\.

## Appendix EAdditional Crypto Results

The main paper keeps the Coinbase exchange table focused on BTC\. This appendix reports the remaining coins so the reader can see whether the same story survives beyond one crypto pair\. Table[10](https://arxiv.org/html/2610.21081#A5.T10)reports ETH, BNB, XRP, and USDT across all horizons\. Within each symbol and horizon, bold marks the best accuracy or macro\-F1 value and underlining marks the second\-best value under the same standard held\-out test protocol as the main table\. A partial rule separates the two proposed non\-Gaussian models from the standard baselines\. The broader pattern is deliberately mixed: linear and attention baselines remain strong on several standard benchmark cells, while StochLOB \(alpha\-stable\) wins the BNB short\-horizon cell and is competitive on ETH, XRP, and USDT\. StochLOB \(jump\-Poisson\) is frequently near the top, consistent with the main claim that market\-shaped score corruption need not sacrifice standard predictive quality to improve stress\-regime behavior\. Table[11](https://arxiv.org/html/2610.21081#A5.T11)provides the per\-class precision, recall, and F1 breakdown for BTC atk=10k\{=\}10, illustrating how each model distributes errors among the three trend classes\. Realized BTC stress\-window diagnostics are reported in the main paper in Figure[1](https://arxiv.org/html/2610.21081#S1.F1)and Table[2](https://arxiv.org/html/2610.21081#S6.T2); the complete stress\-regime grid is reported in Table[9](https://arxiv.org/html/2610.21081#A5.T9)\.

Table 9:Complete BTC extreme\-window macro\-F1 grid underlying Table[2](https://arxiv.org/html/2610.21081#S6.T2)\. Each row uses the top 10% held\-out windows for the named regime and horizon\. The table reports the better of the two heavy\-tailed StochLOB variants, the matched Gaussian StochLOB baseline, and the best supervised comparator in the same tail subset\.Best F1 in each row is shown in bold\.Rows with negative gains are retained rather than filtered out\.Table 10:Additional crypto held\-out test Accuracy \(Acc\) and Macro\-F1 \(F1\) under the same model scope as the main table\. Bold marks the best value within each symbol and horizon; underlining marks the second\-best value\.Table 11:BTC per\-class precision \(P\), recall \(R\), and F1 atk=10k=10\.
## Appendix FFeishu Portfolio Management Details

### F\.1Models and Prediction Targets

Every Feishu model is trained to predict the same three\-class next\-period mid\-price trend target:

y∈\{down,flat,up\}\.y\\in\\\{\\mathrm\{down\},\\mathrm\{flat\},\\mathrm\{up\}\\\}\.The portfolio experiment does not introduce a separate trading model\. It uses the classifier probabilities

\(P⁡\(down\),P⁡\(flat\),P⁡\(up\)\)\(P\(\\mathrm\{down\}\),P\(\\mathrm\{flat\}\),P\(\\mathrm\{up\}\)\)as alpha signals\. This makes the downstream transfer test direct: if the learned probabilities contain useful directional information, a constrained portfolio rule should be able to convert them into positive out\-of\-sample performance\. The experiment is therefore a check of probability usability, not a claim that the paper designs a deployable trading system\.

### F\.2Portfolio Construction

The Feishu portfolio experiment uses the out\-of\-sample release period only for the exploratory portfolio evaluation\. In\-sample data are used as causal warmup history for rolling features such as returns, volatility, RSI, and the 5\-day OFI normalizer\. For each model and each trading daytt, the signal is computed from the historical window ending att−1t\-1\. The portfolio rule receives\(P⁡\(down\),P⁡\(flat\),P⁡\(up\)\)\(P\(\\mathrm\{down\}\),P\(\\mathrm\{flat\}\),P\(\\mathrm\{up\}\)\)and converts the probabilities into positions\. Assets with sufficiently high up probability are eligible for buying, while assets with sufficiently high down probability are eligible for partial selling\.

The backtest uses an initial capital of RMB 50 million\. The same constraints are applied to every model: 100\-share lot size, commission rate10−410^\{\-4\}, stamp\-duty rate5×10−45\\times 10^\{\-4\}on sells, RMB 5 minimum commission, T\+1 sell lock, a 5% cash buffer, and a minimum\-holdings target of 10 names\. These rules matter for Chinese A\-share data\. A signal that is statistically useful but unstable can create high turnover and cost drag; a signal that is too conservative can fail to trade\. The portfolio experiment therefore measures whether the model output remains usable after realistic market constraints\.

The daily rule has three steps\. First, held assets are examined for exit\. A held asset is fully sold whenP⁡\(down\)\>0\.85P\(\\mathrm\{down\}\)\>0\.85and the predicted class is down, subject to the T\+1 sell lock and the minimum\-holdings constraint\. If a held asset is predicted flat and was not already fully sold, the strategy sells 50% of the sellable shares\. Second, new buys are selected from assets withP⁡\(up\)\>0\.75P\(\\mathrm\{up\}\)\>0\.75and predicted class up\. Third, available capital after the 5% cash buffer is allocated equally across the selected buy candidates\. Orders are rounded to 100\-share lots, buy executions use the morning VWAP entry price, and sell executions use the configured close\-price sell mode\. The model therefore controls which assets are traded, while the portfolio engine keeps sizing and market rules common across all methods\.

We also include a passive equal\-weight index\. On each out\-of\-sample day it holds all valid assets equally and applies the same one\-day VWAP\-entry\-to\-close return convention\. It is not an active trading strategy and it does not pay transaction costs in this implementation\. Its purpose is to show whether a learned signal adds value relative to broad market exposure\.

### F\.3Out\-of\-Sample Portfolio Metrics

The main out\-of\-sample portfolio metrics are reported in Table[3](https://arxiv.org/html/2610.21081#S6.T3), and the cumulative\-return comparison is shown in Figure[3](https://arxiv.org/html/2610.21081#S6.F3)\. We keep the numeric table in the main text because it is the economic summary most directly connected to the experimental story\. This appendix gives the construction details needed to interpret those numbers\.

The most important implementation point is that the portfolio engine is shared across models\. A model can improve the backtest only by changing the probability triplet used for buy, hold, and sell decisions; it cannot change position sizing, execution prices, transaction\-cost assumptions, or market constraints\. In the reported run, several evaluated architectures produced mostly flat predictions during the out\-of\-sample window and therefore did not produce meaningful active portfolios under the common thresholds\. These inactive runs are omitted from Table[3](https://arxiv.org/html/2610.21081#S6.T3)so that the main comparison focuses on strategies that actually trade\.

## Appendix GPosterior\-Precision Interpretation ofhth\_\{t\}

###### Proposition 2\(Posterior\-precision interpretation for isotropic Gaussian scale mixtures\)\.

Fix a noise indextt\. SupposeWt\>0W\_\{t\}\>0almost surely and

𝔼\[Wt−d/2−2\]<∞\.\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-d/2\-2\}\\right\]<\\infty\.Letr=∥u∥r=\\lVert u\\rVert\. Then

ht​\(r\)=𝔼⁡\[Wt−1∣u\]=𝔼⁡\[Wt−1∣∥u∥=r\]\.h\_\{t\}\(r\)=\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-1\}\\mid u\\right\]=\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\\right\]\.Moreover,

ht′​\(r\)=−r​Var⁡\(Wt−1∣∥u∥=r\)≤0\.h\_\{t\}^\{\\prime\}\(r\)=\-r\\,\\operatorname\{Var\}\\\!\\left\(W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\\right\)\\leq 0\.Henceht​\(r\)h\_\{t\}\(r\)is the posterior expected corruption precision and is non\-increasing in the residual magnitude\. In particular, residual adaptation is a property of the random scale\-mixture family itself: it is absent whenWtW\_\{t\}is deterministic and is strict when the scale distribution is non\-degenerate\.

###### Proof\.

For fixedww, write

ϕw\(u\)=𝒩\(u;0,wId\)=\(2πw\)−d/2exp\(−∥u∥22​w\)\.\\phi\_\{w\}\(u\)=\\mathcal\{N\}\(u;0,wI\_\{d\}\)=\(2\\pi w\)^\{\-d/2\}\\exp\\\!\\left\(\-\\frac\{\\lVert u\\rVert^\{2\}\}\{2w\}\\right\)\.The component score is

∇u​log​ϕw​\(u\)=−uw,hence∇uϕw​\(u\)=−uw​ϕw​\(u\)\.\\nabla\_\{u\}\\log\\phi\_\{w\}\(u\)=\-\\frac\{u\}\{w\},\\qquad\\text\{hence\}\\qquad\\nabla\_\{u\}\\phi\_\{w\}\(u\)=\-\\frac\{u\}\{w\}\\phi\_\{w\}\(u\)\.Since

qt​\(u\)=𝔼⁡\[ϕWt​\(u\)\],q\_\{t\}\(u\)=\\mathbb\{E\}\\\!\\left\[\\phi\_\{W\_\{t\}\}\(u\)\\right\],the condition

𝔼\[Wt−d/2−2\]<∞\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-d/2\-2\}\\right\]<\\inftyprovides sufficient integrability to interchange differentiation and expectation in the expressions below\. Therefore,

∇uqt​\(u\)=−u​𝔼​\[Wt−1​ϕWt​\(u\)\]\.\\nabla\_\{u\}q\_\{t\}\(u\)=\-u\\,\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-1\}\\phi\_\{W\_\{t\}\}\(u\)\\right\]\.Dividing byqt​\(u\)q\_\{t\}\(u\)gives

∇u​log​qt​\(u\)=−u​𝔼⁡\[Wt−1​ϕWt​\(u\)\]𝔼​\[ϕWt​\(u\)\]\.\\nabla\_\{u\}\\log q\_\{t\}\(u\)=\-u\\,\\frac\{\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-1\}\\phi\_\{W\_\{t\}\}\(u\)\\right\]\}\{\\mathbb\{E\}\\\!\\left\[\\phi\_\{W\_\{t\}\}\(u\)\\right\]\}\.By Bayes’ rule in expectation form,

𝔼⁡\[f⁡\(Wt\)∣u\]=𝔼⁡\[f⁡\(Wt\)​ϕWt​\(u\)\]𝔼​\[ϕWt​\(u\)\]\\mathbb\{E\}\\\!\\left\[f\(W\_\{t\}\)\\mid u\\right\]=\\frac\{\\mathbb\{E\}\\\!\\left\[f\(W\_\{t\}\)\\phi\_\{W\_\{t\}\}\(u\)\\right\]\}\{\\mathbb\{E\}\\\!\\left\[\\phi\_\{W\_\{t\}\}\(u\)\\right\]\}for any integrable functionff\. Takingf⁡\(w\)=w−1f\(w\)=w^\{\-1\}therefore gives

∇u​log​qt​\(u\)=−u​𝔼​\[Wt−1∣u\]\.\\nabla\_\{u\}\\log q\_\{t\}\(u\)=\-u\\,\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-1\}\\mid u\\right\]\.Comparing this identity with

∇u​log​qt​\(u\)=−u​ht​\(∥u∥\)\\nabla\_\{u\}\\log q\_\{t\}\(u\)=\-u\\,h\_\{t\}\(\\lVert u\\rVert\)proves the first equality\. Because the Gaussian components are isotropic, the posterior depends onuuonly throughr=∥u∥r=\\lVert u\\rVert, giving

ht​\(r\)=𝔼⁡\[Wt−1∣∥u∥=r\]\.h\_\{t\}\(r\)=\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\\right\]\.
To establish monotonicity, let

and define

Zt\(s\)=𝔼\[Wt−d/2exp\(−sWt\)\]\.Z\_\{t\}\(s\)=\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-d/2\}\\exp\\\!\\left\(\-\\frac\{s\}\{W\_\{t\}\}\\right\)\\right\]\.Since

qt\(u\)=\(2π\)−d/2Zt\(s\),q\_\{t\}\(u\)=\(2\\pi\)^\{\-d/2\}Z\_\{t\}\(s\),Eq\. \([6](https://arxiv.org/html/2610.21081#S4.E6)\) can be written as

ht​\(r\)=−∂∂s​log⁡Zt​\(s\)\.h\_\{t\}\(r\)=\-\\frac\{\\partial\}\{\\partial s\}\\log Z\_\{t\}\(s\)\.Indeed,

−∂∂s​log⁡Zt​\(s\)=𝔼\[Wt−d/2−1exp\(−s/Wt\)\]𝔼\[Wt−d/2exp\(−s/Wt\)\]=ht​\(r\)\.\-\\frac\{\\partial\}\{\\partial s\}\\log Z\_\{t\}\(s\)=\\frac\{\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-d/2\-1\}\\exp\(\-s/W\_\{t\}\)\\right\]\}\{\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-d/2\}\\exp\(\-s/W\_\{t\}\)\\right\]\}=h\_\{t\}\(r\)\.
The same Bayes weighting gives, fork=1,2k=1,2,

𝔼⁡\[Wt−k∣∥u∥=r\]=𝔼\[Wt−d/2−kexp\(−s/Wt\)\]Zt​\(s\)\.\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-k\}\\mid\\lVert u\\rVert=r\\right\]=\\frac\{\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-d/2\-k\}\\exp\(\-s/W\_\{t\}\)\\right\]\}\{Z\_\{t\}\(s\)\}\.DifferentiatingZtZ\_\{t\}with respect tossgives

Zt′\(s\)=−𝔼\[Wt−d/2−1exp\(−s/Wt\)\]Z\_\{t\}^\{\\prime\}\(s\)=\-\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-d/2\-1\}\\exp\(\-s/W\_\{t\}\)\\right\]and

Zt′′\(s\)=𝔼\[Wt−d/2−2exp\(−s/Wt\)\]\.Z\_\{t\}^\{\\prime\\prime\}\(s\)=\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-d/2\-2\}\\exp\(\-s/W\_\{t\}\)\\right\]\.Hence

−Zt′​\(s\)Zt​\(s\)=𝔼⁡\[Wt−1∣∥u∥=r\]\-\\frac\{Z\_\{t\}^\{\\prime\}\(s\)\}\{Z\_\{t\}\(s\)\}=\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\\right\]and

Zt′′​\(s\)Zt​\(s\)=𝔼⁡\[Wt−2∣∥u∥=r\]\.\\frac\{Z\_\{t\}^\{\\prime\\prime\}\(s\)\}\{Z\_\{t\}\(s\)\}=\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-2\}\\mid\\lVert u\\rVert=r\\right\]\.Now,

∂2∂s2​log⁡Zt​\(s\)=Zt′′​\(s\)Zt​\(s\)−\(Zt′​\(s\)Zt​\(s\)\)2,\\frac\{\\partial^\{2\}\}\{\\partial s^\{2\}\}\\log Z\_\{t\}\(s\)=\\frac\{Z\_\{t\}^\{\\prime\\prime\}\(s\)\}\{Z\_\{t\}\(s\)\}\-\\left\(\\frac\{Z\_\{t\}^\{\\prime\}\(s\)\}\{Z\_\{t\}\(s\)\}\\right\)^\{2\},so

∂2∂s2​log⁡Zt​\(s\)=Var⁡\(Wt−1∣∥u∥=r\)\.\\frac\{\\partial^\{2\}\}\{\\partial s^\{2\}\}\\log Z\_\{t\}\(s\)=\\operatorname\{Var\}\\\!\\left\(W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\\right\)\.Since

ht​\(r\)=−∂∂s​log⁡Zt​\(s\),h\_\{t\}\(r\)=\-\\frac\{\\partial\}\{\\partial s\}\\log Z\_\{t\}\(s\),it follows that

∂ht∂s=−Var⁡\(Wt−1∣∥u∥=r\)≤0\.\\frac\{\\partial h\_\{t\}\}\{\\partial s\}=\-\\operatorname\{Var\}\\\!\\left\(W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\\right\)\\leq 0\.Finally, usings=r2/2s=r^\{2\}/2,

ht′​\(r\)=r​∂ht∂s=−r​Var⁡\(Wt−1∣∥u∥=r\)≤0\.h\_\{t\}^\{\\prime\}\(r\)=r\\frac\{\\partial h\_\{t\}\}\{\\partial s\}=\-r\\,\\operatorname\{Var\}\\\!\\left\(W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\\right\)\\leq 0\.IfWtW\_\{t\}is non\-degenerate, the Gaussian likelihood is strictly positive for everyWt\>0W\_\{t\}\>0, so posterior weighting cannot collapse the scale to a single value\. The posterior variance is therefore strictly positive andht′​\(r\)<0h\_\{t\}^\{\\prime\}\(r\)<0forr\>0r\>0\. Thushth\_\{t\}is strictly decreasing for every genuinely random scale mixture\. ∎

Proposition[2](https://arxiv.org/html/2610.21081#Thmproposition2)provides an intuitive interpretation of the functionhth\_\{t\}and, consequently, of the score target in Eq\. \([7](https://arxiv.org/html/2610.21081#S4.E7)\)\. In particular, it shows thatht​\(r\)h\_\{t\}\(r\)is the posterior expected corruption precision,𝔼⁡\[Wt−1∣∥u∥=r\]\\mathbb\{E\}\[W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\], given the observed residual magnitude\. Gaussian scale mixtures are a classical latent\-variable construction for symmetric non\-Gaussian and heavy\-tailed distributions\[[Andrews and Mallows, 1974](https://arxiv.org/html/2610.21081#bib.bib40),[West, 1987](https://arxiv.org/html/2610.21081#bib.bib41)\]\. The identity that a marginal score is obtained by posterior averaging of latent component scores is a standard latent\-variable score identity\[[Louis, 1982](https://arxiv.org/html/2610.21081#bib.bib16)\]; in score\-based terms, it is closely connected to the conditional\-score identities underlying denoising score matching\[[Vincent, 2011](https://arxiv.org/html/2610.21081#bib.bib15)\]and modern score\-based diffusion/SDE formulations\[[Song et al\., 2021](https://arxiv.org/html/2610.21081#bib.bib14)\]\. The posterior\-precision interpretation is also related in spirit to empirical\-Bayes score identities such as Tweedie’s formula\[[Efron, 2011](https://arxiv.org/html/2610.21081#bib.bib43)\]; here, however, the latent quantity being inferred is the corruption precisionWt−1W\_\{t\}^\{\-1\}rather than an unknown clean mean\. The result is not specific to one non\-Gaussian corruption law: it holds for the whole isotropic Gaussian scale\-mixture family\. Proposition[3](https://arxiv.org/html/2610.21081#Thmproposition3)gives a complementary uncertainty interpretation of the same latent\-scale posterior, after which the corollaries separate the family\-level cases that are relevant to the corruptions used in StochLOB\.

###### Proposition 3\(Posterior uncertainty of the latent component score\)\.

Under the conditions of Proposition[2](https://arxiv.org/html/2610.21081#Thmproposition2), define the score of the Gaussian component associated with the latent scaleWtW\_\{t\}as

St​\(u,Wt\):=∇u​log​ϕWt​\(u\)=−Wt−1​u\.S\_\{t\}\(u,W\_\{t\}\):=\\nabla\_\{u\}\\log\\phi\_\{W\_\{t\}\}\(u\)=\-W\_\{t\}^\{\-1\}u\.Then the marginal score is the posterior mean of this latent component score,

𝔼⁡\[St​\(u,Wt\)∣u\]=∇u​log​qt​\(u\)=−u​ht​\(r\),r=∥u∥,\\mathbb\{E\}\\\!\\left\[S\_\{t\}\(u,W\_\{t\}\)\\mid u\\right\]=\\nabla\_\{u\}\\log q\_\{t\}\(u\)=\-u\\,h\_\{t\}\(r\),\\qquad r=\\lVert u\\rVert,and its posterior covariance is

Cov⁡\(St​\(u,Wt\)∣u\)=Var⁡\(Wt−1∣∥u∥=r\)​u​u⊤\.\\operatorname\{Cov\}\\\!\\left\(S\_\{t\}\(u,W\_\{t\}\)\\mid u\\right\)=\\operatorname\{Var\}\\\!\\left\(W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\\right\)uu^\{\\top\}\.Consequently, define the scalar posterior score uncertainty by

𝒰t​\(r\):=tr⁡\[Cov⁡\(St​\(u,Wt\)∣u\)\]\.\\mathcal\{U\}\_\{t\}\(r\):=\\operatorname\{tr\}\\\!\\left\[\\operatorname\{Cov\}\\\!\\left\(S\_\{t\}\(u,W\_\{t\}\)\\mid u\\right\)\\right\]\.Then

𝒰t​\(r\)=r2​Var⁡\(Wt−1∣∥u∥=r\)=−r​ht′​\(r\)≥0\.\\mathcal\{U\}\_\{t\}\(r\)=r^\{2\}\\operatorname\{Var\}\\\!\\left\(W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\\right\)=\-r\\,h\_\{t\}^\{\\prime\}\(r\)\\geq 0\.Thus the same posterior uncertainty in corruption precision that makeshth\_\{t\}residual\-adaptive also measures uncertainty about the component\-level restoring score\.

###### Proof\.

By the component\-score identity above,

St​\(u,Wt\)=−Wt−1​u\.S\_\{t\}\(u,W\_\{t\}\)=\-W\_\{t\}^\{\-1\}u\.Conditioning on the observed residualuuand applying Proposition[2](https://arxiv.org/html/2610.21081#Thmproposition2)gives

𝔼⁡\[St​\(u,Wt\)∣u\]=−u​𝔼​\[Wt−1∣u\]=−u​ht​\(r\)=∇u​log​qt​\(u\)\.\\mathbb\{E\}\\\!\\left\[S\_\{t\}\(u,W\_\{t\}\)\\mid u\\right\]=\-u\\,\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-1\}\\mid u\\right\]=\-u\\,h\_\{t\}\(r\)=\\nabla\_\{u\}\\log q\_\{t\}\(u\)\.Sinceuuis fixed under this conditional distribution,

Cov⁡\(St​\(u,Wt\)∣u\)\\displaystyle\\operatorname\{Cov\}\\\!\\left\(S\_\{t\}\(u,W\_\{t\}\)\\mid u\\right\)=𝔼⁡\[\(Wt−1−ht​\(r\)\)2∣u\]​u​u⊤\\displaystyle=\\mathbb\{E\}\\\!\\left\[\\bigl\(W\_\{t\}^\{\-1\}\-h\_\{t\}\(r\)\\bigr\)^\{2\}\\mid u\\right\]uu^\{\\top\}=Var⁡\(Wt−1∣u\)​u​u⊤\\displaystyle=\\operatorname\{Var\}\\\!\\left\(W\_\{t\}^\{\-1\}\\mid u\\right\)uu^\{\\top\}=Var⁡\(Wt−1∣∥u∥=r\)​u​u⊤,\\displaystyle=\\operatorname\{Var\}\\\!\\left\(W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\\right\)uu^\{\\top\},where the last equality again follows from isotropy\. Taking the trace and using

tr⁡\(u​u⊤\)=∥u∥2=r2\\operatorname\{tr\}\(uu^\{\\top\}\)=\\lVert u\\rVert^\{2\}=r^\{2\}gives

𝒰t​\(r\)=r2​Var⁡\(Wt−1∣∥u∥=r\)\.\\mathcal\{U\}\_\{t\}\(r\)=r^\{2\}\\operatorname\{Var\}\\\!\\left\(W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\\right\)\.Finally, Proposition[2](https://arxiv.org/html/2610.21081#Thmproposition2)gives

ht′​\(r\)=−r​Var⁡\(Wt−1∣∥u∥=r\),h\_\{t\}^\{\\prime\}\(r\)=\-r\\,\\operatorname\{Var\}\\\!\\left\(W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\\right\),so

𝒰t​\(r\)=−r​ht′​\(r\)≥0\.\\mathcal\{U\}\_\{t\}\(r\)=\-r\\,h\_\{t\}^\{\\prime\}\(r\)\\geq 0\.This is posterior uncertainty induced by the latent corruption scaleWtW\_\{t\}; it does not represent parameter uncertainty in the learned score network\. ∎

###### Corollary 1\(Deterministic\-scale corruption\)\.

SupposeWt=wtW\_\{t\}=w\_\{t\}deterministically for somewt\>0w\_\{t\}\>0\. Then

ht​\(r\)=1wt,ht′​\(r\)=0,h\_\{t\}\(r\)=\\frac\{1\}\{w\_\{t\}\},\\qquad h\_\{t\}^\{\\prime\}\(r\)=0,and

∇u​log​qt​\(u\)=−uwt\.\\nabla\_\{u\}\\log q\_\{t\}\(u\)=\-\\frac\{u\}\{w\_\{t\}\}\.Thus deterministic\-scale Gaussian corruption uses a fixed restoring precision, independent of the residual magnitude\. StochLOB \(Gaussian\) is the particular casewt=σt2w\_\{t\}=\\sigma\_\{t\}^\{2\}\.

###### Proof\.

IfWt=wtW\_\{t\}=w\_\{t\}almost surely, thenWt−1=wt−1W\_\{t\}^\{\-1\}=w\_\{t\}^\{\-1\}almost surely\. Proposition[2](https://arxiv.org/html/2610.21081#Thmproposition2)therefore gives

ht​\(r\)=𝔼⁡\[Wt−1∣∥u∥=r\]=wt−1,h\_\{t\}\(r\)=\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\\right\]=w\_\{t\}^\{\-1\},soht′​\(r\)=0h\_\{t\}^\{\\prime\}\(r\)=0\. Substituting into Eq\. \([5](https://arxiv.org/html/2610.21081#S4.E5)\) gives the score expression\. ∎

###### Corollary 2\(Non\-degenerate random\-scale corruption\)\.

Suppose the scale distribution ofWtW\_\{t\}is non\-degenerate\. Then, for everyr\>0r\>0,

ht′​\(r\)<0\.h\_\{t\}^\{\\prime\}\(r\)<0\.Hence every genuinely random isotropic Gaussian scale mixture has a strictly residual\-adaptive posterior precision: as the residual magnitude increases, the per\-unit restoring precision decreases\. The jump\-Poisson and alpha\-stable scale\-mixture corruptions used in StochLOB are examples of this random\-scale family\.

###### Proof\.

By Proposition[2](https://arxiv.org/html/2610.21081#Thmproposition2),

ht′​\(r\)=−r​Var⁡\(Wt−1∣∥u∥=r\)\.h\_\{t\}^\{\\prime\}\(r\)=\-r\\,\\operatorname\{Var\}\\\!\\left\(W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\\right\)\.For a non\-degenerate scale distribution, the posterior distribution ofWt−1W\_\{t\}^\{\-1\}remains non\-degenerate for every finite residual because the Gaussian likelihood is strictly positive for everyWt\>0W\_\{t\}\>0\. Hence the posterior variance is strictly positive, andht′​\(r\)<0h\_\{t\}^\{\\prime\}\(r\)<0forr\>0r\>0\. ∎

The resulting reduction in effective precision for larger residuals is consistent with the classical robustness interpretation of heavy\-tailed scale\-mixture models, such as Student\-tterror models, in which extreme residuals exert less influence than under Gaussian errors\[[Lange et al\., 1989](https://arxiv.org/html/2610.21081#bib.bib42)\]\. Corollary[2](https://arxiv.org/html/2610.21081#Thmcorollary2)strengthens this intuition for the present setting by establishing strict residual adaptation for every non\-degenerate isotropic Gaussian scale mixture satisfying the proposition’s regularity conditions\.

###### Corollary 3\(Bounded random\-scale corruption\)\.

Suppose the scale distribution is non\-degenerate and satisfies

0<Wt≤Wt,max0<W\_\{t\}\\leq W\_\{t,\\max\}almost surely for some finiteWt,maxW\_\{t,\\max\}\. Then

ht​\(r\)≥1Wt,max,h\_\{t\}\(r\)\\geq\\frac\{1\}\{W\_\{t,\\max\}\},and, forr\>0r\>0,

ht′​\(r\)<0\.h\_\{t\}^\{\\prime\}\(r\)<0\.Thus bounded random\-scale corruption retains residual adaptation but has a positive precision floor\. If the scale distribution also places positive probability atWt,maxW\_\{t,\\max\}, then

ht​\(r\)⟶1Wt,maxas​r→∞\.h\_\{t\}\(r\)\\longrightarrow\\frac\{1\}\{W\_\{t,\\max\}\}\\qquad\\text\{as \}r\\to\\infty\.

###### Proof\.

SinceWt≤Wt,maxW\_\{t\}\\leq W\_\{t,\\max\}, taking reciprocals gives

Wt−1≥Wt,max−1\.W\_\{t\}^\{\-1\}\\geq W\_\{t,\\max\}^\{\-1\}\.Therefore, by Proposition[2](https://arxiv.org/html/2610.21081#Thmproposition2),

ht​\(r\)=𝔼⁡\[Wt−1∣∥u∥=r\]≥1Wt,max\.h\_\{t\}\(r\)=\\mathbb\{E\}\\\!\\left\[W\_\{t\}^\{\-1\}\\mid\\lVert u\\rVert=r\\right\]\\geq\\frac\{1\}\{W\_\{t,\\max\}\}\.Strict decrease follows from Corollary[2](https://arxiv.org/html/2610.21081#Thmcorollary2)\.

For the limiting statement, assume the scale distribution places positive probability atWt,maxW\_\{t,\\max\}\. For any smaller scalew<Wt,maxw<W\_\{t,\\max\}, the Gaussian likelihood ratio relative to the maximal\-scale component is

𝒩⁡\(u,0,w​Id\)𝒩⁡\(u,0,Wt,max​Id\)=\(Wt,maxw\)d/2​exp⁡\[−r22​\(1w−1Wt,max\)\]\.\\frac\{\\mathcal\{N\}\(u;0,wI\_\{d\}\)\}\{\\mathcal\{N\}\(u;0,W\_\{t,\\max\}I\_\{d\}\)\}=\\left\(\\frac\{W\_\{t,\\max\}\}\{w\}\\right\)^\{d/2\}\\exp\\\!\\left\[\-\\frac\{r^\{2\}\}\{2\}\\left\(\\frac\{1\}\{w\}\-\\frac\{1\}\{W\_\{t,\\max\}\}\\right\)\\right\]\.Sincew<Wt,maxw<W\_\{t,\\max\}, this ratio tends to zero asr→∞r\\to\\infty\. Consequently, increasingly large residuals are dominated by the maximal\-scale component, and the posterior expected precision converges toWt,max−1W\_\{t,\\max\}^\{\-1\}\. ∎

###### Corollary 4\(Polynomially heavy\-tailed radial corruption\)\.

Suppose the isotropic marginal corruption density can be written as

qt​\(u\)=gt​\(r\),r=∥u∥,q\_\{t\}\(u\)=g\_\{t\}\(r\),\\qquad r=\\lVert u\\rVert,and, for somem\>dm\>d,

gt​\(r\)=Lt​\(r\)​r−mg\_\{t\}\(r\)=L\_\{t\}\(r\)r^\{\-m\}for all sufficiently largerr, whereLt​\(r\)\>0L\_\{t\}\(r\)\>0is differentiable and satisfies

r​Lt′​\(r\)Lt​\(r\)⟶0\.\\frac\{rL\_\{t\}^\{\\prime\}\(r\)\}\{L\_\{t\}\(r\)\}\\longrightarrow 0\.Then

ht​\(r\)∼mr2,h\_\{t\}\(r\)\\sim\\frac\{m\}\{r^\{2\}\},and consequently

‖∇u​log​qt​\(u\)‖=r​ht​\(r\)∼mr⟶0\.\\left\\lVert\\nabla\_\{u\}\\log q\_\{t\}\(u\)\\right\\rVert=r\\,h\_\{t\}\(r\)\\sim\\frac\{m\}\{r\}\\longrightarrow 0\.Thus polynomially heavy\-tailed radial corruptions have the stronger far\-tail property that the absolute restoring score eventually decreases toward zero\.

###### Proof\.

For an isotropic density,

∇u​log​qt​\(u\)=gt′​\(r\)gt​\(r\)​ur\.\\nabla\_\{u\}\\log q\_\{t\}\(u\)=\\frac\{g\_\{t\}^\{\\prime\}\(r\)\}\{g\_\{t\}\(r\)\}\\frac\{u\}\{r\}\.Comparing with Eq\. \([5](https://arxiv.org/html/2610.21081#S4.E5)\) gives

ht​\(r\)=−gt′​\(r\)r​gt​\(r\)\.h\_\{t\}\(r\)=\-\\frac\{g\_\{t\}^\{\\prime\}\(r\)\}\{r\\,g\_\{t\}\(r\)\}\.Sincegt​\(r\)=Lt​\(r\)​r−mg\_\{t\}\(r\)=L\_\{t\}\(r\)r^\{\-m\},

gt′​\(r\)gt​\(r\)=Lt′​\(r\)Lt​\(r\)−mr=−m\+o⁡\(1\)r\.\\frac\{g\_\{t\}^\{\\prime\}\(r\)\}\{g\_\{t\}\(r\)\}=\\frac\{L\_\{t\}^\{\\prime\}\(r\)\}\{L\_\{t\}\(r\)\}\-\\frac\{m\}\{r\}=\-\\frac\{m\+o\(1\)\}\{r\}\.Therefore,

ht​\(r\)∼mr2,h\_\{t\}\(r\)\\sim\\frac\{m\}\{r^\{2\}\},and multiplying byrrgives the stated score\-magnitude asymptotic\. ∎

The family\-level results above include the corruption choices used in StochLOB as particular examples\. For StochLOB \(Gaussian\),Wt=σt2W\_\{t\}=\\sigma\_\{t\}^\{2\}deterministically, so Corollary[1](https://arxiv.org/html/2610.21081#Thmcorollary1)gives the fixed precisionht​\(r\)=σt−2h\_\{t\}\(r\)=\\sigma\_\{t\}^\{\-2\}\.

For StochLOB \(jump\-Poisson\),

Wt=σt2\+∑j=1NtSj,Nt∼Poisson⁡\(Λt\),Sj∼Gamma⁡\(κ,θ\)\.W\_\{t\}=\\sigma\_\{t\}^\{2\}\+\\sum\_\{j=1\}^\{N\_\{t\}\}S\_\{j\},\\qquad N\_\{t\}\\sim\\operatorname\{Poisson\}\(\\Lambda\_\{t\}\),\\qquad S\_\{j\}\\sim\\operatorname\{Gamma\}\(\\kappa,\\theta\)\.For a nontrivial jump stepΛt\>0\\Lambda\_\{t\}\>0, this scale distribution is non\-degenerate, so Corollary[2](https://arxiv.org/html/2610.21081#Thmcorollary2)givesht′​\(r\)<0h\_\{t\}^\{\\prime\}\(r\)<0forr\>0r\>0\. In addition,Wt≥σt2W\_\{t\}\\geq\\sigma\_\{t\}^\{2\}almost surely, so

0<ht​\(r\)<1σt20<h\_\{t\}\(r\)<\\frac\{1\}\{\\sigma\_\{t\}^\{2\}\}for every finite residual\. Thus the jump\-Poisson corruption is one concrete random\-scale construction in which the effective posterior precision falls below the fixed Gaussian value as the residual grows\.

For the ideal, unclipped isotropicα\\alpha\-stable corruption, the connection between stable laws and Gaussian scale mixtures is classical\[[West, 1987](https://arxiv.org/html/2610.21081#bib.bib41)\], while the standard far\-tail density behavior\[[Samorodnitsky and Taqqu, 1994](https://arxiv.org/html/2610.21081#bib.bib18)\]is

qt​\(u\)∼Ct​∥u∥−\(d\+α\),∥u∥→∞\.q\_\{t\}\(u\)\\sim C\_\{t\}\\lVert u\\rVert^\{\-\(d\+\\alpha\)\},\\qquad\\lVert u\\rVert\\to\\infty\.The corresponding smooth density asymptotics for multidimensionalα\\alpha\-stable laws\[[Zaigraev, 2006](https://arxiv.org/html/2610.21081#bib.bib19)\]give the derivative behavior required by Corollary[4](https://arxiv.org/html/2610.21081#Thmcorollary4), withm=d\+αm=d\+\\alpha\. Hence

ht​\(r\)∼d\+αr2,h\_\{t\}\(r\)\\sim\\frac\{d\+\\alpha\}\{r^\{2\}\},and

‖∇u​log​qt​\(u\)‖=r​ht​\(r\)∼d\+αr⟶0\.\\left\\lVert\\nabla\_\{u\}\\log q\_\{t\}\(u\)\\right\\rVert=r\\,h\_\{t\}\(r\)\\sim\\frac\{d\+\\alpha\}\{r\}\\longrightarrow 0\.Thus the ideal alpha\-stable construction is an example of the stronger polynomial\-tail geometry in which both the per\-unit restoring precision and the absolute restoring score weaken in the far tail\.

For the clipped alpha\-stable corruption implemented in StochLOB,

Aclip=min⁡\(A,Q0\.999​\(A\)\),Wt=σt2​Aclip,A\_\{\\mathrm\{clip\}\}=\\min\\\!\\left\(A,Q\_\{0\.999\}\(A\)\\right\),\\qquad W\_\{t\}=\\sigma\_\{t\}^\{2\}A\_\{\\mathrm\{clip\}\},and

Wt,max:=σt2​Q0\.999​\(A\)\.W\_\{t,\\max\}:=\\sigma\_\{t\}^\{2\}Q\_\{0\.999\}\(A\)\.The resulting scale is non\-degenerate, bounded above byWt,maxW\_\{t,\\max\}, and places positive probability at the maximal scale because of clipping\. Corollary[3](https://arxiv.org/html/2610.21081#Thmcorollary3)therefore gives

ht′​\(r\)<0,ht​\(r\)≥1Wt,max,h\_\{t\}^\{\\prime\}\(r\)<0,\\qquad h\_\{t\}\(r\)\\geq\\frac\{1\}\{W\_\{t,\\max\}\},with

ht​\(r\)⟶1Wt,maxas​r→∞\.h\_\{t\}\(r\)\\longrightarrow\\frac\{1\}\{W\_\{t,\\max\}\}\\qquad\\text\{as \}r\\to\\infty\.Thus clipping preserves residual\-adaptive downweighting but imposes a positive precision floor in the extreme tail\.

These statements provide a theoretical mechanism that complements the paper’s empirical stress results rather than replacing them\. They do not imply that every non\-Gaussian corruption must improve macro\-F1\. Instead, they show that the proposed non\-Gaussian objectives possess a family\-level residual\-adaptation mechanism that the Gaussian baseline cannot possess\. This interpretation is complementary to recent work that replaces Gaussian perturbations by heavy\-tailed or Lévy alternatives in denoising score matching and diffusion models\[[Deasy et al\., 2021](https://arxiv.org/html/2610.21081#bib.bib21),[Yoon et al\., 2023](https://arxiv.org/html/2610.21081#bib.bib20),[Pandey et al\., 2025](https://arxiv.org/html/2610.21081#bib.bib22),[Shariatian et al\., 2025](https://arxiv.org/html/2610.21081#bib.bib23)\]: rather than proposing a new generative reverse process, the analysis here isolates the posterior\-precision mechanism induced by the latent scale in the supervised StochLOB objective\. Proposition[3](https://arxiv.org/html/2610.21081#Thmproposition3)further shows that, when the corruption scale is random, the same latent\-scale posterior also induces uncertainty about the component\-level restoring score, with this posterior score uncertainty determined by the variance of the latent corruption precision\. Thus the non\-Gaussian scale\-mixture objectives differ from the Gaussian baseline not only in how their effective restoring precision adapts to the observed residual, but also in allowing uncertainty over the latent corruption scale underlying that restoration\. The stress experiments determine whether these structural properties are useful for the observed LOB prediction problem\.

### G\.1Mathematical Rationale for Future Work

The future directions in Section[7](https://arxiv.org/html/2610.21081#S7)follow from two related facts\. First, the current score target is a posterior precision\. Second, robustness transfer from a training corruption law to an unseen stress law is controlled by both the size and the direction of the shocks covered by the training law\.

#### From scalar precision to stress coverage\.

At a fixed noise scale, the present model uses an isotropic scale mixture

qt​\(u\)=∫𝒩⁡\(u,0,w​Id\)​pt​\(𝑑w\),∇u​log​qt​\(u\)=−u​𝔼​\[Wt−1∣u\]\.q\_\{t\}\(u\)=\\int\\mathcal\{N\}\(u;0,wI\_\{d\}\)\\,p\_\{t\}\(dw\),\\qquad\\nabla\_\{u\}\\log q\_\{t\}\(u\)=\-u\\,\\mathbb\{E\}\[W\_\{t\}^\{\-1\}\\mid u\]\.This identifies the quantity that future work should generalize: the scalar posterior precisionht​\(∥u∥\)h\_\{t\}\(\\lVert u\\rVert\)\. The next question is whether a corruption lawqtq\_\{t\}that is useful during training also covers a deployment stress lawν\\nu\. LetL⁡\(u\)≥0L\(u\)\\geq 0be the loss incurred under a perturbationuu, and assume the perturbation is a nuisance for the label,

p⁡\(y∣x0\+u\)=p⁡\(y∣x0\)\.p\(y\\mid x\_\{0\}\+u\)=p\(y\\mid x\_\{0\}\)\.Ifν≪qt\\nu\\ll q\_\{t\}, Cauchy–Schwarz gives the second\-order transfer bound, familiar from importance weighting\[[Cortes et al\., 2010](https://arxiv.org/html/2610.21081#bib.bib37)\]:

𝔼ν\[L\]=∫L\(u\)ν⁡\(u\)qt​\(u\)qt\(u\)du≤𝔼qt​\[L2\]1\+χ2\(ν∥qt\),1\+χ2\(ν∥qt\)=∫ν​\(u\)2qt​\(u\)du\.\\mathbb\{E\}\_\{\\nu\}\[L\]=\\int L\(u\)\\frac\{\\nu\(u\)\}\{q\_\{t\}\(u\)\}q\_\{t\}\(u\)\\,du\\leq\\sqrt\{\\mathbb\{E\}\_\{q\_\{t\}\}\[L^\{2\}\]\}\\,\\sqrt\{1\+\\chi^\{2\}\(\\nu\\\|q\_\{t\}\)\},\\qquad 1\+\\chi^\{2\}\(\\nu\\\|q\_\{t\}\)=\\int\\frac\{\\nu\(u\)^\{2\}\}\{q\_\{t\}\(u\)\}\\,du\.Thus the training objective can transfer to stress only when two quantities are controlled: a training\-side second moment and the coverage constant1\+χ2\(ν∥qt\)1\+\\chi^\{2\}\(\\nu\\\|q\_\{t\}\)\. This is a sufficient condition, not a claim that largeχ2\\chi^\{2\}necessarily implies poor accuracy\. The same constant governs the sample size required for importance sampling, where it is known to grow with an intrinsic dimension of the mismatch\[[Agapiou et al\., 2017](https://arxiv.org/html/2610.21081#bib.bib36)\]\.

The fixed\-scale statement matches the implementation because noise levels are sampled from a finite grid\. If a scalettis sampled with probabilityπt\\pi\_\{t\}and loss weightwtw\_\{t\}, a weighted average objective controls the per\-scale loss up to the factor\(πt​wt\)−1\(\\pi\_\{t\}w\_\{t\}\)^\{\-1\}\. For a continuous schedule, an analogous deployed statement would require a finite average or supremum of1\+χ2\(ν∥qt\)1\+\\chi^\{2\}\(\\nu\\\|q\_\{t\}\)overtt\.

#### Radial–angular factorization\.

Writeu=r​θu=r\\theta, wherer\>0r\>0andθ∈𝕊d−1\\theta\\in\\mathbb\{S\}^\{d\-1\}\. Suppose the stress law and corruption law have regularly varying tails

ν⁡\(r​θ\)=r−\(d\+αν\)​Lν​\(r\)​ψν​\(θ\)​\(1\+o⁡\(1\)\),qt​\(r​θ\)=r−\(d\+αq\)​Lq​\(r\)​ψq​\(θ\)​\(1\+o⁡\(1\)\),\\nu\(r\\theta\)=r^\{\-\(d\+\\alpha\_\{\\nu\}\)\}L\_\{\\nu\}\(r\)\\psi\_\{\\nu\}\(\\theta\)\(1\+o\(1\)\),\\qquad q\_\{t\}\(r\\theta\)=r^\{\-\(d\+\\alpha\_\{q\}\)\}L\_\{q\}\(r\)\\psi\_\{q\}\(\\theta\)\(1\+o\(1\)\),uniformly inθ\\theta, following the standard radial–angular form of multivariate regular variation\[[Bingham et al\., 1987](https://arxiv.org/html/2610.21081#bib.bib44),[Resnick, 2007](https://arxiv.org/html/2610.21081#bib.bib45),[Basrak et al\., 2002](https://arxiv.org/html/2610.21081#bib.bib46)\]\. Hereαν\\alpha\_\{\\nu\}andαq\\alpha\_\{q\}are tail indices,LνL\_\{\\nu\}andLqL\_\{q\}are slowly varying radial factors, andψν,ψq\\psi\_\{\\nu\},\\psi\_\{q\}are angular densities with respect to the uniform probability measure on the sphere\. Splitting the integral atr0r\_\{0\}gives

1\+χ2\(ν∥qt\)−C0\(r0\)=\(1\+o\(1\)\)Irad\(r0\)Iang,1\+\\chi^\{2\}\(\\nu\\\|q\_\{t\}\)\-C\_\{0\}\(r\_\{0\}\)=\(1\+o\(1\)\)\\,I\_\{\\mathrm\{rad\}\}\(r\_\{0\}\)\\,I\_\{\\mathrm\{ang\}\},where

Irad​\(r0\)\\displaystyle I\_\{\\mathrm\{rad\}\}\(r\_\{0\}\)=\|𝕊d−1\|​∫r0∞r−1−\(2​αν−αq\)​Lν​\(r\)2Lq​\(r\)​dr,\\displaystyle=\|\\mathbb\{S\}^\{d\-1\}\|\\int\_\{r\_\{0\}\}^\{\\infty\}r^\{\-1\-\(2\\alpha\_\{\\nu\}\-\\alpha\_\{q\}\)\}\\frac\{L\_\{\\nu\}\(r\)^\{2\}\}\{L\_\{q\}\(r\)\}\\,dr,Iang\\displaystyle I\_\{\\mathrm\{ang\}\}=∫𝕊d−1ψν​\(θ\)2ψq​\(θ\)​dσ​\(θ\)\\displaystyle=\\int\_\{\\mathbb\{S\}^\{d\-1\}\}\\frac\{\\psi\_\{\\nu\}\(\\theta\)^\{2\}\}\{\\psi\_\{q\}\(\\theta\)\}\\,d\\sigma\(\\theta\)=exp\(D2\(ψν∥ψq\)\)\.\\displaystyle=\\exp\\\!\\left\(D\_\{2\}\(\\psi\_\{\\nu\}\\\|\\psi\_\{q\}\)\\right\)\.The angular term is the exponentiated order\-2 Rényi divergence\[[Rényi, 1961](https://arxiv.org/html/2610.21081#bib.bib48),[van Erven and Harremoës, 2014](https://arxiv.org/html/2610.21081#bib.bib49)\]\. The radial factor is finite whenαq<2​αν\\alpha\_\{q\}<2\\alpha\_\{\\nu\}and divergent whenαq\>2​αν\\alpha\_\{q\}\>2\\alpha\_\{\\nu\}; the critical case depends on the slowly varying term\. This explains why Gaussian corruption is a weak stress proxy for heavy\-tailed LOB residuals: a Gaussianqtq\_\{t\}decays exponentially, while a heavy\-tailedν\\nudecays polynomially, soν2/qt\\nu^\{2\}/q\_\{t\}is not integrable in the far tail\.

The factorization also shows a limitation of the current isotropic design\. Heavy tails improve radial coverage, but they do not automatically cover the directions in which market stress occurs\. If the corruption allocates mass uniformly over directions while realized stress lies near a low\-dimensional signed order\-flow subspace, the angular termIangI\_\{\\mathrm\{ang\}\}can make the bound numerically vacuous even when the radial tail index is adequate\.

#### High\-dimensional angular penalty\.

The scale of this effect is large for LOB windows\. Letν\\nube an elliptical stress law with shape matrixΣν\\Sigma\_\{\\nu\}, eigenvaluesλ1,…,λd\\lambda\_\{1\},\\ldots,\\lambda\_\{d\}normalized to unit mean, and tail indexαν\\alpha\_\{\\nu\}\[[Cambanis et al\., 1981](https://arxiv.org/html/2610.21081#bib.bib50),[Hult and Lindskog, 2002](https://arxiv.org/html/2610.21081#bib.bib52)\]\. Against isotropic corruption, define

Q⁡\(θ\)=θ⊤​Σν−1​θ,pν=d\+αν2,K⁡\(s\)=log⁡𝔼Unif​\[Q​\(θ\)−s\]\.Q\(\\theta\)=\\theta^\{\\top\}\\Sigma\_\{\\nu\}^\{\-1\}\\theta,\\qquad p\_\{\\nu\}=\\frac\{d\+\\alpha\_\{\\nu\}\}\{2\},\\qquad K\(s\)=\\log\\mathbb\{E\}\_\{\\operatorname\{Unif\}\}\\\!\\left\[Q\(\\theta\)^\{\-s\}\\right\]\.The angular divergence has the exact representation

D2\(ψν∥Unif\)=K\(2pν\)−2K\(pν\)\.D\_\{2\}\(\\psi\_\{\\nu\}\\\|\\operatorname\{Unif\}\)=K\(2p\_\{\\nu\}\)\-2K\(p\_\{\\nu\}\)\.For a nearly spherical spectrum, withCV⁡\(λ\)=std⁡\(λ\)/mean⁡\(λ\)\\operatorname\{CV\}\(\\lambda\)=\\operatorname\{std\}\(\\lambda\)/\\operatorname\{mean\}\(\\lambda\), the cumulant expansion gives

D2\(ψν∥Unif\)=\(d\+αν\)22​\(d\+2\)CV2\(λ\)\(1\+o\(1\)\)=d2CV2\(λ\)\(1\+O\(d−1\)\)\.D\_\{2\}\(\\psi\_\{\\nu\}\\\|\\operatorname\{Unif\}\)=\\frac\{\(d\+\\alpha\_\{\\nu\}\)^\{2\}\}\{2\(d\+2\)\}\\operatorname\{CV\}^\{2\}\(\\lambda\)\\,\(1\+o\(1\)\)=\\frac\{d\}\{2\}\\operatorname\{CV\}^\{2\}\(\\lambda\)\(1\+O\(d^\{\-1\}\)\)\.The coefficient is proportional to dimension\. For the60×3160\\times 31crypto LOB window used in this paper,d=1860d=1860, so keeping this angular exponent near order one requires approximatelyCV\(λ\)≲d−1/2≈2\.3%\\operatorname\{CV\}\(\\lambda\)\\lesssim d^\{\-1/2\}\\approx 2\.3\\%\. A 20% spectral dispersion can multiply1\+χ2\(ν∥qt\)1\+\\chi^\{2\}\(\\nu\\\|q\_\{t\}\)by about101610^\{16\}, which inflates the square\-root transfer bound by about10810^\{8\}\.

Table 12:Angular penalty atd=1860d=1860andαν=1\.62\\alpha\_\{\\nu\}=1\.62\. The symmetric spectrum places half the coordinates at1\+CV1\+\\operatorname\{CV\}and half at1−CV1\-\\operatorname\{CV\}; the skewed spectrum places the excess on 5% of the coordinates\. The angular factor multiplies the coverage constant, and its square root multiplies the transfer bound\.Table[12](https://arxiv.org/html/2610.21081#A7.T12)makes the design implication explicit: future corruption laws should not only be heavy\-tailed; they should also estimate the spectral geometry of realized stress, for which nonparametric angular\-measure estimators exist\[[Einmahl et al\., 2001](https://arxiv.org/html/2610.21081#bib.bib47)\]\.

#### State\-dependent and level\-correlated shocks\.

Letzzdenote the observed market state before corruption; queue\-reactive models show that order\-flow intensities depend strongly on this state\[[Huang et al\., 2015](https://arxiv.org/html/2610.21081#bib.bib51)\]\. Replace the scalar varianceWt​IdW\_\{t\}I\_\{d\}by a positive\-definite random covarianceΣt\\Sigma\_\{t\}\. Student\-ttandα\\alpha\-stable diffusion kernels are special cases of this scale\-mixture construction\[[Pandey et al\., 2025](https://arxiv.org/html/2610.21081#bib.bib22),[Shariatian et al\., 2025](https://arxiv.org/html/2610.21081#bib.bib23)\]:

qt​\(u∣z\)=∫𝒩⁡\(u,0,Σ\)​pt​\(𝑑Σ∣z\)\.q\_\{t\}\(u\\mid z\)=\\int\\mathcal\{N\}\(u;0,\\Sigma\)\\,p\_\{t\}\(d\\Sigma\\mid z\)\.The score becomes

∇ulogqt\(u∣z\)=−𝔼\[Σt−1∣u,z\]u\.\\nabla\_\{u\}\\log q\_\{t\}\(u\\mid z\)=\-\\mathbb\{E\}\\\!\\left\[\\Sigma\_\{t\}^\{\-1\}\\mid u,z\\right\]u\.Thusht​\(r\)​Idh\_\{t\}\(r\)I\_\{d\}is replaced by the posterior precision matrix𝔼\[Σt−1∣u,z\]\\mathbb\{E\}\[\\Sigma\_\{t\}^\{\-1\}\\mid u,z\]\. This extension keeps the same posterior\-score principle, but lets the corruption focus probability mass on the level pairs, assets, and temporal coordinates where stress is empirically concentrated\.

#### Signed empirical jump replay\.

Real order\-book jumps are not only large; they are signed and structured\. This can be represented by a random jump vectorJtJ\_\{t\}and covarianceΣt\\Sigma\_\{t\}:

qt​\(u∣z\)=∬𝒩⁡\(u−j,Σ\)​pt​\(𝑑j,𝑑Σ∣z\)\.q\_\{t\}\(u\\mid z\)=\\iint\\mathcal\{N\}\(u\-j,\\Sigma\)\\,p\_\{t\}\(dj,d\\Sigma\\mid z\)\.Bayes’ rule gives the score target

∇ulogqt\(u∣z\)=−𝔼\[Σt−1\(u−Jt\)∣u,z\]\.\\nabla\_\{u\}\\log q\_\{t\}\(u\\mid z\)=\-\\mathbb\{E\}\\\!\\left\[\\Sigma\_\{t\}^\{\-1\}\(u\-J\_\{t\}\)\\mid u,z\\right\]\.The current model is the special caseJt=0J\_\{t\}=0andΣt=Wt​Id\\Sigma\_\{t\}=W\_\{t\}I\_\{d\}\. An empirical jump\-replay extension would estimatept​\(d​j,d​Σ∣z\)p\_\{t\}\(dj,d\\Sigma\\mid z\)from observed order\-book shock residuals, so the auxiliary task would learn not only how much to denoise a large residual, but also which signed shock pattern made that residual plausible\. In the coverage notation above, this is a way to reduce both the radial mismatch and the angular Rényi penalty\.

#### Ablations and seed\-level uncertainty\.

The training objective can be written as

ℒ⁡\(θ,λs\)=ℒCE​\(θ\)\+λs​ℒscore​\(θ\),\\mathcal\{L\}\(\\theta;\\lambda\_\{s\}\)=\\mathcal\{L\}\_\{\\mathrm\{CE\}\}\(\\theta\)\+\\lambda\_\{s\}\\mathcal\{L\}\_\{\\mathrm\{score\}\}\(\\theta\),whereλs\\lambda\_\{s\}controls score matching\. For any evaluation metricMM, the marginal effect of the score loss can be measured by

Δs=M⁡\(λs\)−M⁡\(0\)\.\\Delta\_\{s\}=M\(\\lambda\_\{s\}\)\-M\(0\)\.Acrossmmrandom seeds, the uncertainty of either effect can be reported as

Δ¯±tm−1,0\.975​σ^Δm\.\\bar\{\\Delta\}\\pm t\_\{m\-1,0\.975\}\\frac\{\\widehat\{\\sigma\}\_\{\\Delta\}\}\{\\sqrt\{m\}\}\.This is the direct way to quantify the contribution of the score target\. The same design can compare temporal readouts by treating the readout family as a controlled factor and reporting paired seed\-level differences inMM\.

#### Cost\-aware portfolio thresholds\.

Letπτ,t\\pi\_\{\\tau,t\}be the portfolio generated from model probabilities using threshold vectorτ\\tau, letrt\+1r\_\{t\+1\}be next\-period asset returns, and letct​\(Δ​πτ,t\)c\_\{t\}\(\\Delta\\pi\_\{\\tau,t\}\)denote transaction costs\. A validation\-tuned threshold should solve

τ⋆=arg⁡maxτ​1T​∑t=1T\[πτ,t⊤​rt\+1−ct​\(Δ​πτ,t\)−γ2​πτ,t⊤​Ωt​πτ,t\],\\tau^\{\\star\}=\\arg\\max\_\{\\tau\}\\frac\{1\}\{T\}\\sum\_\{t=1\}^\{T\}\\left\[\\pi\_\{\\tau,t\}^\{\\top\}r\_\{t\+1\}\-c\_\{t\}\(\\Delta\\pi\_\{\\tau,t\}\)\-\\frac\{\\gamma\}\{2\}\\pi\_\{\\tau,t\}^\{\\top\}\\Omega\_\{t\}\\pi\_\{\\tau,t\}\\right\],This is the single\-period mean–variance objective with transaction costs studied by[Gârleanu and Pedersen \[2013\]](https://arxiv.org/html/2610.21081#bib.bib53)and[Boyd et al\. \[2017\]](https://arxiv.org/html/2610.21081#bib.bib54); the threshold vectorτ\\taureplaces their continuous trade variable\. HereΩt\\Omega\_\{t\}is a risk estimate andγ≥0\\gamma\\geq 0controls risk aversion\. This makes the Feishu extension explicit: the classifier supplies probabilities, but the final trading rule should be selected by expected net utility under market frictions, not by fixed cutoffs chosen independently of costs\.