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

Arshia Abolghasemi; Seyed Reza Tavakoli; Parsa Naderi; Nazanin Mirzaei; Behrad Jahani; Saba Yousefzadeh; Tahereh Javaheri; Mohammad Hosein Rohban

cs.LGOct 15, 2026

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

Authors: Arshia Abolghasemi, Seyed Reza Tavakoli, Parsa Naderi, Nazanin Mirzaei, Behrad Jahani, Saba Yousefzadeh, Tahereh Javaheri, Mohammad Hosein Rohban,

Organizations: Dataism Laboratory for Quantitative Finance · Sharif University of Technology · Amirkabir University of Technology · Boston University ·

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 study STOCHLOB, 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-Lévy noise through a compound-Poisson variance mixture, and clipped $α$ -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 factor $h_t(r)$ 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.

Figures & tables

Figure 1: Main stress result on chronologically held-out Coinbase BTC windows at horizon

k=100

(

31{,}842

test 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 are

95\%

circular block-bootstrap confidence intervals.

Figure 2: StochLOB workflow. The classifier pools the unperturbed sequence representation for three trend logits. The training-only score decoder receives the noisy

T\times F

window, timestep embedding, and per-timestep trunk context, and predicts a

T\times F

score field. At inference, the corruption sampler and score decoder are disabled.

	$k=10$		$k=20$
Model	Acc	F1	Acc	F1
Logistic regression	0.6607 $\pm$ 0.0007	0.6591 $\pm$ 0.0007	0.6061 $\pm$ 0.0049	0.6089 $\pm$ 0.0048
Linear VAR	0.6735 $\pm$ 0.0009	0.6710 $\pm$ 0.0008	0.6205 $\pm$ 0.0017	0.6231 $\pm$ 0.0015
DeepLOB (CNN-LSTM)	0.6338 $\pm$ 0.0167	0.6284 $\pm$ 0.0142	0.5591 $\pm$ 0.0128	0.5535 $\pm$ 0.0079
BiN-CTABL	0.6679 $\pm$ 0.0045	0.6659 $\pm$ 0.0041	0.6097 $\pm$ 0.0027	0.6110 $\pm$ 0.0024
TLOB (Transformer)	0.6783 $\pm$ 0.0068	0.6754 $\pm$ 0.0054	0.6127 $\pm$ 0.0076	0.6142 $\pm$ 0.0068

Table 1: BTC OFI trend prediction on chronologically held-out windows. Accuracy (Acc) and macro-F1 are reported as mean

\pm

std over five seeds (

13,21,42,84,168

). Bold marks the best mean result in each horizon/metric column; underlining marks the second best.

Regime	Best StochLOB	$k$	StochLOB F1	Gaussian F1	Best supervised	Gain vs. Gaussian	Gain vs. supervised
Stress	alpha-stable	10	0.712	0.703	TLOB 0.708	+0.008	+0.004
Stress	alpha-stable	100	0.647	0.604	LogReg 0.614	+0.043	+0.034
High variance	jump-Poisson	20	0.686	0.680	LinVAR 0.651	+0.006	+0.034
High variance	alpha-stable	50	0.658	0.603	LinVAR 0.645	+0.055	+0.014
High variance	alpha-stable	100	0.659	0.613	LinVAR 0.628	+0.046	+0.030
Jump residual	alpha-stable	100	0.637	0.567	LinVAR 0.597	+0.070	+0.039

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 reports the full regime-by-horizon grid.

Model	CAGR(%)	Sharpe	Max DD(%)	Return(%)	Vol.(%)	Calmar	Win(%)	Avg. Hold.
LogReg	28.03	0.69	$-$ 28.41	28.03	46.28	0.99	47.72	167.3
LinVAR	$-$ 38.66	$-$ 0.71	$-$ 49.75	$-$ 38.66	52.35	$-$ 0.78	43.98	11.3
DeepLOB	31.72	0.68	$-$ 36.49	31.72	82.61	0.87	48.55	50.1
TLOB	17.56	0.55	$-$ 29.04	17.56	46.28	0.60	53.11	961.5
StochLOB (jump-Poisson)	38.78	1.04	$-$ 23.20	38.78	36.64	1.67	54.36	1861.0

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.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Snapshots	Raw start (UTC)	Raw end (UTC)	Test start (UTC)
BNB	213,679	2026-06-18 23:42:10	2026-07-13 23:10:30	2026-07-10 05:24:30
BTC	213,335	2026-06-18 23:42:10	2026-07-13 23:10:40	2026-07-10 05:24:10
ETH	213,593	2026-06-18 23:42:10	2026-07-13 23:10:30	2026-07-10 05:28:10
USDT	213,448	2026-06-18 23:42:10	2026-07-13 23:10:40	2026-07-10 05:27:10
XRP	213,752	2026-06-18 23:42:10	2026-07-13 23:10:30	2026-07-10 05:27:40

Appendix

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.

Block	Features
Microstructure	spread/mid; log total bid-depth over total ask-depth; one-step log mid-return
Trade flow	log buy volume; log sell volume; signed trade imbalance; log trade count; VWAP deviation from mid
Activity proxies	log trade count; spread/mid; absolute one-step log mid-return

Appendix

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.

$k$	Real time	$\alpha_{\mathrm{lab}}$	Train	Val.	Test
10	100 s	0.000093	147,218	31,831	31,932
20	200 s	0.000189	147,208	31,821	31,922
50	500 s	0.000362	147,178	31,791	31,892
100	1000 s	0.000543	147,087	31,741	31,842

Appendix

Table 6: BTC split sizes and train-calibrated label thresholds. Other crypto symbols use the same protocol with symbol-specific thresholds.

Symbol	BTC	ETH	BNB	XRP	USDT
$\alpha$ used for $\alpha$ -stable corruption	1.687	1.731	1.900	1.615	1.685

Appendix

Table 7: Coinbase alpha-stable corruption indices from train-only OFI tail calibration.

Panel of Figure 4 — 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.

Figure 5: Log-log complementary CDF of pooled absolute crypto OFI, normalized by symbol median. The top 5% Hill/Pareto fit gives $\hat{\alpha}=1.62$ , supporting heavy-tailed OFI corruption.

Block	Feishu protocol
Universe	2,306 Chinese A-share symbols
Rows	1,606,720 asset-day observations
Features	240 intraday OFI and 19 daily features
Slots	24 ten-minute morning/afternoon slots
History	50 trading days per model input
Splits	756,787 train; 189,882 validation; 542,612 out-of-sample

Appendix

Table 8: Feishu data and evaluation protocol.

Regime	$k$	Best heavy-tailed StochLOB	StochLOB F1	Gaussian F1	Best supervised	Gain vs Gaussian	Gain vs supervised	$n_{\mathrm{tail}}$
Stress	10	alpha-stable	0.712	0.703	TLOB 0.708	+0.008	+0.004	3194
Stress	20	alpha-stable	0.655	0.663	LinVAR 0.648	-0.007	+0.007	3193
Stress	50	jump-Poisson	0.610	0.592	TLOB 0.611	+0.019	-0.001	3190
Stress	100	alpha-stable	0.647	0.604	LogReg 0.614	+0.043	+0.034	3185
High variance	10	alpha-stable	0.695	0.665	TLOB 0.697	+0.031	-0.002	3194
High variance	20	jump-Poisson	0.686	0.680	LinVAR 0.651	+0.006	+0.034	3193

Appendix

Table 9: Complete BTC extreme-window macro-F1 grid underlying Table 2 . 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.

		$k=10$		$k=20$		$k=50$		$k=100$
Symbol	Model	Acc	F1	Acc	F1	Acc	F1	Acc	F1
ETH	LogReg	0.6669	0.6653	0.6304	0.6334	0.5975	0.5999	0.5573	0.5614
ETH	LinVAR	0.6913	0.6876	0.6518	0.6550	0.6033	0.6074	0.5686	0.5734
ETH	DeepLOB	0.6615	0.6543	0.5752	0.5754	0.5471	0.5464	0.4793	0.4829
ETH	BiN-CTABL	0.6857	0.6820	0.6467	0.6484	0.5771	0.5792	0.5464	0.5501
ETH	TLOB	0.6840	0.6793	0.6389	0.6412	0.5606	0.5624	0.5826	0.5782

Appendix

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.

	StochLOB (jump-Poisson)			StochLOB (alpha-stable)
Class ( $k=10$ )	P	R	F1	P	R	F1
down	0.7023	0.6188	0.6579	0.7516	0.6052	0.6705
stationary	0.6685	0.7389	0.7019	0.6646	0.7655	0.7115
up	0.6805	0.6679	0.6741	0.6805	0.6794	0.6800

Appendix

Table 11: BTC per-class precision (P), recall (R), and F1 at

k=10

$\operatorname{CV}(\lambda)$	$D_{2}$	$\exp(D_{2})$	$D_{2}$	$\exp(D_{2})$	$\exp(D_{2}/2)$ (sym.)
	symmetric spectrum		skewed spectrum
0.01	0.09	$1.1$	0.09	$1.1$	$1.05$
0.05	2.32	$1.0\times 10^{1}$	2.37	$1.1\times 10^{1}$	$3.2$
0.10	9.26	$1.1\times 10^{4}$	10.0	$2.2\times 10^{4}$	$1.0\times 10^{2}$
0.20	36.6	$7.9\times 10^{15}$	47.1	$2.9\times 10^{20}$	$8.9\times 10^{7}$

Appendix

Table 12: Angular penalty at

d=1860

and

\alpha_{\nu}=1.62

. The symmetric spectrum places half the coordinates at

1+\operatorname{CV}

and half at

1-\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.

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Open paper AskPDF arXiv

Modeling limit order book (LOB) dynamics is central to market forecasting and execution-strategy evaluation. While diffusion models achieve strong empirical performance, existing LOB snapshot models typically lack a learned recurrent encoder and formal statistical guarantees. We introduce LOBGrad, a conditional DDPM framework for probabilistic LOB forecasting that combines an autoregressive RNN encoder with a diffusion-based generator. In particular, we establish an excess-risk oracle inequality for the one-step DDPM noise-prediction objective under exponential mixing and regularity assumptions. We further show that, under contractive dynamics, the dependence on the conditioning-window length can be reduced to a dependence on the latent-state dimension. Experiments on BTC/USDT, GOOGL, and META show that LOBGrad improves on its encoder-free diffusion ablation across all reported queue metrics and obtains competitive results against other baselines. It also reproduces the key stylized facts of LOB microstructure, and yields reasonably calibrated queue-based predictive intervals. A stylized downstream task illustrates the potential value of the generated forecasts.

Abstract

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