OrderFlowDiT: Transport-Regularized Diffusion Transformer for Probabilistic Limit-Order Supply Forecasting

Jung Hun Phee; Tucker Balch

cs.LGOct 15, 2026

OrderFlowDiT: Transport-Regularized Diffusion Transformer for Probabilistic Limit-Order Supply Forecasting

Authors: Jung Hun Phee, Tucker Balch

Organizations: Georgia Institute of Technology, Atlanta, Georgia, USA · Emory University, Atlanta, Georgia, USA

Abstract

We study the conditional distribution of limit-order volume submitted over the next five minutes, given the current order book and recent order flow. The task requires capturing price placement, total volume, and predictive uncertainty jointly. Coordinate-wise diffusion losses do not explicitly account for price displacement, while normalizing fields to unit mass removes information about total volume. Agreement on selected summary statistics also does not establish fidelity to the joint field distribution. We introduce OrderFlowDiT, a conditional Diffusion Transformer for unnormalized bid and ask volume-at-price fields, trained with an unbalanced optimal-transport (UOT) auxiliary loss. We also develop a full-field evaluation framework that combines a transport cost sensitive to price placement and total volume with proper scoring rules and calibration diagnostics. In the primary comparison on 1950 validation conditions across 30 U.S. equities, adding UOT reduces full-field transport cost by 45.8% and the transport energy score by 26.1% relative to training with mean squared error alone. Both measures improve for every stock. These findings support incorporating price geometry and volume into generative modeling, with field accuracy and predictive uncertainty evaluated together.

Figures & tables

Figure 1. The data at one forecast boundary, within $300$ ticks of the mid-price: the book snapshot $X_{k}$ , the observed field $A_{k}^{-}$ , and the target field $A_{k}^{+}$ . Ask volume is drawn upward and bid volume downward; both are non-negative, unnormalized share counts, and the two flow panels share a vertical scale. This view holds $53\%$ of the submitted volume and $1.2\%$ of the resting depth; the rest lies farther from the mid. Three panels side by side: the book snapshot, the observed submission field, and the target submission field, each plotted against price offset from the mid-price over a range of three hundred ticks either side.

Panel of Figure 2 — Figure 2. Forecast timing in the contemporaneous-snapshot setting. All windows are left-closed and right-open; the target begins at the forecast boundary. A timeline with two adjacent windows of equal duration. Observed submission flow occupies the window before the forecast boundary, and future submission flow occupies the window after it. The book snapshot at the boundary contains only events with earlier timestamps.

Panel of Figure 3 — Figure 3. OrderFlowDiT-Base. Every field shares the one-tick, mid-anchored price grid of Section 3.3 , so a bin refers to the same price in the noisy target $Z_{n}$ and in all six conditioning channels of $C_{k}$ : book depth $X_{k}$ , the two best-quote indicators, and observed flow $A_{k}^{-}$ . Patches of $P=16$ bins are embedded separately for target and conditioning and summed; bins padding the grid to whole patches are masked. Base-MSE trains on the masked velocity loss alone, whereas the Base-UOT term is evaluated on the decoded field $\hat{A}$ in shares rather than in velocity space. GRPO (Section 4.4 ) fine-tunes this same network. A left-to-right schematic of the conditional diffusion transformer. On the left, eight horizontal strips stacked over a shared mid-anchored price axis show the two noisy target channels above the six conditioning channels: bid and ask book depth, bid and ask best-quote indicators, and bid and ask observed flow. A highlighted vertical slice marks one patch of sixteen price bins. Separate patch embeddings for target and conditioning are summed with a positional encoding to form a sequence of price tokens, which pass through twelve transformer blocks and a linear head that outputs a two-channel velocity field. A global conditioning vector, built from the diffusion step, summary features, instrument identity, window duration and the mean condition token, modulates each block through AdaLN-Zero. On the right, the velocity feeds the masked mean-squared-error objective, and a second branch decodes it to a predicted field in shares that is compared with the realized field by an unbalanced transport term used only by the UOT variant. A dashed arrow returning to the target channels marks the fifty-step DDIM sampling loop.

Quantity	Role
$d_{\mathrm{full}}$	Price placement and volume cost
$\mathrm{ES}_{T}$	Proper transport-based score
$\mathrm{ES}_{2}$	Strictly proper joint-field score
Volume CRPS	Strictly proper side-volume score
Sided $W_{1}$	Shape diagnostic in ticks
NTVE	Relative-volume diagnostic

Table 1. Evaluation quantities and their roles. Costs, errors, scores, and RI are lower-is-better. Propriety refers to population scores on the domains stated in the text.

Model	$R$	$d_{\mathrm{full}}/10^{6}$	$\mathrm{ES}_{T}$	$\mathrm{ES}_{2}$	CRPS	Sided $W_{1}$	NTVE	$\mathrm{MSE}_{\log}$
Skew-normal	1	450.515	14.995	0.794	0.317	733.904	0.328	0.722
Skew-normal mixture	1	109.510	4.154	1.452	0.317	191.888	0.328	2.433
Empirical	4	95.698	2.018	0.496	0.283	166.599	0.756	0.934
Base-MSE	4	110.829	2.152	0.414	0.460	149.220	1.058	0.656
Base-UOT	4	60.074	1.591	0.387	0.283	123.552	0.447	0.599
Base-GRPO	4	62.025	1.494	0.409	0.243	107.705	0.382	0.549

Table 2. Main comparison on

1{,}950

conditions. All seven metrics are lower-is-better; bold marks the lowest displayed value in each column. The full-field cost is reported in

10^{6}

\cdot

ticks and sided

W_{1}

in ticks. CRPS denotes log-total-volume CRPS. The mixture uses

K_{\max}=2

. The

R=1

rows are point-mass forecasts.

Figure 4. Realized and Large-UOT submission fields within $100$ ticks of the mid-price on identical axes: ask shares upward, bid downward. The condition is nearest the median Large-UOT energy score among mid-session conditions on this instrument. Total volumes are $1.94$ M and $1.49$ M shares, respectively; only roughly a quarter of each field’s volume lies in the displayed range. Two panels on identical axes spanning one hundred ticks either side of the mid-price. The left panel is the realized submission field, with ask volume above a zero line and bid volume below it; the right panel is one draw from the Large-UOT model. Both show an ask mode just above the mid-price and bid volume spread further below it, of comparable height and width.

Quantity	Base-UOT	Base-GRPO
$\mathrm{ES}_{T}$	1.591	1.494
Target distance $T_{T}$	2.897	2.510
Inter-draw distance $D_{T}$	2.611	2.033
$\mathrm{ES}_{2}$	0.387	0.409
Target distance $T_{2}$	0.722	0.686
Inter-draw distance $D_{2}$	0.670	0.553

Table 3. GRPO comparison on the same lag0 conditions (

R=4

). For each energy-score geometry

j\in\{T,2\}

T_{j}

is the mean target distance and

D_{j}

the mean off-diagonal inter-draw distance, so

\mathrm{ES}_{j}=T_{j}-D_{j}/2

. Interior-rank coverage has nominal value

0.600

. Distance components and coverage are diagnostics, without a common lower-is-better direction.

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Probabilistic Limit Order Book Forecasting with Conditional Diffusion Models

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Department of Statistics and Data Science, Tsinghua University, Beijing, China

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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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