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

# OrderFlowDiT: Transport-Regularized Diffusion Transformer for Probabilistic Limit-Order Supply Forecasting
Source: [https://arxiv.org/html/2610.22727](https://arxiv.org/html/2610.22727)
andTucker BalchAffiliation:Emory University,Atlanta,Georgia,USA

2026

###### 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 on1,9501\{,\}950validation conditions across3030U\.S\. equities, adding UOT reduces full\-field transport cost by45\.8%45\.8\\%and the transport energy score by26\.1%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\.

###### Keywords:

limit order book, world model, generative models, optimal transport, evaluation, market microstructure

## 1\.Introduction

Liquidity in limit\-order\-book \(LOB\) markets is dynamic: trading over a finite horizon depends on incoming liquidity as well as the orders currently resting in the book\. Models of optimal execution show that liquidity replenishment rates affect trading schedules and execution costs\([Obizhaeva and Wang, 2013](https://arxiv.org/html/2610.22727#bib.bib22)\)\. Empirical studies document state\-dependent limit\-order submissions and the recovery of liquidity following trades and cancellations\([Biais et al\., 1995](https://arxiv.org/html/2610.22727#bib.bib2);[Lo and Hall, 2015](https://arxiv.org/html/2610.22727#bib.bib17)\)\. Moreover, estimated quote responses to incoming limit orders vary with their size and price aggressiveness, highlighting the importance of both submitted volume and price placement\([Hautsch and Huang, 2012](https://arxiv.org/html/2610.22727#bib.bib12)\)\. A forecast of the book at the end of a horizon, however, does not separately identify the new volume submitted during that interval, because submissions can be offset by cancellations and executions\. These observations motivate forecasting how much new limit\-order volume will arrive and where it will be submitted, as one component of the future trading environment\.

Forecasting future order supply requires characterizing uncertainty in both submitted volume and its price allocation\. Empirical order\-flow analysis documents interdependence between order sizes and arrival dynamics\([Rambaldi et al\., 2017](https://arxiv.org/html/2610.22727#bib.bib26)\)\. A single expected submission profile cannot express this uncertainty: two predictive distributions can have the same mean field while assigning different probabilities to low submitted volume within a given price region\. This motivates representing forecasts as predictive distributions, rather than point predictions alone\([Gneiting et al\., 2007](https://arxiv.org/html/2610.22727#bib.bib10)\)\. Moreover, marginal distributions at individual prices do not determine how volumes vary jointly across prices and sides; constructing multivariate forecast scenarios requires accounting for dependence as well as the marginal distributions\([Schefzik et al\., 2013](https://arxiv.org/html/2610.22727#bib.bib28)\)\. Total\-volume forecasts omit price allocation, whereas unit\-mass price profiles discard absolute volume\. We therefore seek a conditional joint distribution over unnormalized bid and ask volume\-at\-price fields, allowing total side volumes, price\-localized submissions, and bid–ask submission asymmetries to be examined within the same generated scenarios\.

Specifically, we model the conditional joint distribution of bid and ask limit\-order submission fields over the next five minutes, given the current LOB snapshot and recent order flow\. Using both book state and flow history is consistent with queue\-reactive Hawkes models that explicitly combine these sources of information\([Wu et al\., 2019](https://arxiv.org/html/2610.22727#bib.bib35)\)\. Each field coordinate records the cumulative volume of new limit orders submitted on one side within a price bin during the forecast window\. We fix the price grid relative to the mid\-price observed at the forecast origin and retain absolute submitted volume without unit\-mass normalization\. Unlike message\-level generators coupled to a book simulator\([Nagy et al\., 2023](https://arxiv.org/html/2610.22727#bib.bib20)\), our target aggregates submissions without retaining event order\. Since order\-book dynamics also involve cancellations and executions\([Huang et al\., 2015](https://arxiv.org/html/2610.22727#bib.bib14)\), the submission fields alone do not determine the subsequent book state or remaining liquidity\. We therefore position this conditional order\-supply model as a potential component of a financial world model, rather than as a complete simulator of order\-book evolution\.

Learning these submission fields calls for an objective sensitive to both price displacement and differences in submitted volume\. We use a conditional diffusion model with a denoising mean squared error \(MSE\) objective\([Ho et al\., 2020](https://arxiv.org/html/2610.22727#bib.bib13)\)\. This objective penalizes errors coordinatewise, without explicitly assigning costs to displacement between price levels\. Optimal transport provides a geometry\-aware comparison by matching mass across locations according to a ground cost\([Feydy et al\., 2019](https://arxiv.org/html/2610.22727#bib.bib8)\)\. However, balanced transport requires equal total mass, whereas predicted and observed submission fields can have different total volumes\. Unbalanced optimal transport \(UOT\) relaxes the marginal constraints through divergence penalties, allowing spatial displacement and mass discrepancies to be considered jointly\([Séjourné et al\., 2019](https://arxiv.org/html/2610.22727#bib.bib29)\)\. We therefore augment diffusion MSE with a debiased unbalanced Sinkhorn divergence on decoded denoised fields, introducing an auxiliary training signal that explicitly accounts for price placement and unequal submitted volume\.

Evaluating these forecasts requires assessing not only discrepancies between generated and observed fields, but also the quality of the predictive distribution\. Comparisons of market statistics, such as those included in LOB\-Bench\([Nagy et al\., 2025](https://arxiv.org/html/2610.22727#bib.bib21)\), provide useful measures of realism, but agreement on selected summaries alone does not determine the conditional joint distribution of submission fields\. Even when full fields are compared, averaging distances from individual generated samples to the realization can favor overly concentrated forecasts rather than faithful representations of uncertainty\([Gneiting and Raftery, 2007](https://arxiv.org/html/2610.22727#bib.bib11)\)\. Proper scoring rules instead evaluate predictive distributions and are optimized in expectation by the data\-generating distribution\([Gneiting and Raftery, 2007](https://arxiv.org/html/2610.22727#bib.bib11)\)\. We therefore combine mass\-aware full\-field transport comparisons with proper scoring rules and calibration diagnostics\([Gneiting et al\., 2007](https://arxiv.org/html/2610.22727#bib.bib10)\), evaluating forecasts against their corresponding realizations at matched forecast origins and horizons\.

To address this task, we introduce OrderFlowDiT, a conditional Diffusion Transformer for the joint generation of unnormalized bid and ask limit\-order submission fields, together with a training objective and evaluation framework tailored to price allocation, submitted volume, and predictive uncertainty\. Our contributions are threefold:

1. \(1\)We incorporate an unbalanced optimal\-transport \(UOT\) auxiliary objective into conditional diffusion training\. A debiased unbalanced Sinkhorn divergence on decoded denoised fields augments the diffusion mean squared error \(MSE\), providing a field\-level training signal that explicitly accounts for price displacement and differences in total submitted volume\.
2. \(2\)We construct a full\-field evaluation framework that combines a mass\-aware transport cost with transport and joint\-field energy scores, the continuous ranked probability score \(CRPS\) for side volumes, and calibration diagnostics\. These complementary measures assess field accuracy and predictive uncertainty at matched forecast origins and horizons\.
3. \(3\)We compare MSE\-only and MSE\+UOT training with statistical baselines across1,9501\{,\}950forecast conditions from3030U\.S\. equities, reporting pooled and per\-stock results\. We examine accuracy–diversity tradeoffs and volume calibration in a Group Relative Policy Optimization \(GRPO\) fine\-tuned variant, and assess a larger training configuration that jointly increases model capacity, training data, and optimization budget\.

Across 1,950 validation conditions on 30 U\.S\. equities, Base\-UOT lowers full\-field transport cost by45\.8%45\.8\\%and transport energy score by26\.1%26\.1\\%versus Base\-MSE, with both improving for every stock\. Joint\-field energy and log\-side\-volume CRPS also improve\. The evaluated GRPO configuration further improves transport energy and volume CRPS but worsens full\-field cost, joint\-field energy, and pooled volume\-rank calibration while reducing ensemble dispersion, highlighting trade\-offs across distributional metrics\.

## 2\.Related Work

#### Market simulation and conditional order\-flow generation\.

Market simulation relies on a model of the order flow that drives changes in liquidity\. Agent\-based platforms such as ABIDES generate this flow through interactions among trading agents and an exchange\([Byrd et al\., 2020](https://arxiv.org/html/2610.22727#bib.bib4)\), while queue\-reactive models specify event intensities conditional on book state\([Huang et al\., 2015](https://arxiv.org/html/2610.22727#bib.bib14)\)\. Learning aggregate trading behavior directly from historical data provides a complementary approach, as in the conditional generators Stock\-GAN and World Agent\([Li et al\., 2020](https://arxiv.org/html/2610.22727#bib.bib16);[Coletta et al\., 2022](https://arxiv.org/html/2610.22727#bib.bib5)\)\. Autoregressive models learn the corresponding message sequences\([Nagy et al\., 2023](https://arxiv.org/html/2610.22727#bib.bib20)\), and MarS integrates order and order\-batch generation within an interactive simulator\([Li et al\., 2025](https://arxiv.org/html/2610.22727#bib.bib15)\)\. At the level of book states, DiffVolume uses conditional diffusion to generate future resting\-volume snapshots\([Wang and Ventre, 2025](https://arxiv.org/html/2610.22727#bib.bib34)\)\. Our work addresses future liquidity provision through the conditional generation of new limit\-order volume accumulated at each price over a five\-minute window\. Given the observed book and recent flow, the model produces fields that preserve price placement and total submitted volume while abstracting from within\-window event sequences\.

#### Optimal transport for generative training\.

Optimal transport incorporates the geometry of the data into distributional comparisons through a ground cost\([Peyré and Cuturi, 2019](https://arxiv.org/html/2610.22727#bib.bib24)\)\. Entropic regularization makes these comparisons computationally tractable, enabling generative models to be trained with Sinkhorn\-based losses\([Genevay et al\., 2018](https://arxiv.org/html/2610.22727#bib.bib9)\)\. For sequential data, COT\-GAN further imposes causal constraints on transport couplings to respect temporal structure\([Xu et al\., 2020](https://arxiv.org/html/2610.22727#bib.bib36)\)\. These approaches compare distributions of generated and observed samples\. Our training objective applies transport within each prediction–target pair, treating submitted volumes as measures supported on the price grid\. Because total submitted volume varies across fields, this comparison motivates an unbalanced formulation that accommodates unequal mass\([Piccoli and Rossi, 2014](https://arxiv.org/html/2610.22727#bib.bib25)\)\. Debiased unbalanced Sinkhorn divergences provide differentiable losses for this setting\([Séjourné et al\., 2019](https://arxiv.org/html/2610.22727#bib.bib29)\)\. We incorporate such a divergence as an auxiliary objective alongside the diffusion denoising loss, making training sensitive to both price displacement and volume mismatch\.

#### Market realism and predictive\-distribution evaluation\.

Market simulators are commonly evaluated by their ability to reproduce stylized facts of returns and order\-book dynamics\([Cont, 2001](https://arxiv.org/html/2610.22727#bib.bib6);[Vyetrenko et al\., 2020](https://arxiv.org/html/2610.22727#bib.bib33)\)\. LOB\-Bench formalizes this approach throughL1L\_\{1\}and Wasserstein\-1 comparisons of conditional and unconditional market\-statistic distributions, complemented by a learned discriminator and event\-response diagnostics\([Nagy et al\., 2025](https://arxiv.org/html/2610.22727#bib.bib21)\)\. Agreement on selected scalar summaries alone does not determine the joint distribution of volume across prices\. We complement these diagnostics by comparing complete, unnormalized submission fields at matched forecast boundaries, with a transport\-plus\-volume cost that retains price geometry and total volume\. A low average cost across generated draws, however, can favor concentrated forecasts without correctly representing predictive uncertainty\. Proper scoring rules provide a principled basis for evaluating the predictive distribution\([Gneiting and Raftery, 2007](https://arxiv.org/html/2610.22727#bib.bib11)\), and fair ensemble estimators account for finite sampling\([Ferro, 2014](https://arxiv.org/html/2610.22727#bib.bib7)\)\. We therefore pair the field cost with transport and joint\-field energy scores, supplemented by the continuous ranked probability score \(CRPS\) and rank diagnostics for total volume\. Section[5](https://arxiv.org/html/2610.22727#S5)specifies the support and propriety conditions for each score\.

#### Reward\-based diffusion fine\-tuning\.

Denoising diffusion policy optimization \(DDPO\) treats iterative denoising as a sequential decision process, enabling policy\-gradient updates from rewards assigned to generated samples\([Black et al\., 2024](https://arxiv.org/html/2610.22727#bib.bib3)\)\. Group Relative Policy Optimization \(GRPO\), introduced for mathematical reasoning in language models, estimates advantages from rewards normalized within groups of outputs sampled under the same condition\([Shao et al\., 2024](https://arxiv.org/html/2610.22727#bib.bib30)\)\. DanceGRPO applies this approach to diffusion and rectified\-flow models for image and video generation\([Xue et al\., 2025](https://arxiv.org/html/2610.22727#bib.bib37)\)\. We apply group\-relative fine\-tuning to conditional order\-flow generation, using a reward that combines price displacement, volume error, and diversity among generated fields\. Our evaluation examines whether improvements under the training reward extend to distributional accuracy and volume calibration\.

## 3\.Problem Formulation and Data Representation

We study the conditional distribution of new limit\-order volume submitted during the next time window\. The prediction target is a pair of volume\-at\-price fields, one for each side of the book, conditioned on an observed book snapshot and recent order flow\.

### 3\.1\.Book state and submission fields

The book state records resting liquidity at an instant\. Letθ\>0\\theta\>0be the tick size ands∈\{\+1,−1\}s\\in\\\{\+1,\-1\\\}index bids and asks, respectively\. We writeXt​\(p,s\)X\_\{t\}\(p,s\)for the number of shares resting at pricep∈θ​ℤp\\in\\theta\\mathbb\{Z\}on sidessimmediately before timett, reconstructed from the preceding message history\. Its best bidbtb\_\{t\}and best askata\_\{t\}define the mid\-pricemt=\(bt\+at\)/2m\_\{t\}=\(b\_\{t\}\+a\_\{t\}\)/2\.

The submission field records new limit\-order volume accumulated over an interval\. For each new submissionii, lettit\_\{i\},pip\_\{i\},sis\_\{i\}, andqiq\_\{i\}denote its arrival time, price, side, and volume in shares\. For an intervalWW, define

\(1\)AsW\(p\)=∑i:ti∈Wqi1\{si=s,pi=p\},A^\{W\}\_\{s\}\(p\)=\\sum\_\{i:\\,t\_\{i\}\\in W\}q\_\{i\}\\,\\mathbf\{1\}\\\{s\_\{i\}=s,\\ p\_\{i\}=p\\\},soAW=\(A\+1W,A−1W\)A^\{W\}=\(A^\{W\}\_\{\+1\},A^\{W\}\_\{\-1\}\)is a pair of non\-negative, unnormalized measures on the price grid\. The side massMs​\(AW\)=∑pAsW​\(p\)M\_\{s\}\(A^\{W\}\)=\\sum\_\{p\}A^\{W\}\_\{s\}\(p\)is the total submitted volume on sidess\. Retaining volumes in shares preserves both price placement and the amount of liquidity submitted\.

Each submission contributes its volume even if it is later canceled or executed\. Aggregation discards order identities and within\-window event order\. Bid and ask supports may overlap: as quotes move, a price can receive submissions on different sides at different times\. Cancellation and execution flows are outside the target, so a generated submission field alone does not determine the next book state\.

### 3\.2\.Conditional forecasting task

At each forecast boundaryτk\\tau\_\{k\}, the observed field summarizes submissions in the preceding window, and the target covers the following window\. For a common durationΔ​w\\Delta w, these fields are

\(2\)Ak−=A\[τk−Δw,τk\),Ak\+=A\[τk,τk\+Δw\)\.A\_\{k\}^\{\-\}=A^\{\[\\tau\_\{k\}\-\\Delta w,\\,\\tau\_\{k\}\)\},\\qquad A\_\{k\}^\{\+\}=A^\{\[\\tau\_\{k\},\\,\\tau\_\{k\}\+\\Delta w\)\}\.We useΔ​w=300\\Delta w=300seconds \(five minutes\) in our experiments\. Figure[1](https://arxiv.org/html/2610.22727#acmlabel1)showsXkX\_\{k\},Ak−A\_\{k\}^\{\-\}andAk\+A\_\{k\}^\{\+\}at one boundary\.

![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.](https://arxiv.org/html/2610.22727v1/figures/data_fields_moderate_MSFT_20260610_1150.png)Figure 1\.The data at one forecast boundary, within300300ticks of the mid\-price: the book snapshotXkX\_\{k\}, the observed fieldAk−A\_\{k\}^\{\-\}, and the target fieldAk\+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 holds53%53\\%of the submitted volume and1\.2%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\.In addition toAk−A\_\{k\}^\{\-\}, the conditioning information comprises the book snapshotXk≡XτkX\_\{k\}\\equiv X\_\{\\tau\_\{k\}\}, summary features𝐡k\\mathbf\{h\}\_\{k\}for its bid–ask spread and the time of session, and the instrument identitysymk\\mathrm\{sym\}\_\{k\}\. BothXkX\_\{k\}andAk−A\_\{k\}^\{\-\}use messages strictly beforeτk\\tau\_\{k\}\. A new submission at exactlyτk\\tau\_\{k\}contributes only to the target fieldAk\+A\_\{k\}^\{\+\}\(Figure[2](https://arxiv.org/html/2610.22727#acmlabel2)\)\.

Observed flowAk\-Future flowAk\+\-τk⁢Δwτk\+τk⁢ΔwSnapshot : events beforeXkτkFigure 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\.The forecasting task is to learn the conditional distribution

\(3\)p\(Ak\+\|Xk,Ak−,𝐡k,symk,Δw\)\.p\\\!\\left\(A\_\{k\}^\{\+\}\\,\\middle\|\\,X\_\{k\},A\_\{k\}^\{\-\},\\mathbf\{h\}\_\{k\},\\mathrm\{sym\}\_\{k\},\\Delta w\\right\)\.This distribution jointly describes submitted volume across prices and both sides of the book\. For each condition, a generative model producesRRsampled fields representing possible future outcomes\. We compare this ensemble with the realized fieldAk\+A\_\{k\}^\{\+\}using the evaluation measures in Section[5](https://arxiv.org/html/2610.22727#S5)\.

### 3\.3\.Mid\-relative representation and preprocessing

We represent the book, observed flow, and future submission field on a common price grid anchored to the observed snapshot\. Letqk=⌊mτk/θ⌋q\_\{k\}=\\lfloor m\_\{\\tau\_\{k\}\}/\\theta\\rfloorbe the snapshot mid\-price rounded down to an integer tick\. The grid containsB=2​K\+1B=2K\+1bins, with

\(4\)pk,b=θ\(qk\+b−K\),b=0,…,B−1\.p\_\{k,b\}=\\theta\(q\_\{k\}\+b\-K\),\\qquad b=0,\\ldots,B\-1\.Thus each bin refers to the same price across all fields in a prediction example\. We useK=5000K=5000, givingB=10001B=10001bins at one\-tick spacing\. The model predicts within this finite grid\. Volume outside the grid is recorded for coverage diagnostics, and evaluation retains the full support of the realized target\.

For numerical conditioning, we apply a logarithmic transform followed by standardization:

\(5\)T⁡\(v\)=log⁡\(1\+v\)−μσ~,σ~=max⁡\{σ,10−6\}\.T\(v\)=\\frac\{\\log\(1\+v\)\-\\mu\}\{\\tilde\{\\sigma\}\},\\qquad\\tilde\{\\sigma\}=\\max\\\{\\sigma,10^\{\-6\}\\\}\.The meanμ\\muand standard deviationσ\\sigmaare estimated separately for bid and ask book depth and submission flow, using training cells including zeros\. Generated values are decoded through the inverse transform and clipped to non\-negative share volumes before evaluation\. No unit\-mass normalization is applied, so total submitted volume remains part of the prediction\.

On this grid, the transformed targetY=T⁡\(Ak\+\)∈ℝ2×BY=T\(A\_\{k\}^\{\+\}\)\\in\\mathbb\{R\}^\{2\\times B\}contains bid and ask submission flow\. The conditioning tensorCk∈ℝ6×BC\_\{k\}\\in\\mathbb\{R\}^\{6\\times B\}combines transformed bid and ask book depth, two indicator channels marking the best quotes, and transformed bid and ask observed flow\. The summary vector𝐡k\\mathbf\{h\}\_\{k\}contains the spread in ticks and sine–cosine features for the time of session, standardized using training statistics\. Instrument identity and the window duration provide additional global conditioning inputs\.

Data coverage, chronological splits, and the evaluation protocol are given in Section[6](https://arxiv.org/html/2610.22727#S6)\.

## 4\.Conditional Order\-Flow Model

OrderFlowDiT generates the next\-window submission field conditioned on the current book and recent flow, using the representation in Section[3\.3](https://arxiv.org/html/2610.22727#S3.SS3)\. Its MSE and UOT variants share a conditional Diffusion Transformer; the UOT variant adds a transport objective on decoded fields\. We first define the diffusion model and its MSE loss, then introduce the transport objective, sampling, and additional GRPO fine\-tuning\. Figure[3](https://arxiv.org/html/2610.22727#acmlabel3)illustrates the Base configuration used in the primary comparison\.

bidZnZ\_\{n\}askZnZ\_\{n\}bidXkX\_\{k\}askXkX\_\{k\}bid quoteask quotebidAk−A\_\{k\}^\{\-\}askAk−A\_\{k\}^\{\-\}qkq\_\{k\}= snapshot mid bin ;B=10001B=10001bins, one tick wideZnZ\_\{n\}\(2×B2\\times B\) andCkC\_\{k\}\(6×B6\\times B\) share this mid\-anchored gridpatch ofP=16P\{=\}16binsEtgtE\_\{\\mathrm\{tgt\}\}EcondE\_\{\\mathrm\{cond\}\}\+\+\+PE\+\\,\\mathrm\{PE\}each patch⋯\\cdotsL=626L=626tokens,d=512d=512DiT block,×12\\times\\,12self\-attention\+\+MLPover theLLprice tokenslinear headunpatchifyvelocityvφv\_\{\\varphi\}global conditioning vectorstepnn⋅\\cdot𝐡k\\mathbf\{h\}\_\{k\}\(spread, time of session\)⋅\\cdotsymk\\mathrm\{sym\}\_\{k\}⋅\\cdotlog⁡Δ​w\\log\\Delta w⋅\\cdotmean cond\. tokenAdaLN\-Zeroshifts, scales, gatestraining objectivesℒMSE\\mathcal\{L\}\_\{\\mathrm\{MSE\}\}againstvvmasked valid binsY^=αn​Zn−σn​vφ\\hat\{Y\}=\\alpha\_\{n\}Z\_\{n\}\-\\sigma\_\{n\}v\_\{\\varphi\}A^=T−1​\(Y^\)\\hat\{A\}=T^\{\-1\}\(\\hat\{Y\}\)in sharesλUOT​αn2​S¯​\(A^s,Ak,s\+\)\\lambda\_\{\\mathrm\{UOT\}\}\\alpha\_\{n\}^\{2\}\\,\\bar\{S\}\(\\hat\{A\}\_\{s\},A^\{\+\}\_\{k,s\}\)unbalanced transportrealizedAk\+A\_\{k\}^\{\+\}Base\-UOT onlysampling:5050DDIM steps \(η=0\\eta=0\); independent initial noise per draw gives theRR\-member ensembleFigure 3\.OrderFlowDiT\-Base\. Every field shares the one\-tick, mid\-anchored price grid of Section[3\.3](https://arxiv.org/html/2610.22727#S3.SS3), so a bin refers to the same price in the noisy targetZnZ\_\{n\}and in all six conditioning channels ofCkC\_\{k\}: book depthXkX\_\{k\}, the two best\-quote indicators, and observed flowAk−A\_\{k\}^\{\-\}\. Patches ofP=16P=16bins 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 fieldA^\\hat\{A\}in shares rather than in velocity space\. GRPO \(Section[4\.4](https://arxiv.org/html/2610.22727#S4.SS4)\) 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\.### 4\.1\.Conditional Diffusion Model

WithY=T⁡\(Ak\+\)Y=T\(A\_\{k\}^\{\+\}\)and fixed conditioningck=\(Ck,𝐡k,symk,Δ​w\)c\_\{k\}=\(C\_\{k\},\\mathbf\{h\}\_\{k\},\\mathrm\{sym\}\_\{k\},\\Delta w\), training samplesnnuniformly from a10001000\-level cosine schedule and independent noiseε∼𝒩⁡\(0,I\)\\varepsilon\\sim\\mathcal\{N\}\(0,I\)to form

\(6\)Zn=αn​Y\+σn​ε,v=αn​ε−σn​Y,Z\_\{n\}=\\alpha\_\{n\}Y\+\\sigma\_\{n\}\\varepsilon,\\qquad v=\\alpha\_\{n\}\\varepsilon\-\\sigma\_\{n\}Y,whereαn2\+σn2=1\\alpha\_\{n\}^\{2\}\+\\sigma\_\{n\}^\{2\}=1\.

Our price\-axis DiT\([Peebles and Xie, 2023](https://arxiv.org/html/2610.22727#bib.bib23)\)sums separate linear embeddings of aligned, non\-overlappingPP\-bin patches fromZnZ\_\{n\}andCkC\_\{k\}with fixed sinusoidal positional encodings \(Figure[3](https://arxiv.org/html/2610.22727#acmlabel3)\)\. Bid and ask enter jointly as channels; padding is masked\. AdaLN\-Zero uses summed embeddings ofnn,𝐡k\\mathbf\{h\}\_\{k\},symk\\mathrm\{sym\}\_\{k\},log⁡Δ​w\\log\\Delta w, and the mean valid conditioning token\. A linear head predicts the velocity fieldvφ​\(Zn,n,ck\)v\_\{\\varphi\}\(Z\_\{n\},n,c\_\{k\}\)\.

MSE training minimizes the expected masked velocity loss

\(7\)ℒMSE=∑s,bmb​\[vφ​\(Zn,n,ck\)−v\]s,b22​∑bmb,\\mathcal\{L\}\_\{\\mathrm\{MSE\}\}=\\frac\{\\sum\_\{s,b\}m\_\{b\}\\,\[v\_\{\\varphi\}\(Z\_\{n\},n,c\_\{k\}\)\-v\]\_\{s,b\}^\{\\,2\}\}\{2\\sum\_\{b\}m\_\{b\}\},wherembm\_\{b\}indicates valid price bins\. Architecture and training settings are given in Section[6\.3](https://arxiv.org/html/2610.22727#S6.SS3)\.

### 4\.2\.Training with an Unbalanced Transport Objective

The velocity MSE compares predictions coordinatewise\. To account explicitly for price displacement and unequal submitted volumes, we add an unbalanced optimal\-transport \(UOT\) objective on decoded fields\. For each noisy example, the implied clean prediction isY^=αn​Zn−σn​vφ​\(Zn,n,ck\)\\hat\{Y\}=\\alpha\_\{n\}Z\_\{n\}\-\\sigma\_\{n\}v\_\{\\varphi\}\(Z\_\{n\},n,c\_\{k\}\)\. We decode it toA^=T−1​\(Y^\)\\hat\{A\}=T^\{\-1\}\(\\hat\{Y\}\)in shares using the decoder in Section[4\.3](https://arxiv.org/html/2610.22727#S4.SS3)and compare each side with the realized in\-grid field\.

For non\-negative volume vectorsu,vu,v, define entropic UOT with relaxed marginals:

\(8\)𝒰ϵ,ρ\(u,v\)=minπ≥0\{\\displaystyle\\mathcal\{U\}\_\{\\epsilon,\\rho\}\(u,v\)=\\min\_\{\\pi\\geq 0\}\\ \\big\\\{⟨C,π⟩\+ϵKL\(π∥u⊗v\)\\displaystyle\\langle C,\\pi\\rangle\+\\epsilon\\,\\mathrm\{KL\}\(\\pi\\\|u\\otimes v\)\+ρKL\(π𝟏∥u\)\+ρKL\(π⊤𝟏∥v\)\}\.\\displaystyle\+\\rho\\,\\mathrm\{KL\}\(\\pi\\mathbf\{1\}\\\|u\)\+\\rho\\,\\mathrm\{KL\}\(\\pi^\{\\top\}\\mathbf\{1\}\\\|v\)\\big\\\}\.The ground costCi​j=\|i−j\|C\_\{ij\}=\|i\-j\|measures displacement in ticks\. The marginal penalties allow the transport plan to accommodate unequal side volumes;KL\\mathrm\{KL\}denotes generalized KL divergence, including its linear mass terms\. We use entropic regularizationϵ=4\\epsilon=4and marginal penaltyρ=20\\rho=20\. Following Séjourné et al\.\([Séjourné et al\., 2019](https://arxiv.org/html/2610.22727#bib.bib29)\), we remove entropic self\-bias:

\(9\)S⁡\(u,v\)=\\displaystyle S\(u,v\)=\{\}𝒰⁡\(u,v\)−12​𝒰​\(u,u\)−12​𝒰​\(v,v\)\\displaystyle\\mathcal\{U\}\(u,v\)\-\\tfrac\{1\}\{2\}\\mathcal\{U\}\(u,u\)\-\\tfrac\{1\}\{2\}\\mathcal\{U\}\(v,v\)\+ϵ2​\[M⁡\(u\)−M⁡\(v\)\]2,\\displaystyle\+\\tfrac\{\\epsilon\}\{2\}\\,\[M\(u\)\-M\(v\)\]^\{2\},whereM⁡\(u\)=∑bubM\(u\)=\\sum\_\{b\}u\_\{b\}and𝒰\\mathcal\{U\}abbreviates𝒰ϵ,ρ\\mathcal\{U\}\_\{\\epsilon,\\rho\}\. To reduce variation in loss scale across windows with different volumes, we divide by the pair’s mean mass:S¯​\(u,v\)=S⁡\(u,v\)/max⁡\{\[M⁡\(u\)\+M⁡\(v\)\]/2,1\}\\bar\{S\}\(u,v\)=S\(u,v\)/\\max\\\{\[M\(u\)\+M\(v\)\]/2,1\\\}\. This scales the scalar loss while retaining both input fields in shares\. The combined training objective is

\(10\)ℒ=𝔼⁡\[ℒMSE\+λUOT​αn22​∑sS¯​\(A^s,Ak,s\+\)\],\\mathcal\{L\}=\\mathbb\{E\}\\\!\\left\[\\mathcal\{L\}\_\{\\mathrm\{MSE\}\}\+\\frac\{\\lambda\_\{\\mathrm\{UOT\}\}\\alpha\_\{n\}^\{2\}\}\{2\}\\sum\_\{s\}\\bar\{S\}\(\\hat\{A\}\_\{s\},A^\{\+\}\_\{k,s\}\)\\right\],with expectation over training examples, noise levels, and noise realizations\. The UOT variant usesλUOT=0\.1\\lambda\_\{\\mathrm\{UOT\}\}=0\.1; setting it to zero recovers MSE\-only training\. The factorαn2\\alpha\_\{n\}^\{2\}reduces the auxiliary contribution at high noise levels, where clean\-field estimates are less reliable\.

On the uniform grid, Sinkhorn kernel products use FFT convolution, with detached dual potentials obtained from log\-domain iterations\. Evaluation uses the distinct field distances and scoring rules in Section[5](https://arxiv.org/html/2610.22727#S5)\. The Base variants share a training split and optimizer schedule; budgets and variant\-specific checkpoint\-selection rules are given in Section[6\.3](https://arxiv.org/html/2610.22727#S6.SS3)\.

### 4\.3\.Sampling and Predictive Ensembles

MSE and UOT ensembles useRRindependent Gaussian initializations, each sampled with5050DDIM steps\([Song et al\., 2021](https://arxiv.org/html/2610.22727#bib.bib31)\)\(η=0\\eta=0\), fixed conditioning, and exponential moving average \(EMA\) weights\. Final valuesyydecode toT−1​\(y\)=max⁡\{exp⁡\(z\)−1,0\}T^\{\-1\}\(y\)=\\max\\\{\\exp\(z\)\-1,0\\\}withz=min⁡\{σ~​y\+μ,32\}z=\\min\\\{\\tilde\{\\sigma\}y\+\\mu,32\\\}\. Fields retain non\-negative, real\-valued share volumes without unit\-mass normalization and map back to absolute price ticks; non\-positive\-price bins are excluded and their mass recorded\.

### 4\.4\.GRPO Fine\-Tuning

We further adapt Base\-UOT with GRPO\([Shao et al\., 2024](https://arxiv.org/html/2610.22727#bib.bib30)\), optimizing a reward on groups of generated fields while retaining the architecture and representation\. This*Base\-GRPO*variant treats stochastic DDIM with2020steps andη=1\\eta=1as a policy, drawingG=8G=8decoded fieldsUgU\_\{g\}per condition\. For the realized fieldAk\+A\_\{k\}^\{\+\}, define the target\-dependent cost

\(11\)ℓk​\(U,V\)=12​∑s\[ws​\(U,V\)κ\+βmass​\|Ms​\(U\)−Ms​\(V\)\|Ms​\(Ak\+\)\],\\ell\_\{k\}\(U,V\)=\\frac\{1\}\{2\}\\sum\_\{s\}\\left\[\\frac\{w\_\{s\}\(U,V\)\}\{\\kappa\}\+\\beta\_\{\\mathrm\{mass\}\}\\frac\{\|M\_\{s\}\(U\)\-M\_\{s\}\(V\)\|\}\{M\_\{s\}\(A\_\{k\}^\{\+\}\)\}\\right\],wherewsw\_\{s\}is normalized\-shapeW1W\_\{1\}in ticks \(Section[5\.1](https://arxiv.org/html/2610.22727#S5.SS1)\),κ=50\\kappa=50, andβmass=1\\beta\_\{\\mathrm\{mass\}\}=1\. For the reward, shape cost is zero when both sides are empty andB/2B/2ticks when exactly one is empty; the mass term is omitted whenMs​\(Ak\+\)=0M\_\{s\}\(A\_\{k\}^\{\+\}\)=0\. Each draw receives reward

\(12\)rg=−ℓk​\(Ug,Ak\+\)\+12​\(G−1\)​∑g′≠gℓk​\(Ug,Ug′\)\.r\_\{g\}=\-\\ell\_\{k\}\(U\_\{g\},A\_\{k\}^\{\+\}\)\+\\frac\{1\}\{2\(G\-1\)\}\\sum\_\{g^\{\\prime\}\\neq g\}\\ell\_\{k\}\(U\_\{g\},U\_\{g^\{\\prime\}\}\)\.The first term rewards agreement with the realization; the second encourages separation between draws\. Group\-standardized rewards define the advantages\. Updates use a clipped policy\-ratio surrogate and a KL penalty to the frozen Base\-UOT model, with log\-ratios and KL terms normalized by the number of valid scalar entries\.

## 5\.Full\-Field and Distributional Evaluation

We evaluate both the accuracy of individual submission fields and the quality of their predictive distribution\. At forecast boundaryτk\\tau\_\{k\}, independent model drawsA^k\(1\),…,A^k\(R\)\\widehat\{A\}\_\{k\}^\{\(1\)\},\\ldots,\\widehat\{A\}\_\{k\}^\{\(R\)\}are compared with the realizationAk\+A\_\{k\}^\{\+\}\. These paired field comparisons complement summary\-statistic benchmarks such as LOB\-Bench\([Nagy et al\., 2025](https://arxiv.org/html/2610.22727#bib.bib21)\)\. All comparisons retain share volumes and the full supplied price support, including realized flow outside the model’s grid\. A common price translation leaves the scores unchanged, so absolute ticks and the shared mid\-relative frame give identical values\. Table[1](https://arxiv.org/html/2610.22727#S5.T1)summarizes the eight quantities; sampling and reporting follow Section[6\.4](https://arxiv.org/html/2610.22727#S6.SS4)\.

### 5\.1\.Full\-Field Accuracy

For generic bid–ask fieldsU,VU,V, writej=p/θ∈ℤj=p/\\theta\\in\\mathbb\{Z\}for the price tick andMs​\(U\)=∑jUs​\(j\)M\_\{s\}\(U\)=\\sum\_\{j\}U\_\{s\}\(j\)for side mass\. When this mass is positive, letFU,sF\_\{U,s\}be the CDF ofUs/Ms​\(U\)U\_\{s\}/M\_\{s\}\(U\)\. The one\-dimensional Wasserstein shape discrepancy is

\(13\)ws​\(U,V\)=∑j∈ℤ\|FU,s​\(j\)−FV,s​\(j\)\|\.w\_\{s\}\(U,V\)=\\sum\_\{j\\in\\mathbb\{Z\}\}\\big\|F\_\{U,s\}\(j\)\-F\_\{V,s\}\(j\)\\big\|\.This exact CDF expression measures price displacement in ticks without a transport solver\([Santambrogio, 2015](https://arxiv.org/html/2610.22727#bib.bib27)\)\. Normalization removes total volume, so fields with the same shape have zero discrepancy even when their masses differ\.

To retain both placement and volume, setms=Ms​\(U\)m\_\{s\}=M\_\{s\}\(U\)andns=Ms​\(V\)n\_\{s\}=M\_\{s\}\(V\)and define

\(14\)dfull,s​\(U,V\)\\displaystyle d\_\{\\mathrm\{full\},s\}\(U,V\)=min⁡\(ms,ns\)​ws​\(U,V\)\+λ​\|ms−ns\|,\\displaystyle=\\min\(m\_\{s\},n\_\{s\}\)\\,w\_\{s\}\(U,V\)\+\\lambda\|m\_\{s\}\-n\_\{s\}\|,\(15\)dfull​\(U,V\)\\displaystyle d\_\{\\mathrm\{full\}\}\(U,V\)=∑sdfull,s​\(U,V\)\.\\displaystyle=\\sum\_\{s\}d\_\{\\mathrm\{full\},s\}\(U,V\)\.The first term transports the common mass between the normalized shapes; the second charges for residual volume\. Withλ=50\\lambda=50ticks, the cost has units of share⋅\\cdotticks\. An equal\-mass displacement byδ\\deltaticks costsms​\|δ\|m\_\{s\}\|\\delta\|; a volume change at fixed shape incurs the mass penalty\. If either mass is zero, the transport term is defined as zero, leavingdfull,s=λ​\|ms−ns\|d\_\{\\mathrm\{full\},s\}=\\lambda\|m\_\{s\}\-n\_\{s\}\|\.

This feasible plan leaves total marginal defect\|ms−ns\|\|m\_\{s\}\-n\_\{s\}\|, so its cost upper\-bounds unbalanced OT with aλ\\lambda\-weightedL1L^\{1\}penalty on each marginal discrepancy\([Piccoli and Rossi, 2014](https://arxiv.org/html/2610.22727#bib.bib25)\)\. It need not attain the optimal transport value or satisfy the triangle inequality\. It also differs from the entropic, KL\-relaxed training objective in \([8](https://arxiv.org/html/2610.22727#S4.E8)\)\.

#### Error components\.

Alongside the combined cost, we report sidedW1W\_\{1\}, target\-normalized volume errorNTVEs​\(U,V\)=\|ms−ns\|/ns\\mathrm\{NTVE\}\_\{s\}\(U,V\)=\|m\_\{s\}\-n\_\{s\}\|/n\_\{s\}for a generatedUUand realizationVV, and log\-field errorMSElog​\(U,V\)=ρ2​\(U,V\)2\\mathrm\{MSE\}\_\{\\log\}\(U,V\)=\\rho\_\{2\}\(U,V\)^\{2\}, withρ2\\rho\_\{2\}defined in \([18](https://arxiv.org/html/2610.22727#S5.E18)\)\. SidedW1W\_\{1\}and NTVE are zero when a side is empty in both fields; sidedW1W\_\{1\}is undefined when exactly one is empty and NTVE when the target alone is empty, whereasdfulld\_\{\\mathrm\{full\}\}andMSElog\\mathrm\{MSE\}\_\{\\log\}remain defined for empty fields\.

### 5\.2\.Predictive Distribution Quality

Averaging field costs over draws measures sample fidelity but does not identify the predictive law: concentrated forecasts can achieve small average errors\. Proper scoring rules instead attain their minimum expected value at the true conditional law; strict propriety makes that minimizer unique\([Gneiting and Raftery, 2007](https://arxiv.org/html/2610.22727#bib.bib11)\)\.

For a fixed condition, writeU\(r\)=A^k\(r\)U^\{\(r\)\}=\\widehat\{A\}\_\{k\}^\{\(r\)\}andV=Ak\+V=A\_\{k\}^\{\+\}\. Given a ground distanceρ\\rhoandR≥2R\\geq 2independent draws, the fair energy\-score estimator is\([Ferro, 2014](https://arxiv.org/html/2610.22727#bib.bib7)\)

ES^ρ\\displaystyle\\widehat\{\\mathrm\{ES\}\}\_\{\\rho\}=1R​∑r=1Rρ⁡\(U\(r\),V\)\\displaystyle=\\frac\{1\}\{R\}\\sum\_\{r=1\}^\{R\}\\rho\(U^\{\(r\)\},V\)\(16\)−12​R​\(R−1\)∑r≠r′ρ\(U\(r\),U\(r′\)\)\.\\displaystyle\\quad\-\\frac\{1\}\{2R\(R\-1\)\}\\sum\_\{r\\neq r^\{\\prime\}\}\\rho\(U^\{\(r\)\},U^\{\(r^\{\\prime\}\)\}\)\.The pairwise sum uses ordered pairs; independence makes the estimator unbiased for the population score\. We retain the mean target distanceTρT\_\{\\rho\}and mean distance between distinct drawsDρD\_\{\\rho\}, soES^ρ=Tρ−Dρ/2\\widehat\{\\mathrm\{ES\}\}\_\{\\rho\}=T\_\{\\rho\}\-D\_\{\\rho\}/2\.

#### Transport energy score\.

To combine price geometry with volume uncertainty, setgs​\(U\)=log⁡\(1\+Ms​\(U\)\)g\_\{s\}\(U\)=\\log\(1\+M\_\{s\}\(U\)\)and use

\(17\)ρT​\(U,V\)=12​∑s\[ws​\(U,V\)κ\+β​\|gs​\(U\)−gs​\(V\)\|\],\\rho\_\{T\}\(U,V\)=\\frac\{1\}\{2\}\\sum\_\{s\}\\left\[\\frac\{w\_\{s\}\(U,V\)\}\{\\kappa\}\+\\beta\\big\|g\_\{s\}\(U\)\-g\_\{s\}\(V\)\\big\|\\right\],withκ=50\\kappa=50ticks andβ=1\\beta=1\. The symmetric log\-mass term reduces the dominance of large volumes without exact scale invariance\. On laws with nonempty sides and finite first metric moments,EST\\mathrm\{ES\}\_\{T\}is proper becauseρT\\rho\_\{T\}is a sum ofL1L^\{1\}distances and hence of negative type\([Gneiting and Raftery, 2007](https://arxiv.org/html/2610.22727#bib.bib11);[Székely and Rizzo, 2013](https://arxiv.org/html/2610.22727#bib.bib32)\)\. Its additive side, shape, and mass terms do not identify their dependence, so it is not strictly proper for the joint field law\. The GRPO reward in \([12](https://arxiv.org/html/2610.22727#S4.E12)\) uses a different mass discrepancy and does not inherit this propriety claim\.

For each side, the target and all draws must be either all nonempty or all empty; the latter receives zero score\. Both side scores must be defined\. Mixed empty/nonempty ensembles are excluded, and the propriety statement does not extend to laws mixing empty and nonempty fields\.

#### Joint\-field energy score\.

To assess the joint bid–ask law, definezs​\(U\)​\(j\)=log⁡\(1\+Us​\(j\)\)z\_\{s\}\(U\)\(j\)=\\log\(1\+U\_\{s\}\(j\)\)and

\(18\)ρ2​\(U,V\)=\[12​B​∑s∑j∈ℤ\(zs​\(U\)​\(j\)−zs​\(V\)​\(j\)\)2\]1/2\.\\rho\_\{2\}\(U,V\)=\\left\[\\frac\{1\}\{2B\}\\sum\_\{s\}\\sum\_\{j\\in\\mathbb\{Z\}\}\\big\(z\_\{s\}\(U\)\(j\)\-z\_\{s\}\(V\)\(j\)\\big\)^\{2\}\\right\]^\{1/2\}\.HereB=10001B=10001fixes normalization without truncating support, and the log transform uses raw shares without model\-specific standardization\. The injective log\-field map lies in a separable Hilbert space, whose norm distance has strong negative type\([Lyons, 2013](https://arxiv.org/html/2610.22727#bib.bib18)\)\. Thus the resultingES2\\mathrm\{ES\}\_\{2\}is strictly proper for the joint field law under finite first metric moments, including laws with empty fields\. Its binwise geometry complements transport: the penalty for moving an isolated spike to another bin does not increase with the distance moved\.

#### Volume CRPS\.

Applying \([16](https://arxiv.org/html/2610.22727#S5.Ex1)\) to\|gs​\(U\)−gs​\(V\)\|\|g\_\{s\}\(U\)\-g\_\{s\}\(V\)\|gives the fair continuous ranked probability score \(CRPS\) for log side volume\([Matheson and Winkler, 1976](https://arxiv.org/html/2610.22727#bib.bib19)\); we average the two sides\. Each score is strictly proper for its log\-volume marginal under a finite first moment, including empty sides withgs=0g\_\{s\}=0\. Their average does not identify bid–ask dependence\.

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\.

## 6\.Experimental Setup

### 6\.1\.Data and Experimental Splits

We use LOBSTER full\-depth order\-message data for a fixed panel of3030U\.S\. equities\.111AAPL, AMGN, AMZN, AXP, BA, CAT, CRM, CSCO, CVX, DIS, GS, HD, HON, IBM, JNJ, JPM, KO, MCD, MMM, MRK, MSFT, NKE, NVDA, PG, SHW, TRV, UNH, V, VZ, and WMT\.Each session is restricted to 10:30–16:00 Eastern Time, with a common one\-cent price lattice\. We construct the contemporaneous\-snapshot \(lag0\) conditions of Section[3](https://arxiv.org/html/2610.22727#S3): the book snapshot and preceding five minutes of submission flow condition the forecast of bid and ask submission fields over the nextΔ​w=300\\Delta w=300seconds\.

The Base models train on1010trading days from June 8–22, 2026\. The Large models extend this split with1919earlier days from May 11–June 5, giving2929training days in total\. Training uses window\-grid offsets of00,6060,120120,180180, and240240seconds, yielding96,30096\{,\}300Base and279,270279\{,\}270Large training pairs\. Volume\-transform and summary\-feature statistics are fitted separately for each training split using only its unshifted windows\.

All models are compared on a fixed corpus from June 23, 2026 containing1,9501\{,\}950conditions, with6565per stock\. Evaluation uses the unshifted grid, with target windows starting at 10:35 through 15:55 Eastern Time\. This day also serves for checkpoint selection, so the reported results are validation\-set comparisons following chronological training\. June 24 is reserved in the configured test split and is not used in these comparisons\. The same3030stocks appear in training and evaluation; per\-stock results characterize performance variation across this observed panel\.

### 6\.2\.Experimental Comparisons and Baselines

The primary comparison uses three Base variants of the conditional diffusion model in Section[4\.1](https://arxiv.org/html/2610.22727#S4.SS1)\.*Base\-MSE*trains with the diffusion MSE in \([7](https://arxiv.org/html/2610.22727#S4.E7)\);*Base\-UOT*adds the transport auxiliary objective in \([10](https://arxiv.org/html/2610.22727#S4.E10)\)\. Throughout, UOT denotes this combined MSE\+UOT objective\.*Base\-GRPO*fine\-tunes Base\-UOT as described in Section[4\.4](https://arxiv.org/html/2610.22727#S4.SS4)\. These variants examine how the training objective and fine\-tuning procedure affect field accuracy and predictive uncertainty, alongside the statistical references below\.

An expanded configuration adds*Large\-MSE*and*Large\-UOT*to check whether the MSE/UOT findings persist with greater capacity, more data, and longer training \(Section[6\.3](https://arxiv.org/html/2610.22727#S6.SS3)\)\.

Two parametric baselines fit the observed submission fieldAk−A\_\{k\}^\{\-\}\.*Skew\-normal*fits one skew\-normal distribution per side, using volume weights within5050ticks of that side’s modal price\.*Skew\-normal mixture*fits the full observed support, with BIC selecting one or two components per side \(Kmax=2K\_\{\\max\}=2\) and observed share volume as the effective sample size\. Both fit separately at each forecast boundary and scale the fitted shape to the observed side’s total volume, yielding a deterministic forecast field\. They use recent flow without book\-depth conditioning; their comparison includes differences in fitting support and component limits\.

For a historical distributional reference,*Empirical*samples complete bid–ask fields from the same instrument and five\-minute window index on the Base configuration’s1010training days, using unshifted windows only\. Each selected field is translated from its own snapshot\-mid anchor to the current anchor\. Draws are uniform and independent with replacement\. This baseline uses instrument identity, time of day, and the snapshot anchor; it does not use recent flow or book depth\.

### 6\.3\.Training

#### Diffusion training\.

Base has58\.758\.7M parameters \(hidden widthd=512d=512,1212blocks,88heads\) and trains for30,00030\{,\}000steps at learning rate2×10−42\\times 10^\{\-4\}; Large uses174\.3174\.3M parameters \(d=768d=768,1616blocks,1212heads\) and100,000100\{,\}000steps at10−410^\{\-4\}\. Within each size, MSE and UOT share architecture, data, and optimizer settings\. Both use AdamW with constant learning rates, batch size6464, weight decay0\.010\.01, gradient\-norm clipping at11, mixed precision, and EMA decay0\.99990\.9999\. Both sizes use patch widthP=16P=16, feed\-forward width4​d4d, and dropout0\.10\.1; the grid, diffusion schedule, and UOT coefficients are also shared\. Each learned configuration uses one training seed,4242\.

#### GRPO fine\-tuning\.

Base\-GRPO starts from the selected Base\-UOT EMA weights and uses the same training split\. Each of500500iterations samplesG=8G=8fields for each of1616conditions, followed by one policy\-update epoch\. Updates use AdamW with learning rate10−510^\{\-5\}, zero weight decay, gradient\-norm clipping at11, mixed precision, and EMA decay0\.990\.99\. The policy\-ratio clipping threshold is0\.20\.2and the KL\-penalty coefficient is0\.020\.02\.

### 6\.4\.Evaluation and Reporting Protocol

All models are evaluated on the common lag0 corpus in Section[6\.1](https://arxiv.org/html/2610.22727#S6.SS1), with matched forecast boundaries and realized targets\. Their conditioning inputs follow Section[6\.2](https://arxiv.org/html/2610.22727#S6.SS2)\.

#### Forecast generation\.

Diffusion models generateR=4R=4independent fields per condition from selected EMA weights \(evaluation base seed00\)\. MSE/UOT use5050DDIM steps \(η=0\\eta=0\); Base\-GRPO uses its2020\-step training sampler \(η=1\\eta=1\)\. Empirical also draws four independent fields; each skew\-normal baseline is deterministic \(R=1R=1\)\. Base/Large share the5050\-step sampler, whereas the GRPO comparison combines fine\-tuning and a sampler change\.

#### Scoring and calibration\.

Fields in shares are scored against full\-support targets, including realized flow outside the model grid, using the eight quantities in Section[5](https://arxiv.org/html/2610.22727#S5)\. Field costs and errors average over defined draws; distributional scores use the ensemble jointly\. DeterministicR=1R=1forecasts have zero inter\-draw terms, leaving only target distance\. For GRPO, we retain both energy scores’ target and inter\-draw components\. Volume calibration comparesR=4R=4models using RI and interior\-rank coverage \(nominal0\.60\.6\)\. Ties are broken uniformly at random using reproducible sample\-based seeds\.

#### Aggregation and reporting\.

SidedW1W\_\{1\}and NTVE average defined draws within sides, then equally across sides, requiring both means\. Full\-field cost sums sides; log\-field MSE follows \([18](https://arxiv.org/html/2610.22727#S5.E18)\)\. Per\-stock and pooled means equally weight defined boundaries, with per\-metric valid and excluded counts; volume ranks pool boundary–side pairs\. Raw costs can be dominated by high\-volume windows and are interpreted alongside shape and relative\- or log\-volume metrics\. Comparisons are descriptive, using the checkpoint\-selection day and one training seed per learned configuration\.

## 7\.Results

We assess field accuracy and predictive\-distribution quality on the common1,9501\{,\}950lag0 conditions across3030stocks described in Section[6\.1](https://arxiv.org/html/2610.22727#S6.SS1)\. All field costs and distributional scores are defined on every condition for the eight selected models, with no exclusions\. Volume\-rank diagnostics use3,9003\{,\}900condition–side pairs per model\. As specified in Section[6\.4](https://arxiv.org/html/2610.22727#S6.SS4), these are descriptive comparisons on the checkpoint\-selection day, with one training seed per learned configuration\.

### 7\.1\.Main Comparison with Statistical Baselines

Table 2\.Main comparison on1,9501\{,\}950conditions\. All seven metrics are lower\-is\-better; bold marks the lowest displayed value in each column\. The full\-field cost is reported in10610^\{6\}share⋅\\cdotticks and sidedW1W\_\{1\}in ticks\. CRPS denotes log\-total\-volume CRPS\. The mixture usesKmax=2K\_\{\\max\}=2\. TheR=1R=1rows are point\-mass forecasts\.Base\-UOT improves both field accuracy and predictive\-distribution quality over Base\-MSE \(Table[2](https://arxiv.org/html/2610.22727#S7.T2)\)\. All seven reported metrics decrease, including a45\.8%45\.8\\%reduction in full\-field cost and a26\.1%26\.1\\%reduction in transport energy scoreEST\\mathrm\{ES\}\_\{T\}\. The gains also extend to the joint\-field scoreES2\\mathrm\{ES\}\_\{2\}, whose binwise geometry differs from transport, and to volume CRPS\.

The statistical baselines place these improvements in context\. Base\-UOT outperforms Empirical in full\-field cost and both energy scores, and has the lowestES2\\mathrm\{ES\}\_\{2\}among the six models in Table[2](https://arxiv.org/html/2610.22727#S7.T2)\. Other measures favor different forecasts: the skew\-normal baselines attain the lowest NTVE\. The learned forecasts’ advantages in field accuracy and distributional scores therefore coexist with room for improvement in marginal volume calibration\.

The UOT gains are consistent across the evaluated stock panel\. Relative to Base\-MSE, Base\-UOT lowers both full\-field cost andEST\\mathrm\{ES\}\_\{T\}for every stock\. It also lowers full\-field cost relative to Empirical on2929stocks andES2\\mathrm\{ES\}\_\{2\}on all3030\. Each comparison uses the same6565conditions per stock, showing that the pooled improvements are accompanied by gains across the instruments represented in training and evaluation\.

Figure[4](https://arxiv.org/html/2610.22727#acmlabel4)illustrates a generated field by placing one Large\-UOT draw beside the realization at the same condition\.

![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.](https://arxiv.org/html/2610.22727v1/figures/truth_vs_largeuot_AAPL_20260623_1230.png)Figure 4\.Realized and Large\-UOT submission fields within100100ticks 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 are1\.941\.94M and1\.491\.49M 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\.
### 7\.2\.GRPO: Accuracy, Diversity, and Calibration

Table 3\.GRPO comparison on the same lag0 conditions \(R=4R=4\)\. For each energy\-score geometryj∈\{T,2\}j\\in\\\{T,2\\\},TjT\_\{j\}is the mean target distance andDjD\_\{j\}the mean off\-diagonal inter\-draw distance, soESj=Tj−Dj/2\\mathrm\{ES\}\_\{j\}=T\_\{j\}\-D\_\{j\}/2\. Interior\-rank coverage has nominal value0\.6000\.600\. Distance components and coverage are diagnostics, without a common lower\-is\-better direction\.The GRPO variant improves transport and volume scores, but the gain does not extend to the joint\-field score\. Relative to Base\-UOT, Base\-GRPO lowersEST\\mathrm\{ES\}\_\{T\}by6\.1%6\.1\\%and volume CRPS by14\.0%14\.0\\%, while increasingES2\\mathrm\{ES\}\_\{2\}by5\.7%5\.7\\%and full\-field cost by3\.2%3\.2\\%\. The pooled full\-field cost increases despite decreases in2424of the3030per\-stock means: larger deteriorations elsewhere outweigh those gains\.

The energy\-score decomposition in Table[3](https://arxiv.org/html/2610.22727#S7.T3)explains the disagreement between the two scoring geometries\. Under both distances, generated fields become closer to the realization on average, but also closer to one another\. SinceESj=Tj−Dj/2\\mathrm\{ES\}\_\{j\}=T\_\{j\}\-D\_\{j\}/2, the transport score improves because the reduction in target distance exceeds half the reduction in inter\-draw distance\. For the joint\-field score, the smaller target\-distance improvement is outweighed by the loss of the dispersion contribution\. A lower average distance to the target therefore need not yield a better score for the predictive distribution\.

Volume\-rank diagnostics show a related limitation\. RI rises from0\.2560\.256to0\.2690\.269, while interior\-rank coverage falls from0\.3950\.395to0\.3850\.385, farther below its nominal value of0\.6000\.600\. Thus the improved transport score and volume CRPS coexist with reduced ensemble dispersion and weaker pooled volume calibration\. These results describe the combined change in weights and sampling: Base\-GRPO uses2020DDIM steps withη=1\\eta=1, whereas Base\-UOT uses5050steps withη=0\\eta=0\.

### 7\.3\.Expanded Training Configuration

The UOT advantage over MSE in full\-field cost and both energy scores persists in the expanded configuration\. Relative to Base\-UOT, Large\-UOT reduces full\-field cost by2\.1%2\.1\\%andEST\\mathrm\{ES\}\_\{T\}by17\.5%17\.5\\%\. Large\-MSE nevertheless retains lower NTVE and log\-field MSE than Large\-UOT\. Capacity, training data, optimizer steps, and learning rate change together, so this comparison does not isolate their individual effects\.

## 8\.Conclusion

We presented OrderFlowDiT for conditional generation of unnormalized limit\-order submission fields, together with an evaluation framework that assesses price placement, total volume, and predictive uncertainty\. On1,9501\{,\}950validation conditions across3030U\.S\. equities, Base\-UOT reduces full\-field transport cost by45\.8%45\.8\\%and the transport energy score by26\.1%26\.1\\%relative to Base\-MSE, with gains in both measures for every stock\. The UOT advantage persists when model capacity, training data, and optimization budget increase together\. The GRPO variant illustrates the need for complementary distributional diagnostics: its lower transport energy score coexists with reduced sample diversity, a worse joint\-field score, and weaker volume calibration\. These results support training and evaluation that account jointly for price geometry, submitted volume, and the distribution of possible outcomes\.

The central limitation is that generating submission flow does not complete a financial world model\. An impact model is also needed to connect the current book, order flow, and trading actions to the distribution of subsequent book states and prices\. Our aggregated fields omit cancellations, executions, and event ordering, so they cannot by themselves determine these transitions\. Learning and validating this link, including the market’s response to trading actions, is the next step toward multi\-step simulation and the evaluation of trading decisions\.

## References

- Biais et al\.\(1995\)Bruno Biais, Pierre Hillion, and Chester Spatt\. 1995\.An Empirical Analysis of the Limit Order Book and the Order Flow in the Paris Bourse\.*The Journal of Finance*50, 5 \(1995\), 1655–1689\.[doi:10\.1111/j\.1540\-6261\.1995\.tb05192\.x](https://doi.org/10.1111/j.1540-6261.1995.tb05192.x)
- Black et al\.\(2024\)Kevin Black, Michael Janner, Yilun Du, Ilya Kostrikov, and Sergey Levine\. 2024\.Training Diffusion Models with Reinforcement Learning\. In*The Twelfth International Conference on Learning Representations*\.[https://arxiv\.org/abs/2305\.13301](https://arxiv.org/abs/2305.13301)
- Byrd et al\.\(2020\)David Byrd, Maria Hybinette, and Tucker Hybinette Balch\. 2020\.ABIDES: Towards High\-Fidelity Multi\-Agent Market Simulation\. In*Proc\. 2020 ACM SIGSIM Conf\. on Principles of Advanced Discrete Simulation \(SIGSIM\-PADS\)*\. 11–22\.[doi:10\.1145/3384441\.3395986](https://doi.org/10.1145/3384441.3395986)
- Coletta et al\.\(2022\)Andrea Coletta, Aymeric Moulin, Svitlana Vyetrenko, and Tucker Balch\. 2022\.Learning to Simulate Realistic Limit Order Book Markets from Data as a World Agent\. In*Proc\. 3rd ACM Int\. Conf\. on AI in Finance \(ICAIF\)*\. 428–436\.[doi:10\.1145/3533271\.3561753](https://doi.org/10.1145/3533271.3561753)
- Cont \(2001\)Rama Cont\. 2001\.Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues\.*Quantitative Finance*1, 2 \(2001\), 223–236\.[doi:10\.1080/713665670](https://doi.org/10.1080/713665670)
- Ferro \(2014\)Christopher A\. T\. Ferro\. 2014\.Fair Scores for Ensemble Forecasts\.*Quarterly Journal of the Royal Meteorological Society*140, 683 \(2014\), 1917–1923\.[doi:10\.1002/qj\.2270](https://doi.org/10.1002/qj.2270)
- Feydy et al\.\(2019\)Jean Feydy, Thibault Séjourné, François\-Xavier Vialard, Shun\-ichi Amari, Alain Trouvé, and Gabriel Peyré\. 2019\.Interpolating between Optimal Transport and MMD using Sinkhorn Divergences\. In*Proceedings of the Twenty\-Second International Conference on Artificial Intelligence and Statistics**\(Proceedings of Machine Learning Research, Vol\. 89\)*\. PMLR, 2681–2690\.
- Genevay et al\.\(2018\)Aude Genevay, Gabriel Peyré, and Marco Cuturi\. 2018\.Learning Generative Models with Sinkhorn Divergences\. In*Proceedings of the Twenty\-First International Conference on Artificial Intelligence and Statistics**\(Proceedings of Machine Learning Research, Vol\. 84\)*\. PMLR, 1608–1617\.[https://proceedings\.mlr\.press/v84/genevay18a\.html](https://proceedings.mlr.press/v84/genevay18a.html)
- Gneiting et al\.\(2007\)Tilmann Gneiting, Fadoua Balabdaoui, and Adrian E\. Raftery\. 2007\.Probabilistic Forecasts, Calibration and Sharpness\.*Journal of the Royal Statistical Society: Series B \(Statistical Methodology\)*69, 2 \(2007\), 243–268\.[doi:10\.1111/j\.1467\-9868\.2007\.00587\.x](https://doi.org/10.1111/j.1467-9868.2007.00587.x)
- Gneiting and Raftery \(2007\)Tilmann Gneiting and Adrian E\. Raftery\. 2007\.Strictly Proper Scoring Rules, Prediction, and Estimation\.*J\. Amer\. Statist\. Assoc\.*102, 477 \(2007\), 359–378\.[doi:10\.1198/016214506000001437](https://doi.org/10.1198/016214506000001437)
- Hautsch and Huang \(2012\)Nikolaus Hautsch and Ruihong Huang\. 2012\.The Market Impact of a Limit Order\.*Journal of Economic Dynamics and Control*36, 4 \(2012\), 501–522\.[doi:10\.1016/j\.jedc\.2011\.09\.012](https://doi.org/10.1016/j.jedc.2011.09.012)
- Ho et al\.\(2020\)Jonathan Ho, Ajay Jain, and Pieter Abbeel\. 2020\.Denoising Diffusion Probabilistic Models\. In*Advances in Neural Information Processing Systems*, Vol\. 33\.
- Huang et al\.\(2015\)Weibing Huang, Charles\-Albert Lehalle, and Mathieu Rosenbaum\. 2015\.Simulating and Analyzing Order Book Data: The Queue\-Reactive Model\.*J\. Amer\. Statist\. Assoc\.*110, 509 \(2015\), 107–122\.[doi:10\.1080/01621459\.2014\.982278](https://doi.org/10.1080/01621459.2014.982278)
- Li et al\.\(2025\)Junjie Li, Yang Liu, Weiqing Liu, Shikai Fang, Lewen Wang, Chang Xu, and Jiang Bian\. 2025\.MarS: A Financial Market Simulation Engine Powered by Generative Foundation Model\. In*Int\. Conf\. on Learning Representations \(ICLR\)*\.[https://arxiv\.org/abs/2409\.07486](https://arxiv.org/abs/2409.07486)
- Li et al\.\(2020\)Junyi Li, Xintong Wang, Yaoyang Lin, Arunesh Sinha, and Michael P\. Wellman\. 2020\.Generating Realistic Stock Market Order Streams\.*Proceedings of the AAAI Conference on Artificial Intelligence*34, 1 \(2020\), 727–734\.[doi:10\.1609/aaai\.v34i01\.5415](https://doi.org/10.1609/aaai.v34i01.5415)
- Lo and Hall \(2015\)Danny K\. Lo and Anthony D\. Hall\. 2015\.Resiliency of the Limit Order Book\.*Journal of Economic Dynamics and Control*61 \(2015\), 222–244\.[doi:10\.1016/j\.jedc\.2015\.09\.012](https://doi.org/10.1016/j.jedc.2015.09.012)
- Lyons \(2013\)Russell Lyons\. 2013\.Distance Covariance in Metric Spaces\.*The Annals of Probability*41, 5 \(2013\), 3284–3305\.[doi:10\.1214/12\-AOP803](https://doi.org/10.1214/12-AOP803)
- Matheson and Winkler \(1976\)James E\. Matheson and Robert L\. Winkler\. 1976\.Scoring Rules for Continuous Probability Distributions\.*Management Science*22, 10 \(1976\), 1087–1096\.[doi:10\.1287/mnsc\.22\.10\.1087](https://doi.org/10.1287/mnsc.22.10.1087)
- Nagy et al\.\(2023\)Peer Nagy, Sascha Frey, Silvia Sapora, Kang Li, Anisoara Calinescu, Stefan Zohren, and Jakob Foerster\. 2023\.Generative AI for End\-to\-End Limit Order Book Modelling: A Token\-Level Autoregressive Generative Model of Message Flow Using a Deep State Space Network\. In*Proceedings of the Fourth ACM International Conference on AI in Finance**\(ICAIF ’23\)*\. Association for Computing Machinery, 91–99\.[doi:10\.1145/3604237\.3626898](https://doi.org/10.1145/3604237.3626898)
- Nagy et al\.\(2025\)Peer Nagy, Sascha Yves Frey, Kang Li, Bidipta Sarkar, Svitlana Vyetrenko, Stefan Zohren, Ani Calinescu, and Jakob Nicolaus Foerster\. 2025\.LOB\-Bench: Benchmarking Generative AI for Finance \- an Application to Limit Order Book Data\. In*Proceedings of the 42nd International Conference on Machine Learning**\(Proceedings of Machine Learning Research, Vol\. 267\)*\. PMLR, 45437–45460\.[https://proceedings\.mlr\.press/v267/nagy25a\.html](https://proceedings.mlr.press/v267/nagy25a.html)
- Obizhaeva and Wang \(2013\)Anna A\. Obizhaeva and Jiang Wang\. 2013\.Optimal Trading Strategy and Supply/Demand Dynamics\.*Journal of Financial Markets*16, 1 \(2013\), 1–32\.[doi:10\.1016/j\.finmar\.2012\.09\.001](https://doi.org/10.1016/j.finmar.2012.09.001)
- Peebles and Xie \(2023\)William Peebles and Saining Xie\. 2023\.Scalable Diffusion Models with Transformers\.[https://arxiv\.org/abs/2212\.09748](https://arxiv.org/abs/2212.09748)arXiv:2212\.09748\.
- Peyré and Cuturi \(2019\)Gabriel Peyré and Marco Cuturi\. 2019\.Computational Optimal Transport\.*Foundations and Trends in Machine Learning*11, 5–6 \(2019\), 355–607\.[doi:10\.1561/2200000073](https://doi.org/10.1561/2200000073)
- Piccoli and Rossi \(2014\)Benedetto Piccoli and Francesco Rossi\. 2014\.Generalized Wasserstein Distance and its Application to Transport Equations with Source\.*Archive for Rational Mechanics and Analysis*211, 1 \(2014\), 335–358\.[doi:10\.1007/s00205\-013\-0669\-x](https://doi.org/10.1007/s00205-013-0669-x)arXiv:1206\.3219\.
- Rambaldi et al\.\(2017\)Marcello Rambaldi, Emmanuel Bacry, and Fabrizio Lillo\. 2017\.The Role of Volume in Order Book Dynamics: A Multivariate Hawkes Process Analysis\.*Quantitative Finance*17, 7 \(2017\), 999–1020\.[doi:10\.1080/14697688\.2016\.1260759](https://doi.org/10.1080/14697688.2016.1260759)
- Santambrogio \(2015\)Filippo Santambrogio\. 2015\.*Optimal Transport for Applied Mathematicians*\. Progress in Nonlinear Differential Equations and Their Applications, Vol\. 87\.Birkhäuser\.[doi:10\.1007/978\-3\-319\-20828\-2](https://doi.org/10.1007/978-3-319-20828-2)
- Schefzik et al\.\(2013\)Roman Schefzik, Thordis L\. Thorarinsdottir, and Tilmann Gneiting\. 2013\.Uncertainty Quantification in Complex Simulation Models Using Ensemble Copula Coupling\.*Statist\. Sci\.*28, 4 \(2013\), 616–640\.[doi:10\.1214/13\-STS443](https://doi.org/10.1214/13-STS443)
- Séjourné et al\.\(2019\)Thibault Séjourné, Jean Feydy, François\-Xavier Vialard, Alain Trouvé, and Gabriel Peyré\. 2019\.Sinkhorn Divergences for Unbalanced Optimal Transport\.arXiv:1910\.12958 \[math\.OC\][doi:10\.48550/arXiv\.1910\.12958](https://doi.org/10.48550/arXiv.1910.12958)
- Shao et al\.\(2024\)Zhihong Shao, Peiyi Wang, Qihao Zhu, Runxin Xu, Junxiao Song, Xiao Bi, Haowei Zhang, Mingchuan Zhang, Y\. K\. Li, Y\. Wu, and Daya Guo\. 2024\.DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models\.[https://arxiv\.org/abs/2402\.03300](https://arxiv.org/abs/2402.03300)arXiv preprint arXiv:2402\.03300\.
- Song et al\.\(2021\)Jiaming Song, Chenlin Meng, and Stefano Ermon\. 2021\.Denoising Diffusion Implicit Models\. In*International Conference on Learning Representations*\.[https://arxiv\.org/abs/2010\.02502](https://arxiv.org/abs/2010.02502)
- Székely and Rizzo \(2013\)Gábor J\. Székely and Maria L\. Rizzo\. 2013\.Energy Statistics: A Class of Statistics Based on Distances\.*Journal of Statistical Planning and Inference*143, 8 \(2013\), 1249–1272\.[doi:10\.1016/j\.jspi\.2013\.03\.018](https://doi.org/10.1016/j.jspi.2013.03.018)
- Vyetrenko et al\.\(2020\)Svitlana Vyetrenko, David Byrd, Nick Petosa, Mahmoud Mahfouz, Danial Dervovic, Manuela Veloso, and Tucker Balch\. 2020\.Get Real: Realism Metrics for Robust Limit Order Book Market Simulations\. In*Proc\. 1st ACM Int\. Conf\. on AI in Finance \(ICAIF\)*\.[doi:10\.1145/3383455\.3422561](https://doi.org/10.1145/3383455.3422561)
- Wang and Ventre \(2025\)Zhuohan Wang and Carmine Ventre\. 2025\.DiffVolume: Diffusion Models for Volume Generation in Limit Order Books\. In*Proc\. 6th ACM Int\. Conf\. on AI in Finance \(ICAIF\)*\.[doi:10\.1145/3768292\.3770413](https://doi.org/10.1145/3768292.3770413)
- Wu et al\.\(2019\)Peng Wu, Marcello Rambaldi, Jean\-François Muzy, and Emmanuel Bacry\. 2019\.Queue\-Reactive Hawkes Models for the Order Flow\.arXiv:1901\.08938 \[q\-fin\.TR\][doi:10\.48550/arXiv\.1901\.08938](https://doi.org/10.48550/arXiv.1901.08938)
- Xu et al\.\(2020\)Tianlin Xu, Li Kevin Wenliang, Michael Munn, and Beatrice Acciaio\. 2020\.COT\-GAN: Generating Sequential Data via Causal Optimal Transport\. In*Advances in Neural Information Processing Systems*, Vol\. 33\.[https://proceedings\.neurips\.cc/paper/2020/hash/641d77dd5271fca28764612a028d9c8e\-Abstract\.html](https://proceedings.neurips.cc/paper/2020/hash/641d77dd5271fca28764612a028d9c8e-Abstract.html)
- Xue et al\.\(2025\)Zeyue Xue, Jie Wu, Yu Gao, Fangyuan Kong, Lingting Zhu, Mengzhao Chen, Zhiheng Liu, Wei Liu, Qiushan Guo, Weilin Huang, and Ping Luo\. 2025\.DanceGRPO: Unleashing GRPO on Visual Generation\.[https://arxiv\.org/abs/2505\.07818](https://arxiv.org/abs/2505.07818)arXiv preprint arXiv:2505\.07818\.