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

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.

Autoregressive transformers trained on limit order book (LOB) message streams are increasingly proposed as foundation models for market microstructure. Because the generative process of real markets is unknown, these models can be judged only by the realism of their output, leaving open what they actually learn. We make the question answerable by inverting the setup: we train GPT-style models on order flow from simulators whose latent parameters are known, so that the Bayes-optimal predictor, the generating state, and the exact data entropy serve as references. Measured against them, next-token pretraining on order flow implements amortized Bayesian inference over market regimes. The learned predictive matches the exact Bayes optimum to within a millinat per token and approaches the oracle floor even on a nonstationary family whose latent state drifts within an episode, a regime that requires filtering. The latent regime is linearly decodable from frozen activations, well above baselines, so a frozen-model read-out calibrates simulator parameters more accurately than windowed maximum likelihood at a fraction of the cost. In our experiments these capabilities are not produced by scale: capacity and token volume saturate early, and it is the diversity of training regimes, not the token count, that determines generalization to unseen markets. On real NASDAQ data, zero-shot transfer is set by how well the simulator prior covers real flow: an enriched mixture nearly closes the gap to an in-domain baseline, and pretraining beats from-scratch training at every budget across held-out tickers. Known ground truth turns average next-token loss, which is uninformative near the entropy floor, into an exact comparison against the Bayes-optimal and oracle predictors.

Limit order book (LOB) simulators are most useful to practitioners when they combine realistic market dynamics, computationally efficient sampling, controllable scenario generation, and the ability to generalize beyond the instruments seen during training---properties that existing agent-based and deep generative simulators provide only partially. We present FlowLOB, a conditional flow-matching generator of LOB trajectories, trained on multiple Hong Kong Exchange (HKEX) symbols at three sampling frequencies ($0.1$s, $1$s, $10$s) in tick-relative representation that transfers to unseen instruments. Because flow and diffusion models admit a common formulation, we train both with identical data, architecture, and budget, and sample both through the same fixed-step ODE solvers, yielding a controlled comparison of sampling efficiency and fidelity. Flow matching attains its best quality with only $10$ ODE-solver steps, whereas diffusion needs many more function evaluations to approach the same fidelity. At this efficient operating point, FlowLOB improves realism over baselines, two learned and two agent-based models, in most distributional metrics at the two finer sampling frequencies. We evaluate counterfactual controllability with a distributional test that asks whether changing a scenario condition moves the generated statistic toward the corresponding real tail regime; FlowLOB satisfies this criterion in most tested settings. Both realism and control effects transfer zero-shot on a held-out symbol. We additionally conduct ablation studies on the network architecture and the learning rate.