Sequential Order Book Processing

• Sequential processing of order books is a framework where each order event—limit, market, or cancellation—updates the state sequentially, enabling detailed microstructural analysis.
• The model employs a multidimensional continuous-time Markov chain with Poisson arrivals, facilitating rigorous simulation and empirical validation of liquidity profiles and price diffusion.
• Its robust mathematical foundations, including ergodicity and functional central limit theorem results, offer actionable insights for algorithmic trading and market risk management.

Sequential processing of order books refers to the modeling and analysis of order book dynamics as a sequence of discrete, individual events—limit orders, market orders, and cancellations—each updating the market state one at a time. The rigorous paper of this process forms a bridge between agent-based models of microscopic price formation and the macroscopic, continuous-time descriptions used in stochastic finance. The mathematical foundations, implications for stability and price behavior, and empirical consequences of sequential order processing have been explicated in models that treat the order book as a multidimensional continuous-time Markov chain with Poissonian event arrivals, as well as validated through detailed simulation and comparison to market data (Abergel et al., 2010).

1. Mathematical Structure of Sequential Order Processing

The order book is modeled as a state vector

$$X(t) = (a_1(t), \ldots, a_K(t);\, b_1(t), \ldots, b_K(t))$$

where $a_i(t)$ and $b_i(t)$ are the outstanding volumes at the $i$th ask (above the best bid) and bid (below the best ask) price levels, respectively. Sequential changes to $X(t)$ occur via three fundamental event types:

• Arrival of a market order (removes liquidity at the best quote).
• Arrival of a limit order (adds liquidity at a selected price level).
• Cancellation of a standing limit order.

Each such event constitutes a transition in a high-dimensional state space, and the process is Markovian: future evolution depends only on the current book configuration. The infinitesimal generator for a payoff function $f$,

$$\mathcal{L}f(P) = u\big[\lambda^+(f(P+\Delta P) - f(P)) + \lambda^-(f(P-\Delta P) - f(P))\big]$$

summarizes the event rates and price impact at the best quotes (where $u$ is a probability parameter and $\lambda^\pm$ are market order arrival rates).

These sequential, discrete jumps capture not just price changes but also the depth and resilience profile of the book at each price tick.

2. Poissonian Event Arrivals and Sequentiality

A key modeling assumption is that each order type arrives as an independent Poisson process, rendering the event sequence memoryless and ensuring that at most one event occurs at any instant. Market, limit, and cancellation orders at each level $i$ have respective rates $\lambda^{\ell^+}_i$, $\lambda^{\ell^-}_i$, and cancellation rates (often proportional to the local queue size).

This Poisson structure is fundamental for the well-posedness of sequential processing:

• The chain of book transitions is totally ordered—each event is processed individually, causing a unique, temporally ordered sequence of states (no batch processing or simultaneity).
• The Markov generator can be explicitly written, enabling analysis via probabilistic and Lyapunov function techniques.

Practically, this aligns with high-frequency trading systems, where matching engines process orders one-by-one and enforce time-priority at each price.

3. Stability, Stationarity, and the Role of Cancellations

Stability of the sequentially updated order book hinges on a well-calibrated cancellation mechanism. Positive cancellation rates at all price levels are shown to be sufficient for ergodicity of the Markov chain: the order book achieves a unique, stationary distribution.

A Foster–Lyapunov drift condition is central: $\mathcal{L}V(x) \leq -\beta V(x) + \gamma$ where $V(x)$ is a test function proportional to the total book size, and $\beta > 0$ ensures a negative drift (ordering the process toward a central regime from extreme states).

This drift yields exponential convergence to stationarity: $$\|\mathbb{Q}^t(x, \cdot) - \pi(\cdot)\| \leq R r^t V(x)$$ for $0 < r < 1$, where $\pi$ is the invariant distribution. Cancellations, far from being an ad hoc model feature, critically control book volume and prevent explosive queue growth under repeated sequential additions of limit orders.

4. Price Process Diffusivity: Functional Central Limit Theorem

The Markovian sequential structure supports a rigorous derivation of the large-scale (macroscopic) properties of prices through a functional central limit theorem (FCLT). The cumulative price increment after $n$ sequential events,

$$P_n = \sum_{k=1}^n \eta_k, \quad \eta_k = \Psi(X_{k-1}, e_k) - \mathbb{E}_\mu[\Psi(X_{k-1}, e_k)]$$

(where $\Psi$ maps state transitions to tick-move increments), exhibits weak dependence between increments due to uniform ergodicity. The FCLT gives

$$\frac{P_n - n\mu}{\sqrt{n}} \Rightarrow \sigma B(t)$$

where $B(t)$ is standard Brownian motion and

$$\sigma^2 = \mathbb{E}_\mu[\eta_0^2] + 2 \sum_{n=1}^\infty \mathbb{E}_\mu[\eta_0 \eta_n]$$

is the asymptotic variance. Hence, diffusion-like (Brownian) price movement arises as an emergent property of the sequential, event-driven processing of the order book.

This result holds both in event time and—via stochastic time change—in clock time, consistent with empirical scaling of high-frequency return data.

5. Simulation Algorithms and Empirical Comparison

The sequential Markovian model leads directly to an efficient event-driven simulation paradigm. The algorithm simulates Poisson arrivals for each event type; upon each event, the order book state is updated, and transition probabilities and effects (limit addition, market removal, cancellation) are applied. For realism, both unit and lognormally distributed order sizes are considered.

Empirical validation demonstrates:

• Simulated average depth profiles closely match observed market shapes.
• The simulated spread distribution is bounded, though sometimes narrower than in data.
• Price increments display rapid autocorrelation decay, consistent with real data at high frequency.
• Signature plots (variance of price increments per unit time) show microstructural transition to the long-term variance predicted by the FCLT.

These findings confirm that sequential order processing, as formulated in the Markovian-Poisson framework, robustly generates realistic high-frequency liquidity and price patterns.

6. Macro-Micro Link: Theoretical and Practical Implications

Sequential order processing models based on multidimensional continuous-time Markov chains with Poissonian arrivals provide a rigorous microstructural underpinning for the emergence of macroscopic, diffusive price dynamics. The essential role of cancellations in achieving stationarity and the predictive validity of the framework are substantiated both by theoretical results (ergodicity, FCLT) and by simulation–market data congruence.

This class of models serves as a theoretical foundation for algorithmic trading strategy development, risk management, and the design/analysis of market microstructure, and illustrates how detailed, single-order event processing leads to the well-established scaling laws and liquidity features observed in limit order markets (Abergel et al., 2010).