Stochastic PDE Models of Limit Order Book Dynamics

Summary

The document presents a class of limit order book models using a stochastic partial differential equation for the book relative to the mid-price. Multiplicative noise captures changing order book shape, while a separate stochastic process for the mid-price is linked to order flow. The authors identify conditions that let these dynamics be represented by a finite-dimensional Markov process, which supports more efficient estimation and computation.

Two parsimonious examples are examined: a model with two factors and one with mean-reverting depth. Their parameters are related to price behavior, depth, volume, and order imbalance, with financial interpretations of the model variables. The account says these models reproduce statistical features seen in price changes, market depth, and order flow. It does not provide specific empirical results in the excerpt, and the proposed tractability depends on the stated conditions for finite-dimensional representation.

Key ideas

• A stochastic PDE with multiplicative noise describes the limit order book relative to the mid-price.
• The mid-price follows stochastic dynamics connected to order flow.
• Certain conditions reduce the model to a finite-dimensional Markov process for more efficient computation.
• Two examples use a two-factor structure and mean-reverting order book depth.
• The model links parameters to price changes, depth, volume, and order imbalance.

Tags

# A stochastic partial differential equation model for limit order book dynamics


# A stochastic partial differential equation model for limit order book dynamics









We propose an analytically tractable class of models for the dynamics of a limit order book, described through a stochastic partial differential equation (SPDE) with multiplicative noise for the order book centered at the mid-price, along with stochastic dynamics for the mid-price which is consistent with the order flow dynamics. We provide conditions under which the model admits a finite dimensional realization driven by a (low-dimensional) Markov process, leading to efficient estimation and computation methods. We study two examples of parsimonious models in this class: a two-factor model and a model with mean-reverting order book depth. For each model we analyze in detail the role of different parameters, the dynamics of the price, order book depth, volume and order imbalance, provide an intuitive financial interpretation of the variables involved and show how the model reproduces statistical properties of price changes, market depth and order flow in limit order markets.

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