Deriving a Queue-Volume Master Equation with Depletion Resets

Summary

The document presents a simple stochastic model for the volume of a limit order book queue. Market orders, limit orders, and cancellations are treated as independent Poisson processes with constant rates and unit-sized events. The stated forward equation accounts for transitions between adjacent queue volumes and for depletion, which resets the queue to a new volume drawn from a distribution.

The author applies a small-time-step probability argument and notices that their derivation adds a depletion-rate loss term to the probability at the current volume. The issue raised is a useful distinction: depletion is a state transition out of the current queue volume, so its loss contribution must be considered alongside the reset inflow. The document poses the discrepancy but does not include an answer or establish a resolution. Its scope is a simplified queue model; real order sizes, dependencies, and changing event rates are not discussed.

Key ideas

• The model represents unit-sized limit orders, market orders, and cancellations as independent Poisson events.
• Queue volume changes through transitions to neighboring states.
• Depletion resets the queue volume according to a separate distribution.
• A small-time-step derivation must account for probability leaving the current state through depletion.
• The document raises the derivation issue but does not provide a final resolution.

Tags

# Constant cancellation model for volume of LOB queues


# Constant cancellation model for volume of LOB queues












I have been reading Jean-Philippe Bouchaud's book on stochastic models of LOB queues in Chapter 5, which starts with the simplest model. In this model, market/limit/cancel orders are assumed to be of as small as a lot size and behave as independent Poisson processes with coefficients $\mu, \lambda, \nu$, respectively.

In section 5.3, probability $P(V, t)$ of observing LOB queue of volume $V$ satisfies the following equation: $$\frac{\partial P(V, t)}{\partial t} = -(\lambda + \mu + \nu) P(V, t) + \lambda P(V-1, t) + (\mu + \nu) P(V + 1, t) + J(t) \rho (V)$$

where $J(t)$ is a rate of depleting queue at time $t$ and $\rho(V)$ is a distribution of the next price volume after the depletion.

With my limited understanding of differential equations and such, I tried to convince the above equation as follows:

$$P(V, t + dt) = J(t) dt \rho(V) + \lambda dt P(V-1, t) + (\mu + \nu) dt P(V+1, t) + (1 - J(t) - \lambda - \mu - \nu) dt P(V, t) + O(dt^2)$$ with "law of total probability". However, with the above we get a slightly different equation (omitting $O(dt^2)$ stuff):

$$\frac{\partial P(V, t)}{\partial t} = -(\lambda + \mu + \nu + J(t)) P(V, t) + \lambda P(V-1, t) + (\mu + \nu) P(V + 1, t) + J(t) \rho (V)$$

Could you please help me out with where I might be wrong? Also, I would greatly appreciate if you could point in references to learn more about arguing on the above rigorously.

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