A Simple Model of a Limit Order Book

We formulate a simplified model of a limit order book, in which the arrival process is independent of the current state. We prove a phase transition result: there exist prices κb and κa such that, for any > 0, only finitely many bid (ask) departures occur at prices below κb− (above κa + ), while the interval (κb + , κa − ) infinitely often contains no bids, and infinitely often contains no asks. We derive expressions for κb and κa, which we solve in the case of uniform arrivals. We conjecture the positive recurrence of a modified model, and find the steady-state distribution of the highest bid and of the lowest ask assuming the positive recurrence.