Ergodic transition in a simple model of the continuous double auction

TL;DR

This paper models a continuous double auction as two independent queues, analyzing conditions for ergodicity and identifying three regimes in price behavior, including stability and intermittency in returns.

Contribution

It introduces a phenomenological model linking the auction to queue theory and derives conditions for ergodicity, revealing three distinct regimes of price dynamics.

Findings

Three regimes of price behavior identified

Ergodic regime shows intermittent returns

Non-ergodic regime leads to price stability

Abstract

We study a phenomenological model for the continuous double auction, equivalent to two independent $M/M/1$ queues. The continuous double auction defines a continuous-time random walk for trade prices. The conditions for ergodicity of the auction are derived and, as a consequence, three possible regimes in the behavior of prices and logarithmic returns are observed. In the ergodic regime, prices are unstable and one can observe an intermittent behavior in the logarithmic returns. On the contrary, non-ergodicity triggers stability of prices, even if two different regimes can be seen.