This paper extends the classic limit order book model of Cont and de Larrard (2013) by allowing the arrival rates of limit orders, market orders, and cancellations to be time-dependent, specifically proportional to a common periodic function. The model focuses on the first level of the order book and assumes unit order sizes. The author derives the distribution of the time between price changes, showing that it is a time-changed version of the constant-rate case, with the time-change given by the integral of the intensity function. This leads to a characterization of the tail behavior and moment properties of inter-arrival times, which depend on the relative sizes of the birth and death rates and the growth of the intensity function.

The main theoretical contribution is the analysis of the long-run dynamics of the price process. Under the assumption that the intensity function grows polynomially or decays as 1/t, the rescaled price process converges to a diffusion process with time-dependent volatility, rather than a standard Brownian motion. This provides a mechanism to generate more realistic price dynamics from a microscopic model. The paper also corrects an error in the tail asymptotics of extinction times from the original Cont and de Larrard model.

Empirical analysis using high-frequency data for six stocks (e.g., AAPL, MSFT) tests the model assumptions. The author finds that the proportional rates assumption holds reasonably well for some stocks, and the tail behavior of inter-arrival times is consistent with the theoretical predictions. However, the empirical results also highlight limitations, such as the need to relax the unit volume assumption and the independence of order flows. Overall, the paper provides a rigorous framework for connecting micro-scale order book dynamics to macro-scale price processes with time-dependent volatility.