Statistical properties of local time estimators for ranked particle systems

Investigate the statistical properties of finite-sample, discrete-time estimators of semimartingale local times for the gaps between ranked log market capitalizations, and ascertain how these properties can be applied to the calibration of rank-based particle systems used to model equity markets.

Background

In Section 4.3, the authors introduce an empirical quantity Λ_k to measure the cumulative intensity of rank switching at rank k, motivated by the semimartingale local time of the gap between consecutive ranked log capitalizations. They connect this construction to rank-based diffusion models and volatility-stabilized processes, where calibration often relies on estimators derived from finite-sample and discrete-time data.

Because these models are formulated in continuous time but calibrated on discrete observations, understanding the statistical behavior of local time estimators (e.g., bias, variance, convergence properties) is crucial for robust model calibration. The authors explicitly identify the need to develop theory and methodology for such estimators and their practical use in calibrating ranked particle systems for equity markets.