Geometric Brownian motion with intermittent entries and exits

2026-05-17

AI Summary

This study investigates the long-term evolutionary dynamics of heterogeneous economic agents characterized by asymmetric entry and exit mechanisms. By constructing a generalized geometric Brownian motion model incorporating asymmetric inflow and outflow rates, and employing stochastic process theory alongside non-conservative system analysis, the authors derive the stationary distribution and characterize the dynamics of moments as well as first-passage time properties. The primary contributions include uncovering three distinct regimes of moment evolution governed by the interplay among volatility, drift, and entry–exit rates, demonstrating that the system nonetheless converges to a well-defined stationary distribution, and identifying an optimal exit rate that significantly reduces the first-passage time. These findings quantitatively elucidate the critical influence of entry and exit strategies on systemic dynamics.

Abstract

We study a generalized geometric Brownian motion framework that incorporates both entries of new units and exit mechanisms for the current population, extending earlier stochastic resetting models where these rates are treated as identical. The model captures realistic features observed in many economic observables, which can be explained as market-driven firm entries/exits, worker inflow/outflow, and income growth/loss. This model is not conservative and, despite the asymmetry in the entry and exit rates, we find that the system eventually relaxes to a stationary distribution. Moreover, our analysis reveals three distinct dynamical regimes in the moments of the distribution, arising from the interplay between volatility, drift, entry, and exit rates. We further derive the survival probability and the mean first-passage time associated with the observed variable reaching certain threshold under the competing entry-exit processes. Interestingly, we identify an optimal exit rate that minimizes the mean first-passage time, providing insights into how entry and exit policies can influence the outcome of the system. These results should be useful for understanding the long-run behavior of economic systems in which growth, volatility, entry, and exit jointly shape the evolution of heterogeneous units.

Problem

• geometric Brownian motion
• entry-exit dynamics
• stationary distribution
• mean first-passage time
• economic systems

Innovation

• geometric Brownian motion
• stochastic resetting
• entry-exit dynamics
• mean first-passage time
• non-conservative system