Nonergodicity of reset geometric Brownian motion

We derive. the ensemble-and time-averaged mean-squared displacements (MSD, TAMSD) for Poisson-reset geometric Brownian motion (GBM), in agreement with simulations. We find MSD and TAMSD saturation for frequent resetting, quantify the spread of TAMSDs via the ergodicity-breaking parameter and compute distributions of prices. General MSD-TAMSD nonequivalence proves reset GBM nonergodic.

Automated summary

In this study, we derived ensemble- and time-averaged mean-squared displacements (MSD, TAMSD) for Poisson-reset geometric Brownian motion (GBM), which is in agreement with simulations. We found that MSD and TAMSD reach saturation under frequent resetting, and quantified the spread of TAMSDs using the ergodicity-breaking parameter. We also computed price distributions and proved the general nonequivalence between MSD and TAMSD, indicating that reset GBM is nonergodic.