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#### Scaled geometric Brownian motion features sub- or superexponential ensemble-averaged, but linear time-averaged mean-squared displacements

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##### Citation

Cherstvy, A. G., Vinod, D., Aghion, E., Sokolov, I. M., & Metzler, R. (2021). Scaled
geometric Brownian motion features sub- or superexponential ensemble-averaged, but linear time-averaged mean-squared displacements.* Physical Review E,* *103*(6): 062127. doi:10.1103/PhysRevE.103.062127.

Cite as: https://hdl.handle.net/21.11116/0000-0009-1971-C

##### Abstract

Various mathematical Black-Scholes-Merton-like models of option pricing employ the paradigmatic stochastic process of geometric Brownian motion (GBM). The innate property of such models and of real stock-market prices is the roughly exponential growth of prices with time [on average, in crisis-free times]. We here explore the ensemble- and time averages of a multiplicative-noise stochastic process with power-law-like time-dependent volatility, sigma(t) similar to t(alpha), named scaled GBM (SGBM). For SGBM, the mean-squared displacement (MSD) computed for an ensemble of statistically equivalent trajectories can grow faster than exponentially in time, while the time-averaged MSD (TAMSD)-based on a sliding-window averaging along a single trajectory-is always linear at short lag times Delta. The proportionality factor between these the two averages of the time series is Delta/T at short lag times, where T is the trajectory length, similarly to GBM. This discrepancy of the scaling relations and pronounced nonequivalence of the MSD and TAMSD at Delta/T << 1 is a manifestation of weak ergodicity breaking for standard GBM and for SGBM with s (t)-modulation, the main focus of our analysis. The analytical predictions for the MSD and mean TAMSD for SGBM are in quantitative agreement with the results of stochastic computer simulations.