TY - JOUR

T1 - Scaled geometric Brownian motion features sub- or superexponential ensemble-averaged, but linear time-averaged mean-squared displacements

AU - Cherstvy, Andrey G.

AU - Vinod, Deepak

AU - Aghion, Erez

AU - Sokolov, Igor M.

AU - Metzler, Ralf

N1 - Publisher Copyright:
© 2021 American Physical Society.

PY - 2021/6

Y1 - 2021/6

N2 - 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, σ(t)∼tα, 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 Δ. The proportionality factor between these the two averages of the time series is Δ/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 Δ/T≪1 is a manifestation of weak ergodicity breaking for standard GBM and for SGBM with σ(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.

AB - 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, σ(t)∼tα, 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 Δ. The proportionality factor between these the two averages of the time series is Δ/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 Δ/T≪1 is a manifestation of weak ergodicity breaking for standard GBM and for SGBM with σ(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.

UR - http://www.scopus.com/inward/record.url?scp=85108235727&partnerID=8YFLogxK

U2 - 10.1103/physreve.103.062127

DO - 10.1103/physreve.103.062127

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C2 - 34271619

AN - SCOPUS:85108235727

SN - 2470-0045

VL - 103

JO - Physical Review E

JF - Physical Review E

IS - 6

M1 - 062127

ER -