TY  - JOUR
A1  - Cherstvy, Andrey G.
A1  - Vinod, Deepak
A1  - Aghion, Erez
A1  - Sokolov, Igor M.
A1  - Metzler, Ralf
T1  - Scaled geometric Brownian motion features sub- or superexponential ensemble-averaged, but linear time-averaged mean-squared displacements
T2  - Physical review : E, Statistical, nonlinear and soft matter physics
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, 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.
Y1  - 2021
UR  - https://publishup.uni-potsdam.de/frontdoor/index/index/docId/62540
SN  - 2470-0045
SN  - 2470-0053
VL  - 103
IS  - 6
PB  - American Physical Society
CY  - College Park
ER  -