Physics

Mathematics

Mathematical physics

Multiplicative function

Knowledge management

Geometric Brownian motion

Statistical physics

Mathematical analysis

Ergodicity

Mean squared displacement

Geometry

Computer science

Molecular dynamics

Quantum mechanics

Innovation diffusion

Diffusion process

Scaling

Physical review. E

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, $\ensuremath{\sigma}(t)\ensuremath{\sim}{t}^{\ensuremath{\alpha}}$, 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 $\mathrm{\ensuremath{\Delta}}$. The proportionality factor between these the two averages of the time series is $\mathrm{\ensuremath{\Delta}}/T$ 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 $\mathrm{\ensuremath{\Delta}}/T\ensuremath{\ll}1$ is a manifestation of weak ergodicity breaking for standard GBM and for SGBM with $\ensuremath{\sigma}(t)$-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.

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