Refine
Language
- English (17)
Is part of the Bibliography
- yes (17)
Keywords
- anomalous diffusion (5)
- Levy flights (2)
- diffusion (2)
- fractional Brownian motion (2)
- non-Gaussianity (2)
- stochastic processes (2)
- Bulk-mediated diffusion (1)
- Bulk-mediated diffusion; (1)
- Complete Bernstein function (1)
- Completely monotone function (1)
- Distributed order diffusion-wave equations (1)
- ageing (1)
- scaled Brownian motion (1)
Institute
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.