@article{CherstvyVinodAghionetal.2017,
  author    = {Cherstvy, Andrey G. and Vinod, Deepak and Aghion, Erez and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {Time averaging, ageing and delay analysis of financial time series},
  series = {New journal of physics},
  volume    = {19},
  journal   = {New journal of physics},
  publisher = {IOP},
  address   = {London},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/aa7199},
  pages     = {1 -- 11},
  year      = {2017},
  abstract  = {We introduce three strategies for the analysis of financial time series based on time averaged observables. These comprise the time averaged mean squared displacement (MSD) as well as the ageing and delay time methods for varying fractions of the financial time series. We explore these concepts via statistical analysis of historic time series for several Dow Jones Industrial indices for the period from the 1960s to 2015. Remarkably, we discover a simple universal law for the delay time averaged MSD. The observed features of the financial time series dynamics agree well with our analytical results for the time averaged measurables for geometric Brownian motion, underlying the famed Black-Scholes-Merton model. The concepts we promote here are shown to be useful for financial data analysis and enable one to unveil new universal features of stock market dynamics.},
  language  = {en}
}
@article{CherstvyVinodAghionetal.2017,
  author    = {Cherstvy, Andrey G. and Vinod, Deepak and Aghion, Erez and Chechkin, Aleksei V. and Metzler, Ralf},
  title     = {Time averaging, ageing and delay analysis of financial time series},
  series = {New journal of physics : the open-access journal for physics},
  volume    = {19},
  journal   = {New journal of physics : the open-access journal for physics},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/aa7199},
  pages     = {135 -- 147},
  year      = {2017},
  abstract  = {We introduce three strategies for the analysis of financial time series based on time averaged observables. These comprise the time averaged mean squared displacement (MSD) as well as the ageing and delay time methods for varying fractions of the financial time series. We explore these concepts via statistical analysis of historic time series for several Dow Jones Industrial indices for the period from the 1960s to 2015. Remarkably, we discover a simple universal law for the delay time averaged MSD. The observed features of the financial time series dynamics agree well with our analytical results for the time averaged measurables for geometric Brownian motion, underlying the famed Black-Scholes-Merton model. The concepts we promote here are shown to be useful for financial data analysis and enable one to unveil new universal features of stock market dynamics.},
  language  = {en}
}
@article{CherstvyVinodAghionetal.2021,
  author    = {Cherstvy, Andrey G. and Vinod, Deepak and Aghion, Erez and Sokolov, Igor M. and Metzler, Ralf},
  title     = {Scaled geometric Brownian motion features sub- or superexponential ensemble-averaged, but linear time-averaged mean-squared displacements},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {103},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {6},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {2470-0045},
  doi       = {10.1103/PhysRevE.103.062127},
  pages     = {11},
  year      = {2021},
  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.},
  language  = {en}
}