Uncovering the evolution of nonstationary stochastic variables: The example of asset volume-price fluctuations

Wind Speed Modeling by Nested ARIMA Processes

Wind speed modelling is of increasing interest, both for basic research and for applications, as, e.g., for wind turbine development and strategies to construct large wind power plants. Generally, such modelling is hampered by the non-stationary features of wind speed data that, to a large extent, reflect the turbulent dynamics in the atmosphere. We study how these features can be captured by nested ARIMA models. In this approach, wind speed fluctuations in given time windows are modelled by one stochastic process, and the parameter variation between successive windows by another one. For deriving the wind speed model, we use 20 months of data collected at the FINO1 platform at the North Sea and use a variable transformation that best maps the wind speed onto a Gaussian random variable. We find that wind speed increments can be well reproduced for up to four standard deviations. The distributions of extreme variations, however, strongly deviate from the model predictions.

Modeling correlated bursts by the bursty-get-burstier mechanism

Temporal correlations of time series or event sequences in natural and social phenomena have been characterized by power-law decaying autocorrelation functions with decaying exponent γ. Such temporal correlations can be understood in terms of power-law distributed interevent times with exponent α and/or correlations between interevent times. The latter, often called correlated bursts, has recently been studied by measuring power-law distributed bursty trains with exponent β. A scaling relation between α and γ has been established for the uncorrelated interevent times, while little is known about the effects of correlated interevent times on temporal correlations. In order to study these effects, we devise the bursty-get-burstier model for correlated bursts, by which one can tune the degree of correlations between interevent times, while keeping the same interevent time distribution. We numerically find that sufficiently strong correlations between interevent times could violate the scaling relation between α and γ for the uncorrelated case. A nontrivial dependence of γ on β is also found for some range of α. The implication of our results is discussed in terms of the hierarchical organization of bursty trains at various time scales.

Stochastic modelling of non-stationary financial assets

We model non-stationary volume-price distributions with a log-normal distribution and collect the time series of its two parameters. The time series of the two parameters are shown to be stationary and Markov-like and consequently can be modelled with Langevin equations, which are derived directly from their series of values. Having the evolution equations of the log-normal parameters, we reconstruct the statistics of the first moments of volume-price distributions which fit well the empirical data. Finally, the proposed framework is general enough to study other non-stationary stochastic variables in other research fields, namely, biology, medicine, and geology.

Intraday Seasonalities and Nonstationarity of Trading Volume in Financial Markets: Individual and Cross-Sectional Features

We study the intraday behaviour of the statistical moments of the trading volume of the blue chip equities that composed the Dow Jones Industrial Average index between 2003 and 2014. By splitting that time interval into semesters, we provide a quantitative account of the nonstationary nature of the intraday statistical properties as well. Explicitly, we prove the well-known ∪-shape exhibited by the average trading volume-as well as the volatility of the price fluctuations-experienced a significant change from 2008 (the year of the "subprime" financial crisis) onwards. That has resulted in a faster relaxation after the market opening and relates to a consistent decrease in the convexity of the average trading volume intraday profile. Simultaneously, the last part of the session has become steeper as well, a modification that is likely to have been triggered by the new short-selling rules that were introduced in 2007 by the Securities and Exchange Commission. The combination of both results reveals that the ∪ has been turning into a ⊔. Additionally, the analysis of higher-order cumulants-namely the skewness and the kurtosis-shows that the morning and the afternoon parts of the trading session are each clearly associated with different statistical features and hence dynamical rules. Concretely, we claim that the large initial trading volume is due to wayward stocks whereas the large volume during the last part of the session hinges on a cohesive increase of the trading volume. That dissimilarity between the two parts of the trading session is stressed in periods of higher uproar in the market.