Scale-invariant truncated Lévy process

– We develop a scale-invariant truncated Lévy (STL) process to describe physical systems characterized by correlated stochastic variables. The STL process exhibits Lévy stability for the distribution, and hence shows scaling properties as commonly observed in empirical data; it has the advantage that all moments are finite and so accounts for the empirical scaling of the moments. To test the potential utility of the STL process, we analyze financial data. In recent years, the Lévy process [1] has been proposed to describe the statistical properties of a variety of complex phenomena [2–14], due to scaling behavior in distributions similar to that observed in empirical data. However, the application of the Lévy process to empirical data is limited because, opposite to commonly encountered data, it is characterized by no correlations in the moments. Lévy walks [7] have been proposed to account for the finite moments observed for empirical data. Another way to retain the finite variance is by means of truncated Lévy (TL) flights [15] defined to have a Lévy distribution in the central regime, truncated by a function decaying faster than a Lévy distribution in the tails. However, the TL process with either abrupt [15] or smooth [16] truncation has limitations when applied to empirical data. i) The TL process is introduced for independent and identically distributed (i.i.d.) stochastic variables, while variables describing many physical systems are long-range correlated [17–20], and so are not i.i.d. ii) The distributions for a variety of complex systems, however, are often characterized by regions of scale-invariant behavior, while the TL process tends to the Gaussian distribution and hence does not exhibit scale invariance. Here we introduce a stochastic process which we call the scale-invariant truncated Lévy (STL) process. The STL process might be regarded as a generalization of the truncated Lévy process —due to scaling transformations of the Lévy type, the stochastic variables exhibit scale-invariant behavior in the distributions, and due to truncation even in the moments. We also propose a dynamical mechanism to account for the regimes of different scaling behavior.