Stochastic Simulation of Time Series by Using the Spatial-Temporal Weierstrass Function

We extend the recently considered toy model of Weierstrass (or Levy) walks with varying velocity of the walker [1,2] by introducing a more realistic possibility that the walk can be occasionally intermitted by its momentary localization; the localizations themselves are again described by the Weierstrass (or Levy) process. The direct empirical motivation for developing this combined model is, for example, the dynamics of financial high-frequency time series or meteorological ones. This approach makes it possible to study by efficient stochastic simulations the whole spatial-temporal range. To describe empirical data, which are collected at discrete time-steps, we used in the continuous-time series produced by the model a discretization procedure. We observed that such a procedure constitutes a basis for long-time autocorrelations (of the variation of the walker single-step displacements) which appear to be similar to those observed, e.g., in financial time series [3,4,5,6,7,8], although single steps of the walker within the continuous time are uncorrelated.