Analytic model for transient anomalous diffusion with highly persistent correlations

In recent decades, many stochastic processes have been proposed as models for real world time series data with anomalous spreading, highly persistent correlations, and transient distributional characteristics. We introduce the higher order fractional tempered stable motion as the stochastic integral of the tempered stable motion with respect to a generalized higher order moving average kernel, which provides an analytic model for stochastic processes possessing these characteristics. This stochastic process provides a mathematical model for anomalous diffusion with a transient distribution resembling higher order fractional stable motion on short timescales and higher order fractional Brownian motion in the long run. The specifics of the crossover dynamics from the Lévy stable anomalous diffusion to the Gaussian anomalous diffusion are controlled by explicit parameter values that correspond to physical attributes of the process. It is well suited to modeling anomalous diffusion of any “type” (sub-, super-, regular, or hyperdiffusion) under appropriate parametrizations due to its power-law scaling of variance with respect to time. It is also a useful model for position-velocity-acceleration triples due to its convenient path differentiability and integrability properties. To highlight the potential physical relevance of this model for real world data, we outline its key statistical properties including its covariance structure, memory, and second order self-similarity. We also give an easy to implement elementary method for sample path generation which may be used as a basis for simulation and Monte Carlo studies.

Figure 1

Typical trajectories of $n$-ftsm. Parameters used were (top row) $\alpha =1.5,\beta =1$, and $H=n-0.2$ and (bottom row) $\alpha =1.1,\beta =0.1$, and $H=n-0.2$. In both rows we put (a) and (c) trajectories of 1-ftsm generated using (6.1) and (b) and (d) trajectories of 2-ftsm produced using successive cumulative integration of the 1-ftsm in (a) as per integral relation (3.1) (circles) as well as using (6.1) with the same background noise (common sequence of jumps) as in the $n=1$ case. In the top row, the larger values of $\alpha$ and $\beta$ stifle large jumps in the driving noise making the motion appear continuous, while in the bottom row the relatively smaller values of $\alpha$ and $\beta$ ensure the presence of occassional large jumps in the background noise, leading to occassional jumplike displacements in the integrated motion. However, since the parameter setting used in (c) (in particular, $2G=0.7818<1$) gives antipersistent behavior (subdiffusive spread and short memory) the sample path tends to revert back to its mean after each displacement.

Figure 2

Short and long run convergence of $n$-ftsm: histograms of ${10}^4$ simulated trajectories with corresponding densities of limiting process. Parameters used were $n=3,\alpha =1.2,\beta =0.1,H=2.8,\kappa =50$, and $m=5000$ (see Sec. 6 for details on these parameters). (a) Short run convergence demonstrated by simulating the left hand side of (5.1) with $h=0.001$ (foreground, darker). (b) Long run convergence demonstrated by simulating the left hand side of (5.2) with $h=1000$ (foreground, darker). In both figures, the lighter background histogram is of the $h=1$ case given by (6.1) and the dotted red lines are the density functions of the respective theoretical limiting distributions given in Sec. 5. Good agreement between the empirical histograms and theoretical limiting density functions is seen.