On fractional Lévy processes: tempering, sample path properties and stochastic integration. (English) Zbl 1447.60069
Summary: We define two new classes of stochastic processes, called tempered fractional Lévy process of the first and second kinds (TFLP and TFLP II, respectively). TFLP and TFLP II make up very broad finite-variance, generally non-Gaussian families of transient anomalous diffusion models that are constructed by exponentially tempering the power law kernel in the moving average representation of a fractional Lévy process. Accordingly, the increment processes of TFLP and TFLP II display semi-long range dependence. We establish the sample path properties of TFLP and TFLP II. We further use a flexible framework of tempered fractional derivatives and integrals to develop the theory of stochastic integration with respect to TFLP and TFLP II, which may not be semimartingales depending on the value of the memory parameter and choice of marginal distribution.
MSC:
| 60G22 | Fractional processes, including fractional Brownian motion |
| 60G51 | Processes with independent increments; Lévy processes |
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
| 60H05 | Stochastic integrals |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 60F05 | Central limit and other weak theorems |
Keywords:
Lévy processes; fractional processes; statistical turbulence; anomalous diffusion; stochastic analysisReferences:
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