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Stationary Increment Tempered Fractional Lévy Processes (TFLP) introduced by Boniece  et al.  [On fractional Lévy processes: Tempering, sample path properties and stochastic integration.  J. Stat. Phys. , 2020,  178 , 954–985] are applied to financial data. They are used to model the stochastic drift rate of a mean reverting equation. The new processes are called OU processes with a TFLP drift rate. Expressions for the characteristic functions, variance, skewness and kurtosis at arbitrary horizons are developed. Estimations are conducted for daily return data on ten underlying assets. It is observed that the processes may be consistent with high levels of excess kurtosis in perpetuity. Enquiring further into the possible source of the excess kurtosis it is observed that stochastic drifts that are highly fluctuating with strong mean reversion towards them can generate excess kurtosis at all horizons. Such features may well be characteristic of financial markets and provide an explanation for the persistence of excess kurtosis that has already been documented in the literature. Extensions to matrix tempered multivariate fractional Lévy processes are also considered with estimations reported in the bivariate case.

Stationary increments reverting to a Tempered Fractional Lévy Process (TFLP)