Ornstein-Uhlenbeck (OU) type processes driven by Levy noise are useful in modeling the activity time in the fractal activity time geometric Brownian motion (FATGBM) model for a risky asset. Discrete superpositions of these processes can be constructed to incorporate non-Gaussian marginal distributions and long or short range dependence. While the partial sums of finite superpositions of OU type processes obey the central limit theorem, we show that the partial sums of a large class of infinite long range dependent superpositions are intermittent. We discuss the property of intermittency and behavior of the cumulants for the long-range dependent superpositions of OU type processes. In addition we show an application of finite superpositions in modeling financial time series and superiority of the model at hand compared to the Black-Scholes model when modeling log-returns.
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