Tempered stable Lévy motion driven by stable subordinator

•The subordinated tempered stable Lévy motion with inverse stable subordinator is examined.•We analyze the main statistical properties of the considered system.•We propose the parameter’s estimation procedure.•We analyze the ensemble averaged MSD as a tool of recognition between subdiffusive and diffusive models.•We examine real financial data by using the considered processes.

In this article we propose a new model for financial data description. Combining two independent mechanisms, namely the tempered stable process and inverse stable subordinator, we obtain a new model which captures not only the tempered stable character of the underlying data but also such a property as periods in which the values of an asset stay on the same level. Moreover, we classify our system to the family of subdiffusive processes and investigate its tail behavior. We describe in detail testing and estimation procedures for the proposed model. In the last step we calibrate our model to the real data.