Lévy processes with jumps governed by lower incomplete gamma subordinator and its variations

In this paper, we study the Lévy process time-changed by independent Lévy subordinators, namely, the incomplete gamma subordinator, the $\epsilon$-jumps incomplete gamma subordinator and tempered incomplete gamma subordinator. We derive their important distributional properties such as mean, variance, correlation, tail probabilities and fractional moments. The long-range dependence property of these processes are discussed. An application in insurance domain is studied in detail. Finally, we present the simulated sample paths for the subordinators.