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 $\varepsilon$-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 is discussed. An application in the insurance domain is studied in detail. Finally, we present the simulated sample paths for the subordinators.