Fractional Lévy stable motion time-changed by gamma subordinator

Abstract

In this paper a new stochastic process is introduced by subordinating fractional Lévy stable motion (FLSM) with gamma process. This new process incorporates stochastic volatility in the parent process FLSM. Fractional order moments, tail asymptotic, codifference and persistence of signs long-range dependence of the new process are discussed. A step-by-step procedure for simulations of sample trajectories and estimation of the parameters of the introduced process are given. Our study complements and generalizes the results available on variance-gamma process and fractional Laplace motion in various directions, which are well studied processes in literature.

Keywords:

• fractional Lévy stable motion
• gamma process
• symmetric Lévy stable motion
• subordination

Acknowledgments

The authors would like to acknowledge a support of NCN OPUS Grant No. UMO-2016/21/B/ST1/00929 “Anomalous diffusion processes and their applications in real data modelling”. The authors are grateful to the reviewers for several helpful comments and suggestions, which have led to improvements in the paper.

Additional information

Funding

The authors would like to acknowledge a support of NCN OPUS Grant No. UMO-2016/21/B/ST1/00929 “Anomalous diffusion processes and their applications in real data modelling”. The authors are grateful to the reviewers for several helpful comments and suggestions, which have led to improvements in the paper