Non-Homogeneous Generalized Fractional Skellam Process

TL;DR

This paper introduces the Non-homogeneous Generalized Skellam process and its fractional version, analyzing their distributional properties, dependence structures, differential equations, and applications to financial data.

Contribution

It presents the first comprehensive study of NGSP and NGFSP, including new properties, recurrence relations, and an application to high-frequency financial data.

Findings

Derived probability generating functions and pmfs.

Established dependence and correlation structures.

Applied models to real financial data.

Abstract

This paper introduces the Non-homogeneous Generalized Skellam process (NGSP) and its fractional version NGFSP by time changing it with an independent inverse stable subordinator. We study distributional properties for NGSP and NGFSP including probability generating function, probability mass function (p.m.f.), factorial moments, mean, variance, covariance and correlation structure. Then we investigate the long and short range dependence structures for NGSP and NGFSP, and obtain the governing state differential equations of these processes along with their increment processes. We obtain recurrence relations satisfied by the state probabilities of Non-homogeneous generalized counting process (NGCP), NGSP and NGFSP. The weighted sum representations for these processes are provided. We further obtain martingale and renewal properties along with arrival time distribution for NGSP and NGFSP. An alternative version of NGFSP with a closed-form p.m.f. is introduced along with a discussion of its distributional and asymptotic properties. In addition, we study the running average processes of GCP and GSP which are of independent interest. The p.m.f. of NGSP and the simulated sample paths of GFSP, NGFSP and related processes are plotted. Finally, we discuss an application to a high-frequency financial data set pointing out the advantages of our model compared to existing ones.