  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 characterization and
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.
