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Journal ArticleDOI

Tempered fractional Poisson processes and fractional equations with Z-transform

17 Apr 2020-Stochastic Analysis and Applications (Taylor & Francis)-Vol. 38, Iss: 5, pp 939-957
TL;DR: In this article, the state probabilities of different types of space and time-fractional Poisson processes were derived using z-transform and shown to be similar to the state probability of time fractional poisson processes.
Abstract: In this article, we derive the state probabilities of different type of space- and time-fractional Poisson processes using z-transform. We work on tempered versions of time-fractional Poisson proce...

Summary (2 min read)

1 Introduction

  • In recent years fractional processes are getting increased attention due to their real life applications.
  • Moreover, they have shown that space-fractional Poisson process can also be obtained by time-changing the standard Poisson process with a stable subordinator.
  • Further, it gives more flexibility in modeling of natural phenomena discussed in [6], due to extra parameters which can be picked based on the situation.
  • Moreover, the authors suggest to use z-transform since the z-transform method is more general then method of probability generating functions, and hence it could be applied for solutions of fractional equations which are not probability distributions, see Section 3.6.

2 Preliminaries

  • The authors provide some basic definitions and results to be used further in subsequent sections.
  • The authors assume that there exists a region of convergence (ROC) such that the infinite-series (2.1) converges in ROC.
  • The existence of the inverse imposes restrictions on f(k) for the uniqueness.

3 Fractional Poisson Processes

  • The authors revisit space- and time-fractional Poisson processes using the z-transform approach.
  • Note that z-transform is more general than the probability generating function approach and can be used to solve the difference-differential equations where the solution may not be a probability distribution.
  • Also, the authors introduce and study tempered space-timefractional Poisson processes.
  • Further, to show the importance of z-transform, the authors consider Gegenbauer type fractional difference equations.

3.1 The Time-Fractional Poisson Process

  • The time-fractional Poisson process (TFPP) was first introduced by Laskin (2003) (see [3]).
  • Further, the marginals of TFPP are not infinitely divisible [14].
  • Recently, Aletti et al. (2018) established that TFPP is a martingale with respect to its natural filtration [15].

3.2 The Space-Fractional Poisson Process

  • The identity (3.38) is used in subsequent section.
  • The iterated composition S(n)(t) is also a stable subordinator with parameter α1α2 · · ·αn.

3.3 The Time-Space-Fractional Poisson Process

  • Note that [4] introduced the following time-space-fractional differential equations dβ dtβ.
  • The state probabilities of time-space-fractional Poisson process defined in (3.46) satisfies the equation (3.40).
  • Note that the two z-transforms given in (3.45) and (3.47) are same and hence two representations are equivalent by the uniqueness of z-transform.

3.4 The Tempered Space-Fractional Poisson Process

  • One can also define tempered space-fractional Poisson process by subordinating homogeneous Poisson process with the tempered stable subordinator.
  • Note that tempered stable subordinators are obtained by exponential tempering in the distribution of stable subordinator, see [20] for more details on tempering stable processes.
  • However the integer order moments of SFPP are not finite.
  • The authors have following proposition for the state probabilities of TSFPP.

3.5 The Tempered Time-Space-Fractional Poisson Process

  • The authors introduce and study tempered time-space-fractional Poisson process .
  • Note that this process is non-Markovian due to the time-change component of Yβ,ν(t), which is not a Lévy process.

3.6 Fractional Equation with Gegenbauer Type Fractional Opera-

  • The authors introduce new class of fractional differential equations and their solutions.
  • One can say that Nud (t) is a defective random variable which indicates that there is some positive mass concentrated at ∞.

4 Conclusion

  • The authors introduce and study tempered time-space-fractional Poisson processes, which may provide more flexibility in modeling of real life data.
  • Further, the authors argue that z-transform is more useful than the PGF in solving the difference-differential equations since it is more general and hence may be used in the situations where the solution is not a probability distribution indeed.
  • To support this, the authors work with the Gegenbauer type fractional shift operator.
  • N. Leonenko was supported in particular by Australian Research Council’s Discovery Projects funding scheme (project DP160101366)and by project MTM201571839-P of MINECO, Spain (co-funded with FEDER funds), also known as Acknowledgments.
  • NG would like to thank Council of Scientific and Industrial Research (CSIR) India, for the award of a research fellowship.

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Poi s so n p r o c e s s e s a n d fr ac tio n al e q u a tio ns wi th Z-t r a n sfo r m. S t oc h a s ti c
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h old e r s.

Tempered Fractional Poisson Processes and Fractional
Equations with Z-Transform
Neha Gupta
a
, Arun Kumar
a
, Niko l ai Leo n en ko
b
a
Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, Punjab - 140001, India
b
Cardiff School of Mathemat ic s, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK
Abstract
In this article, we derive the state probabilities of different type of space- and time-fracti ona l
Poi sso n processes using z-transform. We work on tempered versions of time- fr ac ti o n a l Poisson
process and space-fractional Poisson processes. We also introduce Gegenbauer type fractional
differential equations and their solutions using z-transform. Our r esu l t s general i ze and com-
plement the results available on fr a ct i ona l Poisson pr ocesses in several directions.
Key Words: Fractional Poisson process, Z-transform, inverse tempered stable subordinator,
fractional derivatives.
1 Introduction
In recent years fractional processes ar e getting increased att ention due to their real life applica-
tions. For exam p l e fractional Brownian motion (FBM) overcome the limitation s of Brownian
motion in modeling of long-range dependent p h en o m en a occurring in financi a l time series,
Nile ri ver data and fractal analysis etc (see e.g., [1]). Similarly time-fractiona l Poisson process
is h el p fu l in modeling of counting processes where the inter-arrival times are heavy tailed or
arrivals are delayed (see e.g., [2, 3]). In time-fractional Poisson process the waiting times
are Mittag-Leffler (ML) distributed see [3]. Recently, [4] introduced space-fraction a l Pois-
son process by taking a fractional shift operator in place of an integer shift operator in the
governing differential-difference equation of standard Poisson process. Moreover, t h ey have
shown that space-fraction a l Poisson process can also be obtained by time-changing the stan-
dard Poisson process with a stable subordinator. Further, they argue that time-fractional
Poi sso n process and the space-fractional Poisson process are specific cases of the same gener-
alized complete model and hence might be useful in the study of tr a n sport of charge carriers
in semiconductors [5] or applications related to fractional quantum mechanics [6]. In this
article, we extend the space-fractional and time-fractional Poisson p r ocess by considering a
1

tempered time-space-frac ti o n a l Poisson process. We feel a strong motivat i o n to study th ese
processes since tempering introduces a finite moment condition in space-fractional Poisson
process. Further, it gives more flexibility in modeling of natural phenomena discussed in [6],
due to extra parameters which can be picked based on the sit u at io n . Moreover, we suggest
to use z-tr an sf or m since the z-transform method i s more general then method of probabil i ty
generating functio n s, and hence it could be appli ed for s ol u t i o n s of fract i on a l equations which
are not probabil i ty distribution s, see S ect i on 3.6. The governing equations for marginal distri-
butions of Poi sso n and Skellam processes time-changed by inverse subordinators are discussed
in [ 7 ] . For properties of Poisson processes directed by compound Poisson-Gamma subordina-
tors (see [8]).
The rest of the paper is organized as follows. In Section 2, we introduce the z-transform
and the inverse z-transform by indicating their main characteristics. In this section Caputo-
Djrbashian fra ct i o n al derivative, Capu t o tempered fractional derivative and the Riemann-
Liouville tempered fractional derivative are also discussed. Further, main properties of Poisson
process are discussed briefly. Section 3 is devoted to different ki n d o f fract i o n al Poisson
processes. In this section first we rev i si t the time- and space-fractional Poisson processes with
z-transform approach. Our main resu l t s are given in Sectio n s 3.4, 3.5 a n d 3.6. The last section
concludes.
2 Preliminaries
In th i s section, we provide some basic definitions and results to be used further in subsequent
sections.
2.1 The Z-Transform and Its Inverse
The z-transform is a lin ear transformation and can be considered a s an operator, mapping
sequence of scalars into functions of complex variable z. For a function f(k), k Z, th e
bilateral z-transform is defined by (see e.g., [9])
F (z) = Zf(k) =
X
k=−∞
f(k)z
k
, z C. (2.1)
We assume that th er e exists a region of convergence (ROC) such that the infinite-series (2.1)
converges in ROC. Alter n ati vely, i n case where f(k) is defined only for k 0, the (unil at era l )
z-transform is defined as
F (z) = Zf(k) =
X
k=0
f(k)z
k
, z C,
2

where the coefficient of z
k
in this expansion is an inverse given by
f(k) = Z
1
(F (z)). (2.2)
The inverse z-transform is also defined by the complex integral
Z
1
{F (z)} = f(k) =
1
2πi
I
C
F (z)z
k1
dz,
where C is si mp l e closed contour enclosing the origin a n d lying outside the circle |z| = R. The
existence of the inverse imposes rest r i cti o n s on f(k) for the uniqueness. If f(k), k N {0},
is probability distribution, that is, f(k) 0 an d
X
k=0
f(k) = 1,
then the probability generating function (PGF) is defined by
G(u) =
X
k=0
u
k
f(k), |u| 1,
and relates to unilateral z-transform as foll ows G(z
1
) = F (z). The following operational
properties of z-transform are used further for the solution of initial value problem involving
difference equations
Zf(k) = F (z),
Z(f(k m )) = z
m
[F (z) +
1
X
r=m
f(r)z
r
], (2.3)
Z(f(k + m) ) = z
m
[F (z)
m1
X
r=0
f(r)z
r
], m 0. (2.4)
2.2 Fractional Derivatives
The Caputo-Djrbashain (CD) fractional derivative of order β (0, 1], for a function g(t), t 0
is defined as
d
β
dt
β
g(t) =
1
Γ(1 β)
Z
t
0
dg(τ)
(t τ)
β
, β (0, 1 ] . (2.5)
Note that the classes of functions for which the CD derivative is well defined is discussed in
([10], Section s 2.2, 2.3). The Laplace transform (LT) of CD fraction al derivative is given by
(see e.g., [10], p.39)
L
d
β
dt
β
g(t)
=
Z
0
e
st
d
β
dt
β
g(t)dt = s
β
˜g(s) s
β1
g(0
+
), 0 < β 1,
3

where ˜g(s) is the LT of the function g(t), t 0, such that
L(g(t)) = ˜g(s) =
Z
0
e
st
g(t)dt.
The Riemann-Liouville tempered fractional derivative is defined by (see [11])
D
β
t
g(t) = e
νt
D
β
t
[e
νt
g(t)] ν
β
g(t),
where
D
β
t
g(t) =
1
Γ(1 β)
d
dt
Z
t
0
g(u)du
(t u)
β
is the u su a l Riemann-Liouville fractional derivative of order β (0, 1). Furth er , the LT of
Riemann-Liouville tempered fractional derivative is
L
h
D
β
t
g(t)
i
(s) = ((s + ν)
β
ν
β
)˜g(s). (2.6)
The Caputo tempered fractiona l der i vative is defined by (see [11])
d
β
dt
β
g(t) = D
β
t
g(t)
g(0)
Γ(1 β)
Z
t
e
νr
βr
β1
dr. (2.7)
The Laplace tra n sfo r m for the Caputo tempered fractional derivative for a fun ct i on g(t) sat-
isfies
L
d
β
dt
β
g
(s) = ((s + ν)
β
ν
β
)˜g(s) s
1
((s + ν)
β
ν
β
)g(0). (2.8)
2.3 Poisson Process
The homogeneous Poisson process N(t), t 0, with parameter λ > 0 is defined as,
N(t) = max{n : T
1
+ T
2
+ . . . + T
n
t}, t 0 ,
where the inter-arrival times T
1
, T
2
, . . . , T
n
are non-negati ve iid exponential random variables
with mean 1. The probability mass function (PMF) P (k, t) = P(N(t) = k) is given by
P (k, t) =
e
λt
(λt)
k
k!
, k = 0, 1, 2, . . . (2.9)
The PMF of the Poisson pr ocess satisfies the differential-difference equati on o f the form
d
dt
P (k, t) = λ(P (k, t) P (k 1, t)) = λP (k, t) (2.10)
with initial conditions
P (k, 0) = δ
k,0
=
0, k 6= 0,
1, k = 0.
(2.11)
Further, by definition P (l, t) = 0, l > 0 and (1 B) with B a s the backward shift
operator, i.e. B{P (k, t)} = P (k 1, t).
4

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TL;DR: In this article, the Skellam process of order k and its running average was introduced and the marginal probabilities, Levy measures, governing difference-differential equations of the introduced processes were derived.
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References
More filters
Book
01 Jan 1994
TL;DR: Theorems of Stationary Processes with Long Memory Limit Theorems and Estimations of Long Memory-Heuristic Approaches, Forecasting Regression Goodness of Fit Tests, and Robust Estimation of Long memory estimates are presented.
Abstract: Statistical Methods for Long Term Memory Processes covers the diverse statistical methods and applications for data with long-range dependence. Presenting material that previously appeared only in journals, the author provides a concise and effective overview of probabilistic foundations, statistical methods, and applications. The material emphasizes basic principles and practical applications and provides an integrated perspective of both theory and practice. This book explores data sets from a wide range of disciplines, such as hydrology, climatology, telecommunications engineering, and high-precision physical measurement. The data sets are conveniently compiled in the index, and this allows readers to view statistical approaches in a practical context. Statistical Methods for Long Term Memory Processes also supplies S-PLUS programs for the major methods discussed. This feature allows the practitioner to apply long memory processes in daily data analysis. For newcomers to the area, the first three chapters provide the basic knowledge necessary for understanding the remainder of the material. To promote selective reading, the author presents the chapters independently. Combining essential methodologies with real-life applications, this outstanding volume is and indispensable reference for statisticians and scientists who analyze data with long-range dependence.

3,566 citations

MonographDOI
TL;DR: In this paper, the authors introduce sample path properties such as boundedness, continuity, and oscillations, as well as integrability, and absolute continuity of the path in the real line.
Abstract: Stable random variables on the real line Multivariate stable distributions Stable stochastic integrals Dependence structures of multivariate stable distributions Non-linear regression Complex stable stochastic integrals and harmonizable processes Self-similar processes Chentsov random fields Introduction to sample path properties Boundedness, continuity and oscillations Measurability, integrability and absolute continuity Boundedness and continuity via metric entropy Integral representation Historical notes and extensions.

2,611 citations

BookDOI
01 Jan 1997
TL;DR: Panagiotopoulos, O.K.Carpinteri, B. Chiaia, R. Gorenflo, F. Mainardi, and R. Lenormand as mentioned in this paper.
Abstract: A. Carpinteri: Self-Similarity and Fractality in Microcrack Coalescence and Solid Rupture.- B. Chiaia: Experimental Determination of the Fractal Dimension of Microcrack Patterns and Fracture Surfaces.- P.D. Panagiotopoulos, O.K. Panagouli: Fractal Geometry in Contact Mechanics and Numerical Applications.- R. Lenormand: Fractals and Porous Media: from Pore to Geological Scales.- R. Gorenflo, F. Mainardi: Fractional Calculus: Integral and Differential Equations of Fractional Order.- R. Gorenflo: Fractional Calculus: some Numerical Methods.- F. Mainardi: Fractional Calculus: some Basic Problems in Continuum and Statistical Mechanics.

1,389 citations

Book
11 Nov 2014
TL;DR: In this article, the authors present a self-contained, comprehensive treatment of the theory of the Mittag-Leffler functions, ranging from rather elementary matters to the latest research results, treating various situations and processes in viscoelasticity, physics, hydrodynamics, diffusion and wave phenomena.
Abstract: As a result of researchers and scientists increasing interest in pure as well as applied mathematics in non-conventional models, particularly those using fractional calculus, Mittag-Leffler functions have recently caught the interest of the scientific community. Focusing on the theory of the Mittag-Leffler functions, the present volume offers a self-contained, comprehensive treatment, ranging from rather elementary matters to the latest research results. In addition to the theory the authors devote some sections of the work to the applications, treating various situations and processes in viscoelasticity, physics, hydrodynamics, diffusion and wave phenomena, as well as stochastics. In particular the Mittag-Leffler functions allow us to describe phenomena in processes that progress or decay too slowly to be represented by classical functions like the exponential function and its successors. The book is intended for a broad audience, comprising graduate students, university instructors and scientists in the field of pure and applied mathematics, as well as researchers in applied sciences like mathematical physics, theoretical chemistry, bio-mathematics, theory of control and several other related areas.

882 citations

Frequently Asked Questions (16)
Q1. What are the contributions in this paper?

In this article, the authors derive the state probabilities of different type of spaceand time-fractional Poisson processes using z-transform. The authors also introduce Gegenbauer type fractional differential equations and their solutions using z-transform. 

The authors feel a strong motivation to study these processes since tempering introduces a finite moment condition in space-fractional Poisson process. 

The tempered space-fractional Poisson process is defined byNα,µ(t) = N(Sα,µ(t)), α ∈ (0, 1), µ ≥ 0, (3.52)where homogeneous Poisson process N(t) is independent of the tempered stable subordinator Sα,µ(t). 

[4] introduced space-fractional Poisson process by taking a fractional shift operator in place of an integer shift operator in the governing differential-difference equation of standard Poisson process. 

Note that z-transform is more general than the probability generating function approach and can be used to solve the difference-differential equations where the solution may not be a probability distribution. 

The Laplace transform (LT) of CD fractional derivative is given by (see e.g., [10], p.39)L(dβdtβ g(t))=∫∞0e−st dβdtβ g(t)dt = sβ g̃(s)− sβ−1g(0+), 0 < β ≤ 1,where g̃(s) is the LT of the function g(t), t ≥ 0, such thatL(g(t)) = g̃(s) =∫∞0e−stg(t)dt. 

The probability mass function (PMF) P (k, t) = P(N(t) = k) is given byP (k, t) = e−λt(λt)kk! , k = 0, 1, 2, . . . (2.9)The PMF of the Poisson process satisfies the differential-difference equation of the formddt P (k, t) = −λ(P (k, t)− P (k − 1, t)) = −λ∇P (k, t) (2.10)with initial conditionsP (k, 0) = δk,0 = 0, k 6= 0, 1, k = 0. (2.11)Further, by definition P (−l, t) = 0, l > 0 and ∇ ≡ (1 − B) with B as the backward shift operator, i.e. B{P (k, t)} = P (k − 1, t). 

Taking ν = 0 in (3.68) and using (3.65), it follows− ∂∂x hβ,0(x, t) =∂β∂tβ hβ,0(x, t) +t−βΓ(1− β) δ(x), (3.69)which is the governing equation of the density function of inverse β-stable subordiantor, which complements the result obtained in literature (see e.g., [25, 26]). 

Acknowledgments: N. Leonenko was supported in particular by Australian Research Council’s Discovery Projects funding scheme (project DP160101366)and by project MTM201571839-P of MINECO, Spain (co-funded with FEDER funds). 

The Riemann-Liouville tempered fractional derivative is defined by (see [11])D β,ν t g(t) = e −νt D β t [e νtg(t)]− νβg(t),whereD β t g(t) =1Γ(1− β)ddt∫ t0g(u)du(t− u)βis the usual Riemann-Liouville fractional derivative of order β ∈ (0, 1). 

The state probabilities for TSFPP are given byPα,µ(k, t) = (−1)ketµ α∞ ∑m=0µmλαr−m ∞ ∑r=0(−t)rr!(αr m)(αr −mk), k ≥ 0, µ ≥ 0, t ≥ 0. (3.54)Proof. 

in case where f(k) is defined only for k ≥ 0, the (unilateral) z-transform is defined asF (z) = Zf(k) = ∞ ∑k=0f(k)z−k, z ∈ C,where the coefficient of z−k in this expansion is an inverse given byf(k) = Z−1(F (z)). 

The space-fractional Poisson process (SFPP) Nα(t), t ≥ 0, 0 < α < 1 was introduced by [4], as followsNα(t) = N(Sα(t)), t ≥ 0, 0 < α < 1, N(t), t ≥ 0, α = 1. (3.23)The density f(x, 1) of Sα(1) is infinitely differentiable on (0,∞), with the asymptotics given by (see e.g., [18])f(x, 1) ∼ (α x )2−α 2(1−α)√ 2πα(1− α) e−(1−α)(x α ) −α 1−α, as x → 0; (3.24)f(x, 1) ∼ αΓ(1− α)x1+α , as x → ∞. (3.25)Exact forms of the density f(x, 1) in terms of infinite series or integral are discussed in (see e.g. [19], p. 583) and has the following infinite-series representationf(x, 1) = 1π∞ ∑k=1(−1)k+1 Γ(kα + 1)k!1xkα+1 sin (παk) , x > 0. (3.26)Note that from (3.24) and (3.25), the authors havelim x→0 f(x, 1) = f(0, 1) = 0 and lim x→∞ f(x, 1) = f(∞, 1) = 0. (3.27)The PGF of this process isGα(u, t) = EuN α(t) = e−λ α(1−u)αt, |u| ≤ 1, α ∈ (0, 1). 

The following operational properties of z-transform are used further for the solution of initial value problem involving difference equations 

One can also define tempered space-fractional Poisson process (TSFPP) by subordinating homogeneous Poisson process with the tempered stable subordinator. 

one can define the time-space-fractional Poisson process (TSFPP) as followsNαβ (t) = N(Sα(Yβ(t)) = N α(Yβ(t)), t ≥ 0, (3.46)where TSFPP is obtained by subordinating the standard Poisson process N(t) by an independent α-stable subordinator Sα(t) and then by the inverse β-stable subordinator Yβ(t).