Tempered fractional Poisson processes and fractional equations with Z-transform
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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Frequently Asked Questions (16)
Q2. Why do the authors feel a strong motivation to study these processes?
The authors feel a strong motivation to study these processes since tempering introduces a finite moment condition in space-fractional Poisson process.
Q3. what is the tempered 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).
Q4. How did they introduce the space-fractional Poisson process?
[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.
Q5. What is the Laplace transform for the Caputo tempered fractional derivative?
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.
Q6. what is the lt of a Riemann-Liouville fractional?
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.
Q7. what is the lt of the function g(t)?
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).
Q8. What is the inverse tempered stable subordinator?
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]).
Q9. What is the author’s affiliation with the FEDER?
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).
Q10. what is the Riemann-Liouville tempered fractional derivative?
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).
Q11. what is the state probability of a tsfp?
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.
Q12. What is the inverse of the z-transform?
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)).
Q13. what is the density f(x, 1) in terms of infinite series?
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).
Q14. What are the main properties of z-transform?
The following operational properties of z-transform are used further for the solution of initial value problem involving difference equations
Q15. What is the tempered space-fractional Poisson process?
One can also define tempered space-fractional Poisson process (TSFPP) by subordinating homogeneous Poisson process with the tempered stable subordinator.
Q16. what is the state probabilities of time-space-fractional Poisson process?
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).