Stable Lévy process delayed by tempered stable subordinator

Accepted Manuscript Stable Lévy process delayed by tempered stable subordinator J. Gajda, A. Kumar, A. Wyłoma´nska

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S0167-7152(18)30307-9 https://doi.org/10.1016/j.spl.2018.09.008 STAPRO 8330

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Statistics and Probability Letters

Received date : 23 June 2017 Revised date : 7 September 2018 Accepted date : 19 September 2018 Please cite this article as: Gajda J., et al., Stable Lévy process delayed by tempered stable subordinator. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.09.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Stable L´ evy Process Delayed by Tempered Stable Subordinator J. Gajdaa A. Kumarb , and A. Wyloma´ nskac a

Department of Statistics and Econometrics, Faculty of Economic Sciences, University of Warsaw, Dluga 44/50 00-241 Warsaw, Poland b Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, Punjab - 140001, India c Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wybrze˙ze Wyspia´ nskiego 27, 50-370 Wroclaw, Poland

Abstract We consider symmetric stable L´evy motion time-changed by tempered stable subordinator. This process generalizes the normal inverse Gaussian process without drift term, introduced by Barndorff-Nielsen. The asymptotic tail behavior of the density function of this process and corresponding L´evy density is obtained. The governing Fokker-Planck-Kolmogorov equation of the density function of the introduced process in terms of shifted fractional derivative is established. Codifference and asymptotic behavior of the moments are discussed. Further, we also introduce and analyze stable subordinator delayed by tempered stable subordinator.

Key words: Stable L´evy motion; tempered stable subordinator; subordination, Fokker-Planck-Kolmogorov equation.

1

Introduction

In recent years the time-changed stochastic processes have found many interesting applications (Gabaix et al., 2003, Stanislavsky et al., 2008). In general, the time-changed stochastic process is obtained by changing the time of a process (outer process) by some other process (inside process) which plays the role of random model of time. They are also called as the subordinated or delayed processes. The motivation to introduce time-changed processes comes from the fact that they provide a convenient way to get a stochastic model where one required to have some properties of the outer process but at the same time it is required to change some characteristics. The idea of subordination was proposed by Bocher in 1949 (Bochner, 1949). One of the well known time-changed process is the normal inverse Gaussian (NIG) process proposed by Barndorff-Nielsen (Barndorff-Nielsen, 1998). This process arises from the classical Brownian motion with drift time-changed by inverse Gaussian process. The NIG process was proposed to capture stochastic volatility in financial timeseries. In the literature one can find other examples of time-changed processes, like variance Gamma process (Madan et al., 1998) or fractional Brownian motion time-changed by inverse Gaussian process (Wyloma´ nska et al., 2016). In this paper we introduce the symmetric stable L´evy motion time-changed by tempered stable subordinator. In general, subordinator is a non-decreasing, non-negative L´evy process (see Applebaum, 2009, p.52). The proposed time-changed process is a generalization of the NIG process. This comes from the fact that symmetric stable L´evy motion is a generalization of the classical Brownian motion while inverse Gaussian process is a special case of the tempered stable one. Our results generalize the driftless normal inverse Gaussian (NIG) process (Barndorff-Neilsen, 1998) as well as the driftless normal tempered stable L´evy process (see BarndorffNeilsen and Shephard, 2001), which is a generalized version of NIG process. 1

Stable distributions are an important class of probability distributions. They possess heavy tails and have many intriguing mathematical properties, like self-similarity. Despite having interesting properties the applications of these processes are limited some time since even the mean is not finite for some values of stable parameters. To overcome these limitations tempered stable processes are introduced (see e.g. Rosinski, 2007). Tempered stable processes are infinitely divisible, have exponentially decaying tail probabilities and have all moments finite. These properties are obtained on the cost of self-similarity. In this paper we present main properties of the stable L´evy process time-changed by tempered stable subordinator, such as Fourier transform, tail behavior, fractional moments behavior or asymptotics of L´evy density. Moreover we concentrate also on the Fokker-Planck-Kolmogorov type governing equations for probability density function and structure of dependence of the new time-changed process. Because the NIG process, which is a special case of the stable L´evy process delayed by tempered stable subordinator, was applied to financial data description, the new process seems to be also applicable in this area. Many financial data exhibit nonGaussian behavior and heavy-tailed-based processes seem to be more appropriate in this case. The obtained theoretical results can be useful in the estimation problem of the unknown parameters or statistical inference if the proposed time-changed process can be considered as the appropriate one for given data. As the second model we consider also stable subordinator delayed by non-decreasing, non-negative tempered stable L´evy process. The rest of the paper is organized as follows. In Section 2, we introduce stable L´evy motion and tempered stable subordinator (TSS) and indicate their main characteristics. In Section 3, the stable L´evy motion delayed by TSS is defined and its main properties are discussed. Section 4 is devoted to the stable subordinator delayed by TSS.

2

Stable L´ evy motion and TSS

In this section we remind the definition of the stable distributions and processes as well as the tempered stable subordinator (TSS). Moreover, we present main properties of introduced process. The class of stable distributions is governed by four parameters. The family is denoted by S(α, γ, µ, σ), where the first parameter α ∈ (0, 2] is the index of stability, γ ∈ [−1, 1] is the skewness parameter, µ ∈ R is the location parameter and σ > 0 is the shape parameter. Apart from the Gaussian case (α = 2), the Cauchy case (α = 1, γ = 0) and the L´evy case (α = 1/2, γ = ±1) the stable distributions do not posses closed form probability density function (PDF). Mostly stable distributions are represented in terms of their characteristic functions (or Fourier Transforms). A random variable X is said to follow stable distribution with parameters α, γ, µ and σ if its characteristic function φ(u) satisfies (Samorodnitsky and Taqqu, 1994)  −σ α |u|α [1 − iγsign(u) tan( πα )] + iµu, if α 6= 1
2 ln φ(u) = ln E eiXu = (2.1) −σ|u|[1 + iγsign(u) 2 ln |u|] + iµu, if α = 1. π

The stable distributed random variable with the stability parameter α is also called α−stable. In this section we consider the special symmetric case, namely γ = µ = 0. The L´evy process corresponding to the stable distribution is called stable L´evy motion. A stable L´evy motion Sα (t) has stationary independent increments with Sα (t) − Sα (s) (t > s) having stable distribution with parameters α ∈ (0, 2], γ = 0, µ = 0 and σ = (|t − s|)1/α . Further, the characteristic function of Sα (t) (Samorodnitsky and Taqqu, 1994) is given by   α E eiuSα (t) = e−t|u| , α ∈ (0, 2], t > 0. (2.2) 2

For large x right tail of the stable L´evy motion for 0 < α < 2 behaves (Samorodnitsky and Taqqu, 1994) as P(Sα (t) > x) ∼ Cα tx−α , where Cα =

 

(1−α) , 2Γ(2−α) cos( πα 2 )

if α 6= 1

π

if α = 1.

1,

(2.3)

(2.4)

In the next parts of the papers we use the term subordinator. In general, the subordinator it is a non-negative, non-decreasing process (Applebaum, 2009). One can find in the literature many examples of subordinators. Below we introduce the tempered stable subordinator. Let Tλ,β (t) be the TSS with index β ∈ (0, 1) and tempering parameter λ. TSS are obtained by exponential tempering in the distribution of stable processes (see e.g. Rosinski, 2007). TSS Tλ,β (t) has density function β

gλ,β (x, t) = e−λx+λ t fβ (x, t), λ > 0, β ∈ (0, 1),

(2.5)

where fβ (x, t) is the PDF of a β-stable subordinator. Further, the Laplace transform (LT) of gλ,β (x, t) (see Meerschaert et al., 2013) is
  Z ∞ β β Lx gλ,β (x, t) = e−sx gλ,β (x, t)dx = e−t (s+λ) −λ , s ≥ 0. (2.6) 0

The advantage of tempered stable process over a stable process is that its all moments exist and its density is also infinitely divisible. A driftless subordinator D(u) with L´evy density πD and density function f has R∞ the L´evy-Khinchin representation (see e.g. Bertoin, 1996; Sato, 1999), 0 e−st fD(x) (t)dt = e−xΨD (s) , where R∞ ΨD (s) = 0 (1 − e−su )πD (du), is the Laplace exponent. The L´evy density corresponding to a tempered β-stable subordinator is given by (see e.g. Cont and Tankov, 2004, p. 115) πTλ,β (x) =

cβ e−λx , x > 0, xβ+1

(2.7)

R∞ where cβ = β/Γ(1 − β). Further, 0 πTλ,β (x) = ∞ which leads to the conclusion that the sample paths of Tλ,β (t) are strictly increasing with jumps by an application of Theorem 21.3 of Sato (1999). The tail probability of TSS has the following asymptotic behavior P(Tλ,β (t) > x) ∼ d(β, λ, t)

e−λx , as x → ∞, xβ

(2.8)

β

t where d(β, λ, t) = βπ Γ(1 + β) sin(πβ)eλ t . The first two moments of the TSS can be easily obtained with the help of (2.6) and are given by

E(Tλ,β (t)) = βλβ−1 t, E(Tλ,β (t))2 = β(1 − β)λβ−2 t + (βλβ−1 t)2 .

(2.9)

Using (2.9) and with the help of the formula E(XY ) = 12 (E(X 2 ) + E(Y 2 ) − E(X − Y )2 ), we obtain that the autocovariance function for the Tλ,β (t) process is given by Cov(Tλ,β (t), Tλ,β (s)) = β(1 − β)λβ−2 min(t, s).

3

Stable L´ evy motion delayed by TSS

In this section, the stable L´evy motion time-changed with TSS is introduced and its properties are discussed. This process generalizes the well known normal inverse Gaussian (NIG) process introduced by BarndorffNielsen where the NIG process is obtained by time-changing the Brownian motion with drift by an inverse Gaussian process. For α = 2 and β = 1/2, the studied process is a drfitless NIG process (Barndorff-Nielsen, 1998). In this case λ is the parameter. 3

Definition 3.1 (Stable L´evy motion delayed by TSS). Let Sα (t) be a symmetric α-stable process, 0 < α ≤ 2, and Tλ,β (t) be the TSS independent of Sα (t). Then we define a new process Xα,λ,β (t) as follows Xα,λ,β (t) := Sα (Tλ,β (t)), t ≥ 0.

(3.10)

In this case the Sα (t) is called the outside process while Tλ,β (t) - the inside process. The Fourier transform (FT) of Xα,λ,β (t) can be obtained as follows E(eiuXα,λ,β (t) ) = E(eiuSα (Tλ,β (t)) ) = E(E(eiuSα (Tλ,β (t)) |Tλ,β (t)))
α β β α = E(e−Tλ,β (t)|u| ) = e−t (|u| +λ) −λ (using (2.6)).

The PDF hα,λ,β (x, t) of Xα,λ,β (t) can be represented as Z ∞ hα,λ,β (x, t) = fα (x, r)gλ,β (r, t)dr,

(3.11)

(3.12)

0

where fα (·, ·) and gλ,β (·, ·) are the densities of stable L´evy motion and TSS, respectively. However, it is difficult to find the explicit form of the PDF of the introduced process but its asymptotic behavior can be obtained in closed form. Suppose Kν (ω) denotes the modified Bessel function of third kind with index ν, defined by (see e.g. Abramowitz and Stegun, 1992) Z 1 ∞ ν−1 − 1 ω(x+x−1 ) x e 2 dx, ω > 0. (3.13) Kν (ω) = 2 0

For α ∈ (0, 2], we have the following result for the asymptotic tail behaviour.

Proposition 3.1. The tail probability of the process Xα,λ,β (t) follows, as x → ∞ ( Cα βλβ−1 tx−α , for√ α ∈ (0, 2), P(Xα,λ,β (t) > x) ∼ −x 2λ d(β,λ,t) (2λ)β/2 e xβ , for α = 2, 2

(3.14)

where the constants Cα and d(β, λ, t) are defined in (2.4) and (2.8) respectively. Proof. For 0 < α < 2, using (2.3), (2.9) and standard conditioning argument, it follows P(Xα,λ,β (t) > x) = P(Sα (Tλ,β (t)) > x) = E(P(Sα (Tλ,β (t)) > x)|Tλ,β (t)) = E(Cα Tλ,β (t)x−α ) = Cα βλβ−1 tx−α , as x → ∞. For α = 2, the outside process is standard Brownian motion B(t). Hence for large x P(X2,λ,β (t) > x) = P(B(Tλ,β (t)) > x) = P(Tλ,β (t)1/2 B(1) > x) = E(P(Tλ,β (t)1/2 B(1) > x)|B(1)) = E(P(Tλ,β (t) > (x/B(1))2 )|B(1)) ∼ = = = =

2 2 d(β, λ, t) E(B(1)2β e−λx /B(1) ) (using (2.8)) 2β x Z   2 d(β, λ, t) ∞ 2β − 12 y2 + 2λx y2 √ y e dy x2β 2π 0 Z   2 d(β, λ, t) 1 ∞ β−1/2 − 12 u+ 2λx u √ u e du (substituting y 2 = u) x2β 2π 2 0 Z √ d(β, λ, t) √ β+1/2 1 ∞ β−1/2 − 1 x√2λ(w+ 1 ) w dw (substituting u = x √ (x 2λ) w e 2 2λw) 2β 2 0 x 2π √ d(β, λ, t) √ β+1/2 √ (x 2λ) Kβ+1/2 (x 2λ) (using (3.13)) 2β x 2π

√

d(β, λ, t) e−x 2λ (2λ)β/2 , ∼ 2 xβ 4

pπ

which follows using Kν (w) ∼

2e

−ω

w−1/2 as ω → ∞.

Remark 3.1. For α = 2,√ β = 1/2 the introduced process is a driftless NIG process and P(X2,λ,1/2 (t) > x) ∼ d(1/2,λ,t) (2λ)1/4 x−1/2 e−x 2λ as → ∞. Further, for α = 2 and λ ↓ 0, the process is a stable process of index 2 p √ 2β (see Mandelbrot and Taylor, 1967). In this case P(X2,0,β (t) > x) ∼ d(β,0,t) (2)β−1/2 β + 1/2x−2β , which 2π √ follows using (3.15) with the help of asymptotic behavior Kν (ω) ∼ ν2ν−1 ω −ν , ν > 0 for ω ↓ 0. Both results presented in the above Remark are known from the literature (see e.g. Barndorff-Nielsen, 1998; Mandelbrot and Taylor, 1967). Since the outside and inside processes are L´evy processes, the subordinated process is also L´evy. In next result, the L´evy density for the process Xα,λ,β (t) is obtained which in particular for α = 2 and β = 1/2 provide the L´evy density for NIG process. Proposition 3.2. The L´evy density for the process Xα,λ,β (t) is given by ( √ cβ = √2π ( √|y| )−β−1/2 K−β−1/2 (|y| 2λ), for α = 2 2λ ν(dy) β ∼ αcβ Cα Γ(−β) yλ1+α , for 0 < α < 2,

(3.15)

where cβ = β/Γ(1 − β). Proof. The L´evy density νX for a general subordinated process X(t) can be written as (see Huff, 1969) Z ∞ νX (dx) = f (x, t)νT (t)dt, (3.16) 0

where f (·, ·) is the PDF of the outer process and νT is the L´evy density for the inside process. For α = 2, the outside process is Gaussian and hence using (3.16) and (3.13) we obtain Z ∞ Z ∞ 1 − y2 cβ e−λt √ e 2t β+1 dt (using(2.7)) νX (dy) = f (y, t)νT (t)dt = t 2πt 0 0  −β− 12 Z ∞ Z ∞ √ 2 3 1 1 cβ |y| cβ −β−3/2 − 12 ( yt +2λt) √ =√ dt = √ w−β− 2 e− 2 |y| 2λ( w +w) dw t e 2π 0 2π 2λ 0  −β− 12 √ cβ |y| √ K−β−1/2 (|y| 2λ). (3.17) =√ 2π 2λ Further, for 0 < α < 2, νX (dy) =

Z

0

∞

f (y, t)νT (t)dt ∼

Z

0

∞

λβ αCα t cβ e−λt dt = c αC Γ(−β) . β α y 1+α tβ+1 y 1+α

Remark 3.2. For β = 1/2 and α = 2 the result is known in the literature (see Barndorff-Nielsen, 2000). Further, for α = 2, the L´evy density given in (3.17) complement the result available in (see eq. 5.3, p. 9) Barndorff-Neilsen and Shephard (2001). Next, the governing Fokker-Planck-Kolmogorov equation of the density function of the stable L´evy motion delayed by TSS is established. Theorem 3.1. For the process Xα,λ,β (t) defined in (3.10) the one dimensional PDF hα,λ,β (x, t) governs the fractional-type Fokker-Planck-Kolmogorov equation of the form  1/β ∂ λβ − hα,λ,β (x, t) = λhα,λ,β (x, t) − (−∆)α/2 (3.18) x hα,λ,β (x, t), ∂t 5

with the initial conditions hα,λ,β (0, t) = 0 and hα,λ,β (x, 0) = δ(x). Here the operator (−∆)a , a ∈ (0, 1) denotes
∂ 1/β fractional Laplacian, and λβ − ∂t is shifted fractional derivative defined in Beghin (2015) for analytic functions f : R → R and some c ∈ R as,

with

ν j




ν  ∞   X ν ∂ dj f (t) = (−1)j cν−j j f (x), c− ∂t j dx j=0

=

ν > 0,

Γ(ν+1) j!Γ(ν+1−j) .

Proof. Note that   1/β 1/β Z ∞ ∂ ∂ β β hα,λ,β (x, t) = fα (x, r) λ − gλ,β (r, t)dr λ − ∂t ∂t 0   Z ∞ ∂ fα (x, r) λgλ,β (r, t) + = gλ,β (r, t) dr ∂r 0 Z ∞ ∂ = λhα,λ,β (x, t) + fα (x, r) gλ,β (r, t)dr ∂r 0 Z ∞ ∂ = λhα,λ,β (x, t) + fα (x, r)gλ,β (r, t)|∞ − fα (x, r)gλ,β (r, t)dr r=0 ∂r 0 Z ∞ = λhα,λ,β (x, t) − (−∆)α/2 x fα (x, r)gλ,β (r, t)dr 0

= λhα,λ,β (x, t) − (−∆)α/2 x hα,λ,β (x, t).

Moreover, for special case of β parameter we obtain some known results. Remark 3.3. For β = 1/2, the densities h := hα,λ,1/2 (x, t) govern √ ∂h ∂2h −2 λ = (−∆)α/2 x h, 2 ∂t ∂t

(3.19)

which is the governing equation of stable time-changed by inverse Gaussian subordinator. For α = 2 equation (3.19) becomes the governing equation of driftless normal inverse Gaussian (NIG) process (see e.g. Kumar et √ 2 ∂2h al. 2011), that is given by ∂∂t2h − 2 λ ∂h ∂t = − ∂x2 which has a similar form as telegraph equation but it not since the coefficients of first-order time derivative and second order-space derivative are negative. Further, for 2 2 α = 2, β = 1/2 and λ = 0, h represents the density of a symmetric Cauchy process and satisfies ∂∂t2h = − ∂∂xh2 . Proposition 3.3. The marginals of Xα,λ,β (t) are self-decomposable. Proof. Tempered stable distributions are self decomposable (see e.g. Rosi´ nski, 2007). It is well known that if the subordinand U (t) is strictly stable and the subordinator V (t) is self-decomposable, then the subordinated process W (t) = U (V (t)) is self-decomposable, where U (t) and V (t) are independent (see Barndorff-Nielsen et al. 2001). Since the subordinand Sα (t) is symmetric stable it is strictly stable and the subordinator Tλ,β (t) is self-decomposable the subordinated process Xα,λ,β (t) is self-decomposable. Next the dependence structure of the considered process is investigated. Since underlying model for α ∈ (0, 2) does not possess finite second moments, one of the convenient way to study dependence is the codifference, alternative measure for infinite variance processes. The properties of the codifference and its application can be found for instance in (Wyloma´ nska et al. 2015), here only the definition is reminded. 6

Definition 3.2 (Codifference). The codifference for the infinitely divisible process X(t) is defined as (see e.g. Samorodnitsky and Taqqu, 1994) CD(X(t), X(s)) = log Eei(X(t)−X(s)) − log EeiX(t) − log Ee−iX(s) ,

t, s ≥ 0.

(3.20)

Using the above formula one can obtain closed form expression for the codifference for Xα,λ,β (t) process. Remark 3.4. The codifference of the process Xα,λ,β (t) defined in 3.10 is given by the following formula   β (3.21) CD(Xα,λ,β (s), Xα,λ,β (t)) = 2 min (t, s) (1 + λ) − λβ , which is equal to the codifference of the tempered stable subordinator.

The general result for moments of stable distribution is discussed in (Shanbhag and Sreehari, 1977). Using self-decomposability of symmetric stable distributions, it is established that for −1 < q < α < 2 ESαq (t) =

q 2q Γ( 1+q 2 )Γ(1 − α ) q/α t = cq,α tq/α . Γ(1 − 2q )Γ( 12 )

For r > 0, the r-th order moment for tempered stable subordinator has following asymptotic behavior (see e.g. Kumar et al., 2017) E(Tλ,β (t))r ∼ (βλβ−1 t)r , as t → ∞. (3.22) Using self-similarity of stable process and standard conditioning argument for q > 0 we obtain E(Xα,λ,β (t))q = E(Tλ,β (t)q/α Sα (1)q ) = E(Tλ,β (t)q/α )E(Sα (1)q ) ∼ cq,α (βλβ−1 t)q/α , as t → ∞.

4

Stable subordinator delayed by TSS

In this section, we discuss stable subordinator delayed by TSS. Note that stable subordinators are one-sided stable L´evy processes with non-decreasing sample paths. The subordinated process is defined by Yα,λ,β (t) = S˜α (Tλ,β (t)),

(4.1)

α ˜ where S˜α (t) is a stable subordinator with LT E(e−uSα (t) ) = e−tu , α ∈ (0, 1) independent of TSS. The intro
α β β duced process Yα,λ,β (t) is also a non-decreasing L´evy process and has the LT E(e−uYα,λ,β (t) ) = e−t (u +λ) −λ . Here we discuss the PDF and L´evy density of the introduced process. The technique of complex Laplace inversion of the LT is used to obtain the PDF of the process Yα,λ,β (t). Note that the PDF of Yα,λ,β (t) can be obtained by using (3.12), but that will have a complicated form since both inside and outside process don’t have closed form PDF. In next result, the PDF for Yα,λ,β (t) is obtained when 1/α is an integer. It is difficult to find the PDF for all possible α ∈ (0, 1) using this approach, since the branch points varies with α and every time a different contour is required to invert the LT.

˜ α,λ,β (x, t) of Yα,λ,β (t) has the following integral representation for α = Theorem 4.1. The PDF h 1, 2, . . ., Z ∞ β ˜ α,λ,β (x, t) = 1 h e−xy−tr1 (y) cos(βφ) sin(tr1 (y)β sin(βφ))dy, π 0 7

1 2m , m

=

(4.2)

1

1

π π where r1 (y) sin φ = y 2m sin 2m and r1 (y) cos φ = y 2m cos 2m + λ. Further, for α =

˜ α,λ,β (x, t) = 1 h π

Z

λ2m+1

0

1 + π

β

e−xy−tr2 (y)

Z

∞

cos(βθ)

2m+1

e−xy−xλ

1 2m+1 , m

= 1, 2, . . .,

sin(tr2 (y)β sin(βθ))dy

−tr3 (y)β cos(βψ)

sin(tr3 (y)β sin(βψ))dy,

(4.3)

0

1

1

1

π π π where r2 (y) sin θ = y 2m+1 sin( 2m+1 ), r2 (y) cos θ = y 2m+1 cos( 2m+1 ) + λ and r3 (y) sin ψ = (y+λ2m+1 ) 2m+1 sin( 2m+1 ), 1

π r3 (y) cos ψ = (y + λ2m+1 ) 2m+1 cos( 2m+1 ) + λ.

Proof. We have

h i ˜ α,λ,β (x, t) = e−t Lx→u h

(uα +λ)β −λβ




= F (u, t) (say),

(4.4)

where Lx→u represents the Laplace transform with respect to the space variable x. The PDF of Yα,λ,β (t) can be obtained by using the Laplace inversion formula, namely (Schiff, 1999) Z x0 +i∞ ˜ α,λ,β (x, t) = 1 eux F (u, t)du. (4.5) h 2πi x0 −i∞ We consider the case when α =

1 2m , m

= 1, 2, . . . first. For calculating integral in (4.5), consider a closed y

A

R

B

C

I

H

r O

D

E

G

F

r O′

P (x0 , 0)

x

J

Figure 1: Contour ABCDEFGHIJA double-key-hole contour C: ABCDEFGHIJA (Fig. 1) with branch points at origin O = (0, 0) and O0 = (λ2m , 0). In the contour AB and IJ are arcs of a circle of radius R with center at O, BC, DE, FG and HI are line segments parallel to x-axis, CD, GH and EF are arcs of circles with radius r and JA is the line segment from x0 − iy to x0 + iy with x0 > λ2m (see Fig. 1). By residue theorem (Schiff, 1999), Z X 1 eux F (u, t)du = Res eux F (u, t). (4.6) 2πi C The right hand side in (4.6) is 0, since the function has no simple pole. On evaluation we find that integral over AB, HI, DE, FG and HI tend to zero as the radius R goes to ∞ and r goes to zero. The line integral along 8

DE and FG cancels each other, thus leaving integral along BC and HI only. Along BC, we have u = yeiπ , which implies du = −dy and Z

1

β

eux e−t(u 2m +λ) du =

Z

−r

1

β

eux e−t(u 2m +λ) du =

−R

BC

Z

R

e−yx e

r

β  1 iπ −t y 2m e 2m +λ

dy.

Similarly, along HI, we have, u = ye−iπ , which implies du = −dy and Z

1

ux −t(u 2m +λ)β

e e

DE

du = −

Z

R

−yx

e

e

β  −iπ 1 −t y 2m e 2m +λ

dy.

r

Thus, Z

1

ux −t(u 2m +λ)β

e e

du +

BC

=

Z

1

β

eux e−t(u 2m +λ) du HI "  Z R

r

β 1 iπ −t y 2m e 2m +λ

e−yx e

−e

 β # −iπ 1 −t y 2m e 2m +λ

dy.

(4.7)

Equation (4.2) follows by using (4.5) and (4.7) with R → ∞ and r → 0. 1 For α = 2m+1 , a similar contour is considered but with branch point at (0, 0) and (−λ2m+1 , 0). The same steps lead to the desired result. Finally convert the integrands which are complex numbers to polar form.

Corollary 4.1. Putting λ = 0, β = 1 and m = 1 in (4.2) gives the PDF of a Applebaum, 2009, p. 53) and which is given by

1 2 -stable

subordinator (see e.g.

t − 3 − t2 ˜ 1/2,0,1 (x, t) = √ h x 2 e 4x . 2 π Further, for λ = 0 and m = 0 in (4.3), we obtain the PDF of a β-stable subordinator (see e.g. Kumar and Vellaisamy, 2015), that is Z ∞ β ˜ 1,0,β (x, t) = 1 e−xy e−ty cos βπ sin(ty β sin βπ)dy, y > 0. h π 0 √ Proof. Putting λ = 0, β = 1 and m = 1 in (4.2), leads to r1 (y) sin(βφ) = y and r1 (y) cos(βφ) = 0. Thus Z ∞ Z 3 t2 2 ∞ −xu2 t √ ˜ 1/2,0,1 (x, t) = 1 h e−xy sin(t y)dy = ue sin(tu)du = √ x− 2 e− 4x , π 0 π 0 2 π which follows using a result from Abramowitz and Stegun (1992). Further, for m = 0, λ = 0, we have r3 (y) sin ψ = y sin π and r3 (y) cos ψ = y cos π, which give r3 (y) = y and ψ = π. Thus Z 1 ∞ −xy −tyβ cos βπ ˜ h1,0,β (x, t) = e e sin(ty β sin βπ)dy. π 0

Being a composition of two L´evy processes, the introduced process Yα,λ,β (t) is also a L´evy process. In next result, the L´evy density for the introduced process is obtained.

9

Proposition 4.1. The L´evy density for Yα,λ,β (t), where 1/α is an integer, is given by Z 1 ∞ −xy 1 νY (dx) = e r1 (y)β sin(βφ)dy, for α = . π 0 2m Further, for α =

1 2m+1 ,

νY (dx) =

(4.8)

we have 1 π

Z

λ2m+1

e−xy r2 (y)β sin(βθ)dy +

0

1 π

Z

∞

2m+1

e−xy−xλ

r3 (y)β sin(βψ)dy.

(4.9)

0

Proof. For positive L´evy processes (i.e. subordinators) with PDF f (x, t), the L´evy density ν(dx) is given by (see e.g. Barndorff-Nielsen, 2000) 1 ν(dx) = lim f (x, t). (4.10) t↓0 t Since the outside and inside process are strictly increasing the resulting process Yα,λ,β (t) is also strictly increasing L´evy process. The result follows by using (4.2), (4.3) and (4.10) with the result limt↓0 sin(at)/t → a, a 6= 0, and dominated convergence theorem. Corollary 4.2. By taking λ = 0 and m = 0 in (4.9), we obtain the L´evy density corresponding to the β-stable subordinator which is given by (see e.g. Appleabum, 2009, p. 53) νY (dx) =

β 1 , 0 < β < 1. Γ(1 − β) x1+β

Further for m = 1, λ = 0 and β = 1 in (4.8), it yields Z 1 ∞ −xy 1/2 1 1 νY (dx) = e y dy = √ 3/2 , π 0 2 πx which is the L´evy density for 12 -stable subordinator (see Applebaum, 2009, p. 53). Proof. Taking λ = 0 and m = 0 in (4.9), yields Z sin βπ Γ(1 + β) 1 ∞ −xy β νY (dx) = e y sin βπdy = . π 0 π x1+β The result follows by using Euler’s identiy Γ(z)Γ(1 − z) =

π sin πz .

Next, the moments of the process Yα,λ,β (t) are discussed. One approach to find the moments of the process Yα,λ,β (t) is to use self-similarity of the outer process and independence of both the processes. Here a different approach is used to find the general form of the moments using the Lapalce transform. With the help of Fubini’s theorem, for a random variable X > 0, the fractional order moments from Laplace transform F (s) can be obtained as follows Z ∞ n  Z ∞ n Z ∞ n d d d p−1 −sX p−1 −sX p−1 [F (s)]s ds = [Ee ]s ds = E [e ]s ds dsn dsn dsn 0 0  0 Z ∞ = (−1)n E X n e−sX sp−1 ds = (−1)n Γ(p)EX n−p . 0

Thus (n − p)-th order moment, where n is an integer and p is a real number is given by Z (−1)n ∞ dn E(X n−p ) = [F (s)]sp−1 ds. Γ(p) 0 dsn

(4.11)

In definition of the introduced process, outside process is a stable subordinator and hence moments of order q ≥ 1 don’t exist. Next result provides the moments of order q < 1 for the introduced process Yα,λ,β (t). 10

Proposition 4.2. For q ∈ (0, 1), the q-th order moment of Yα,λ,β (t) has the form β

E(Yα,λ,β (t)q ) =

αβtetλ Γ(1 − q)

Z

∞

sα−1−q (sα + λ)β−1 e−t(s

α

+λ)β

ds, 0 < α < 1,

(4.12)

0

which is finite for q < α if λ > 0 and for q < αβ in case λ = 0. Proof. For 0 < q < 1, consider n = 1 and p = 1 − q in (4.11) and which gives the desired result. Denoting R∞ R1 R∞ R∞ α β w(s) = sα−1−q (sα + λ)β−1 e−t(s +λ) , it follows 0 w(s)ds = 0 w(s)ds + 1 w(s)ds. Now, 1 w(s)ds ≤ R α β ∞ (1 + λ)β−1 1 e−t(s +λ) ds < ∞. Further, ( R1 Z 1 Z 1 sα−1−q (sα + λ)β−1 ds, if λ > 0 α−1−q α β−1 R01 αβ−1−q s (s + λ) ds = w(s)ds < s ds, if λ = 0. 0 0 0 Thus the integral in (4.12) is finite if q < α for λ > 0 and if q < αβ for λ = 0.

The asymptotic behavior of q-th order moments for Yα,λ,β (t) can be obtained by using the self-similarity of stable subordinator and asymptotic moments of tempered stable subordinator. Alternatively, one can use Laplace-Erdelyi theorem (Erd´elyi, 1956) in (4.12) to find the asymptotic behaviour of the integral, which further can be used in estimating the parameters. Using Laplace-Erdelyi theorem, we obtain E(Yα,λ,β (t)q ) ∼ Γ(1−q/α) β−1 q/α q/α ) t , as t → ∞. For λ = 0, Yα,λ,β (t) reduces to a stable subordinator of index αβ and Γ(1−q) (βλ its q-th order moments follow easily from (4.12) and is given by E(Yα,0,β (t)q ) =

Γ(1−q/(αβ)) q/(αβ) t . Γ(1−q)  β β



The

codifference of the process Yα,λ,β (t) is given by CD(Yα,λ,β (s), Yα,λ,β (t)) = 2 min (t, s) (1 + λ) − λ , which is similar to the codifference of the tempered stable subordinator and the process Xα,λ,β (t). Using similar ˜ α,λ,β (x, t) is steps as in Theorem 3.1, the governing equation of h 1/β  α ∂ β ˜ α,λ,β (x, t) = λh ˜ α,λ,β (x, t) + ∂ h ˜ α,λ,β (x, t), h λ − ∂t ∂xα where ∂ α /∂xα is Riemann-Liouville (R-L) fractional derivative. The R-L fractional derivative defined as the function with LT uα F (u).

(4.13) ∂α ∂xα f (x)

is

Ackowledgements The work is supported by NCN OPUS grant No. 2016/21/B/ST1/00929. The authors are grateful to the reviewers for several helpful comments and suggestions, which have led to improvements in the paper.

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