[1]Aubain Nzokem

# Self-Decomposable Laws Associated with General Tempered Stable (GTS)
Distribution and their Simulation Applications

<EMAIL_ADDRESS>
###### Abstract

The paper describes the self-decomposable distribution and the background
driving Lévy process (BDLP) associated with the Generalized Tempered Stable
(GTS) distribution. Two distributions are provided: the background driving
Lévy process (BDLP) of the GTS distribution and the self-decomposable
distribution generated by the GTS distribution as BDLP. The derived self-
decomposable distribution and the GTS distribution are used as stationary
distribution in the Ornstein-Uhlenbeck type process. A simulation method,
based on sampling the random integral representation, is applied to mimic the
S&P 500 Index and Bitcoin daily cumulative return process.

###### keywords:

Self-Decomposable, Background Driving Lévy Process (BDLP), Ornstein–Uhlenbeck
Process, Generalized Tempered Stable Distribution

##  1 Introduction

The Tempered stable distribution was introduced to overcome the limitations of
the stable distribution and the normal distribution. It was shown [1] that The
tempered stable process behaves in the short times as a stable process and in
the long term as a Gaussian process. The tails of the Tempered stable
distribution are heavier than the normal distribution and thinner than the
stable distribution [2, 3]. Tempered stable models are used in several areas
of physics [4], inclusing turbulence, plasma physic , solar winds, and their
popularity is still growing in financial applications.
Although the spectral measure[1, 2, 5] provide the analytical properties of
all classes of Tempered stable distributions, the Generalised tempered stable
(GTS) process, which is the most popular subclass, is constructed [6, 7] by
tempering the Lévy measure of a stable distribution. The six-parameter Lévy
measure of the GTS distribution (3) is a product of a tempering or tilting
function ($q(y)$) (1) and a Lévy measure of the $\alpha$-stable distribution
($M_{stable}(dy)$) (2) described as follows.

$\displaystyle q(y)$ $\displaystyle=
e^{-\lambda_{+}y}\boldsymbol{1}_{(0,\infty)}(y)+e^{-\lambda_{-}|y|}\boldsymbol{1}_{(-\infty,0)}(y)$
(1) $\displaystyle M_{stable}(dy)$
$\displaystyle=\left(\frac{\alpha_{+}}{y^{1+\beta_{+}}}\boldsymbol{1}_{(0,\infty)}(y)+\frac{\alpha_{-}}{|y|^{1+\beta_{-}}}\boldsymbol{1}_{(-\infty,0)}(y)\right)dy$
(2) $\displaystyle M(dy)=q(y)M_{stable}(dy)$
$\displaystyle=\left[\frac{\alpha_{+}}{y^{1+\beta_{+}}}e^{-\lambda_{+}y}\boldsymbol{1}_{(0,\infty)}(y)\right]dy+\left[\frac{\alpha_{-}}{|y|^{1+\beta_{-}}}e^{-\lambda_{-}|y|}\boldsymbol{1}_{(-\infty,0)}(y)\right]dy
$ (3)

Where $0\leq\beta_{+}\leq 1$, $0\leq\beta_{-}\leq 1$, $\alpha_{+}\geq 0$,
$\alpha_{-}\geq 0$, $\lambda_{+}\geq 0$ and $\lambda_{-}\geq 0$.

The GTS distribution in (3) is a six-parameter family of infinitely divisible
distribution, which covers several well-known distribution subclasses: the
KoBol distribution [8] is obtained by substituting
$\beta=\beta_{+}=\beta_{-}$, the truncated Lévy flight [7] by substituting
$\lambda=\lambda_{+}=\lambda_{-}$ and $\beta=\beta_{+}=\beta_{-}$, the
bilateral Gamma distributions [9] by substituting $\beta_{+}=\beta_{-}=0$, the
Variance Gamma distributions [10, 11, 12, 13, 14] by substituting
$\alpha=\alpha_{+}=\alpha_{-}$ and $\beta_{+}=\beta_{-}=0$ while the Carr-
Geman-Madan-Yor (CGMY) distributions [15], also called Classical Tempered
Stable Distribution[6] is obtained by substituting
$\alpha=\alpha_{+}=\alpha_{-}$, $G=\lambda_{-}$, $M=\lambda_{+}$ and
$Y=\beta_{+}=\beta_{-}$.
The goal of the present paper is to investigate the self-decomposable
distributions associated with General tempered stable (GTS) distribution and
provide a simulation technique to sample the associated Ornstein-Uhlenbeck
type processes.
There are different methods for simulating the Tempered Sable processes, but
the choice usually depends on the structure of the Lévy measures. For example,
the CGMY process [16], also called the Classical Tempered Stable process, can
be simulated as a time-changed Brownian motion with drift. The method with the
inverse of the Lévy measure, in the series representations [17], was not
suitable to implement in our case. The tail of Lévy measures does not have an
explicit inverse. The simulation method used in the paper is based on the
Ornstein-Uhlenbeck type process, and the classic sampling method of inversion
was applied to the random integral representation. The inverse of the
cumulative probability function was provided by the composite FRFTs algorithm
[18].

The remainder of the text is organized as follows: In section 2, we provide an
overview of the framework for Lévy process, Generalized Tempered Stable (GTS)
Process, Ornstein-Uhlenbeck process, and self-decomposable distribution. In
section 3, we investigate the background-driving Lévy process (BDLP) of the
GTS distribution and the self-decomposition distribution generated by the GTS
distribution as a background-driving lévy process (BDLP). Section 4 presents
the GTS Parameter Estimation and analyses the lévy density and characteristic
functions derived in section 3. In section 4, we develop a simulation method
and apply to sample the S&P 500 Index and Bitcoin daily cumulative return
process.

##  2 Preliminaries

###  2.1 Lévy Framework and Asset Pricing: Overview

Let $(\Omega,\mathscr{F},\left\\{\mathscr{F}_{t}\right\\}_{t\geq
0},\mathbf{P})$ be a filtered probability space, with
$\mathscr{F}=\vee_{t>0}\mathscr{F}_{t}$ and
$\left\\{\mathscr{F}_{t}\right\\}_{t\geq 0}$ is a filtration.
$\mathscr{F}_{t}$ is a $\sigma$-algebra included in $\mathscr{F}$ and for
$\tau<t$, $\mathscr{F}_{\tau}\subseteq\mathscr{F}_{t}$.
A stochastic process $L=\\{L_{t}\\}_{t\geq 0}$ is a Lévy process, if it has
the following properties
(A1): $L_{0}=0$ a.s;
(A2): $L_{t}$ has independent increments, that is, for
$0<t_{1}<t_{2}<\dots<t_{n}$, the random variables $L_{t_{1}}-L_{0}$,
$L_{t_{2}}-L_{t_{1}}$, …, $L_{t_{n}}-L_{t_{n-1}}$ are independent;
(A3): $L_{t}$ has stationary increments, that is, for any
$t_{1}<t_{2}<+\infty$, the probability distribution of $L_{t_{2}}-L_{t_{1}}$
depends only on $t_{2}-t_{1}$ ;
(A4): $L_{t}$ is stochastically continuous: for any $t$ and $\epsilon>0$,
$\lim_{s\to t}P\left(|L_{s}-L_{t}|>\epsilon\right)=0$;
(A5): $c\grave{a}dl\grave{a}g$ paths, that is, $t\mapsto L_{t}$ is a.s right
continuous with left limits

Given a Lévy process $L=\\{L_{t}\\}_{t\geq 0}$ on the filtered probability
space $(\Omega,\mathscr{F},\left\\{\mathscr{F}_{t}\right\\}_{t\geq
0},\mathbf{P})$, we define the asset value process $S=\\{S_{t}\\}_{t\geq 0}$
as $S_{t}=S_{0}e^{L_{t}}$.

######  Theorem 1.

(Lévy & Khintchine representation)
Let $L=\\{L_{t}\\}_{t\geq 0}$ be a Lévy precess on $\mathbb{R}$ with
characteristic triplet $(\gamma,\sigma^{2},M)$. Then for any
$\xi\in\mathbb{R}$ and $t\geq 0$, we have:

$\displaystyle E\left[e^{iL_{t}\xi}\right]$ $\displaystyle=e^{t\varphi(\xi)}$
(4) $\displaystyle\varphi(\xi)= Log\left(Ee^{iL_{1}\xi}\right)$
$\displaystyle=i\gamma\xi-\frac{1}{2}\sigma^{2}\xi^{2}+\int_{-\infty}^{+\infty}\left(e^{i\xi
y}-1-y\xi 1_{|y|\leq 1}\right)M(dy) $

Where $\gamma\in\mathbb{R}$, $\sigma\geq 0$ is the Gaussian coefficient,
$\varphi(\xi)$ is the characteristic exponent and $M$ is the Lévy measure of L
such that

$\displaystyle\int_{-\infty}^{+\infty}Min(1,|y|^{2})M(dy)<+\infty$

For the theorem-proof, see [19, 20, 21]

The Lévy process is completely determined by the Lévy–Khintchine triplet
$(\gamma,\sigma^{2},M)$. The triplet terms suggest that the Lévy process has
three independent components: a linear drift, a Brownian motion, and a Lévy
jump process. When the diffusion term $\sigma=0$, we have a Lévy jump process;
in addition, if $\gamma=0$, we have a pure jump process.

The Lévy-Khintchine representation has a simpler expression when the sample
paths of the Lévy process have finite variation on every compact time interval
almost surely. A Lévy process has bounded variation[22] if and only if

$\displaystyle\sigma^{2}=0\quad\quad\mbox{ and
}\quad\quad\int_{-\infty}^{+\infty}min(1,|x|)M(dx)<+\infty$ (5)

With the assumption of Lévy process with finite variation, the characteristic
exponent $\varphi(\xi)$ in the theorem 1 becomes

$\displaystyle\varphi(\xi)=
Log\left(Ee^{iL_{1}\xi}\right)=i\Gamma\xi+\int_{-\infty}^{+\infty}\left(e^{i\xi
y}-1\right)M(dy)$ (6)

With

$\displaystyle\sigma^{2}=0\quad\quad\Gamma=\gamma+\int_{-1}^{1}xM(dx)$

###  2.2  Generalized Tempered Stable (GTS) Process: Overview

The Lévy measure of the Generalized Tempered Stable (GTS) distribution
($M(dy)$) is defined (9) as a product of a tempering or tilting [23, 1]
function ($q(y)$) (7) and a Lévy measure of the $\alpha$-stable distribution
($M_{stable}(dy)$) (8).

$\displaystyle q(y)$ $\displaystyle=
e^{-\lambda_{+}y}\boldsymbol{1}_{y>0}+e^{-\lambda_{-}|y|}\boldsymbol{1}_{y<0}$
(7) $\displaystyle M_{stable}(dy)$
$\displaystyle=\left(\frac{\alpha_{+}}{y^{1+\beta_{+}}}\boldsymbol{1}_{y>0}+\frac{\alpha_{-}}{|y|^{1+\beta_{-}}}\boldsymbol{1}_{y<0}\right)dy$
(8) $\displaystyle M(dy)=q(y)M_{stable}(dy)$
$\displaystyle=\left(\frac{\alpha_{+}e^{-\lambda_{+}y}}{y^{1+\beta_{+}}}\boldsymbol{1}_{y>0}+\frac{\alpha_{-}e^{-\lambda_{-}|y|}}{|y|^{1+\beta_{-}}}\boldsymbol{1}_{y<0}\right)dy
$ (9)

Where $0\leq\beta_{+}\leq 1$, $0\leq\beta_{-}\leq 1$, $\alpha_{+}\geq 0$,
$\alpha_{-}\geq 0$, $\lambda_{+}\geq 0$ and $\lambda_{-}\geq 0$.

The six parameters that appear have important interpretations. $\beta_{+}$ and
$\beta_{-}$ are the indexes of stability bounded below by 0 and above by 2
[24]. They capture the peakedness of the distribution similarly to the
$\beta$-stable distribution, but the distribution tails are tempered. If
$\beta$ increases (decreases), then the peakedness decreases (increases).
$\alpha_{+}$ and $\alpha_{-}$ are the scale parameters, also called the
process intensity [8]; they determine the arrival rate of jumps for a given
size. $\lambda_{+}$ and $\lambda_{-}$ control the decay rate on the positive
and negative tails. Additionally, $\lambda_{+}$ and $\lambda_{-}$ are also
skewness parameters. If $\lambda_{+}>\lambda_{-}$ ($\lambda_{+}<\lambda_{-}$),
then the distribution is skewed to the left (right), and if
$\lambda_{+}=\lambda_{-}$, then it is symmetric [6, 25]. $\alpha$ and
$\lambda$ are related to the degree of peakedness and thickness of the
distribution. If $\alpha$ increases (decreases), the peakedness and the
thickness decrease (increase). Similarly, If $\lambda$ increases (decreases),
then the peakedness increases (decreases) and the thickness decreases
(increases) [26].

The GTS distribution can be denoted by $X\sim
GTS(\textbf{$\beta_{+}$},\textbf{$\beta_{-}$},\textbf{$\alpha_{+}$},\textbf{$\alpha_{-}$},\textbf{$\lambda_{+}$},\textbf{$\lambda_{-}$})$
and $L_{1}=L^{+}_{1}-L^{-}_{1}$ with $L^{+}_{1}\geq 0$, $L^{-}_{1}\geq 0$.
$L^{+}_{1}\sim
TS(\textbf{$\beta_{+}$},\textbf{$\alpha_{+}$},\textbf{$\lambda_{+}$})$ and
$L^{-}_{1}\sim
TS(\textbf{$\beta_{-}$},\textbf{$\alpha_{-}$},\textbf{$\lambda_{-}$})$.

The activity process of the GTS distribution can be studied from the integral
(10) of the Lévy measure (9). As shown in (10), if $\beta_{+}<0$,
TS($\beta_{+}$, $\alpha_{+}$,$\lambda_{+}$) is of finite activity process and
can be written as a compound Poisson. We have a similar pattern when
$\beta_{-}<0$. The interesting case is when $\beta_{+}\geq 0$. As shown in
(10), if $\beta_{+}\geq 0$, TS($\beta_{+}$, $\alpha_{+}$,$\lambda_{+}$) is of
infinite activity process with infinite jumps in any given time interval. We
have a similar pattern when $0\leq\beta_{-}$:

$\displaystyle\int_{-\infty}^{+\infty}M(dy)=\begin{cases}+\infty&\text{if
}{\beta_{+}\geq 0\vee\beta_{-}\geq 0}\\\
\alpha_{+}{\lambda_{+}}^{\beta_{+}}\Gamma(-\beta_{+},0)+\alpha_{-}{\lambda_{-}}^{\beta_{-}}\Gamma(-\beta_{-},0)&\text{if
}{\beta_{+}<0\wedge\beta_{-}<0}\end{cases}$ (10)

where $\Gamma(s,x)=\int_{x}^{+\infty}y^{s-1}e^{-y}dy$ is the upper incomplete
gamma function. In addition, the variation or smoothness of the process can be
studied through the following integral:

$\displaystyle\int_{-\infty}^{+\infty}min(1,|y|)M(dy)=\int_{-\infty}^{-1}M(dy)+\int_{-1}^{0}|y|M(dy)+\int_{1}^{+\infty}M(dy)+\int_{0}^{1}|y|M(dy)$
$\displaystyle=\alpha_{-}\lambda_{-}^{\beta_{-}}\left[\Gamma(-\beta_{-},\lambda_{-})+\lambda_{-}^{-1}\gamma(1-\beta_{-},\lambda_{-})\right]+\alpha_{+}\lambda_{+}^{\beta_{+}}\left[\Gamma(-\beta_{+},\lambda_{+})+\lambda_{+}^{-1}\gamma(1-\beta_{+},\lambda_{+})\right]$

where $\gamma(s,x)=\int_{0}^{x}y^{s-1}e^{-y}dy$ is lower incomplete gamma
function.
We have

$\displaystyle\int_{-\infty}^{+\infty}min(1,|y|)M(dy)<+\infty\hskip 14.22636pt
\hbox{ if $0\leq\beta_{-}<1$ \& $0\leq\beta_{+}\leq 1$}$ (11)

As shown in (11), GTS($\beta_{+}$, $\beta_{-}$, $\alpha_{+}$,$\alpha_{-}$,
$\lambda_{+}$, $\lambda_{-}$) is a finite variance process, which is not the
case for the Brownian motion process.

By adding the location parameter, we have:

$\displaystyle L=\mu+L_{1}=\mu+L^{+}_{1}-L^{-}_{1}\quad\quad L\sim
GTS(\mu,\beta_{+},\beta_{-},\alpha_{+},\alpha_{-},\lambda_{+},\lambda_{-}) $
(12)

The GTS distribution in (12) is a seven-parameter family of infinitely
divisible distribution, which covers several well-known distribution
subclasses: the KoBol distribution [8] is obtained by substituting
$\beta=\beta_{+}=\beta_{-}$, the truncated Lévy flight [7] by substituting
$\lambda=\lambda_{+}=\lambda_{-}$ and $\beta=\beta_{+}=\beta_{-}$, the
bilateral Gamma distributions [9] by substituting $\beta_{+}=\beta_{-}=0$, the
Variance Gamma distributions [10, 11, 12, 13, 14] by substituting
$\alpha=\alpha_{+}=\alpha_{-}$ and $\beta_{+}=\beta_{-}=0$ while the Carr-
Geman-Madan-Yor (CGMY) distributions [15], also called Classical Tempered
Stable Distribution[6] is obtained by substituting
$\alpha=\alpha_{+}=\alpha_{-}$, $G=\lambda_{-}$, $M=\lambda_{+}$ and
$Y=\beta_{+}=\beta_{-}$.

######  Theorem 2.

Consider a variable $Y\sim
GTS(\textbf{$\mu$},\textbf{$\beta_{+}$},\textbf{$\beta_{-}$},\textbf{$\alpha_{+}$},\textbf{$\alpha_{-}$},\textbf{$\lambda_{+}$},\textbf{$\lambda_{-}$})$,
the characteristic exponents can be written as

$\displaystyle\Psi(\xi)$ $\displaystyle=Log\left(Ee^{iL\xi}\right)=\mu\xi
i+\Psi^{+}(\xi)+\Psi^{-}(-\xi)$ (13)

Where

$\displaystyle\Psi^{+}(\xi)$
$\displaystyle=Log\left(Ee^{iL^{+}_{1}\xi}\right)=\alpha_{+}\Gamma(-\beta_{+})\left((\lambda_{+}-i\xi)^{\beta_{+}}-{\lambda_{+}}^{\beta_{+}}\right)$
$\displaystyle\Psi^{-}(\xi)$
$\displaystyle=Log\left(Ee^{iL^{-}_{1}\xi}\right)=\alpha_{-}\Gamma(-\beta_{-})\left((\lambda_{-}-i\xi)^{\beta_{-}}-{\lambda_{-}}^{\beta_{-}}\right)$

See [27, 28, 29] for theorem 5 proof.

###  2.3 Decomposable Laws: Overview

Consider a sequence $(X_{k}:k=1,2,...)$ of independent random variables, but
not necessarily identically distributed; and another sequence
$(S_{n}:n=1,2,...)$ with $S_{n}=\sum_{k=0}^{n}X_{k}$. We assumed there exist a
sequence $(c_{n}:n=1,2,...)$ and a scaling sequence $(b_{n}:n=1,2,...)$ such
that $\frac{S_{n}-c_{n}}{b_{n}}$ converges in distribution to some random
variable $X$. In other words, the random variable $X$ distribution is the
limiting distribution of a sequence of normalized sums of independent but not
necessarily identically distributed random variables. The class of limit laws
is known as Lévy class L distributions or class L in the literature[30], and
the random variable $X$ is said to belong to Class L. In the classical Central
Limit Theorem (CLT), the asymptotically standard normal is of Class L.
Likewise, the class of stable laws is of Class L. It can be described by four
parameters: a location parameter ($\mu$), a scale parameter ($\sigma>0$), a
stability index ( $0\leq\alpha\leq 2$) and a skewness parameter
($-1\leq\beta\leq 1$). The distribution of any random variable in the class L
is infinitely divisible.

In addition, it is shown [20] that a Class L is also a class of self-
decomposable probability distributions. The distribution of a random variable
$X$ is self-decomposable if for any constant $c$ ($0<c<1$ ), there exists an
independent random variable $X_{c}$ such that

$\displaystyle X\overset{d}{=}cX+X_{c}\hskip 28.45274pt \hbox{with $X$ and
$X_{c}$ are independent variable}$ (14)

where $\overset{d}{=}$ means equality in distribution.
Analytically, a random variable $X$ with a characteristic function $\phi$ is
self-decomposable if

$\displaystyle\forall(0<c<1)\,\exists\rho_{c}\,\forall(t\in\mathbb{R})\quad\quad\phi(t)=\phi(ct)\rho_{c}(t)$
(15)

where $\rho_{c}$ is also a characteristic function.

Class L distributions is also described [31] as distributions of random
integrals. Class L is represented as follows

$\displaystyle L=\left\\{\int_{0}^{+\infty}e^{-t}dY(s),\mbox{with $Y$ is a
L\'{e}vy Process and $E\left[\log(1+|Y(1)|)\right]\leq\infty$ }\right\\}$ (16)

where the Lévy process $Y$ is called the background driving process of a self-
composable distribution. In terms of characteristic functions, the random
integral representation provides another characteristic of the Class L [31].

$\displaystyle X\in\mbox{Class L \quad if only
if}\quad\log(\phi(\xi))=\int_{0}^{\xi}\log(\varphi(u))\frac{du}{u}$ (17)

where $\phi(x)$ and $\varphi(x)$ are characteristic functions of $X$ and
$Y(1)$

The relationship between a self-decomposable variable $(X)$ and its background
driving process can also be featured[32, 31] by the triplet in Lévy-Khintchine
formula (20). Let $\left[A_{X},\sigma^{2}_{X},M_{X}\right]$ and
$\left[A_{Y},\sigma^{2}_{Y},M_{Y}\right]$ are triplets in the Lévy-Khintchine
formula for $X$ and $Y$ respectively. we have

$\displaystyle M_{X}(A)$
$\displaystyle=\int_{0}^{+\infty}M_{Y}(e^{t}A)dt\quad\mbox{ for borel
subsets}\quad A\subset\mathbb{R}\setminus{\\{0\\}}$ (18)
$\displaystyle\sigma_{X}^{2}$
$\displaystyle=\frac{1}{2}\sigma_{Y}^{2}\quad\quad
A_{X}=A_{Y}+\int_{-\infty}^{+\infty}\left(\frac{x}{1+x^{2}}-arctan(x)\right)M_{Y}(dx)$

The class of self-decomposable probability distributions is large and includes
among others[33]: Stable, Gamma, Log-Gamma, Log-Normal, Chi-Squared
($\chi^{2}$), Student-T, Log-Student-T, Fisher–Snedecor and Hyperbolic
distributions.

###  2.4 Ornstein-Uhlenbeck process: Overview

The Ornstein-Uhlenbeck process is a diffusion process introduced by Ornstein
and Uhlenbeck [34] to model the stochastic behavior of the velocity of a
particle undergoing Brownian motion. The Ornstein-Uhlenbeck diffusion
$X=\\{X(t)\\}_{t\geq 0}$ is the solution of the Langevin Stochastic
Differential Equation (SDE) (19)

$\displaystyle dX_{t}=-\lambda X_{t}dt+dB_{t}. $ (19)

where $\lambda>0$ and $B=\\{B_{t},t\geq 0\\}$ is a Brownian motion.

The Ornstein-Uhlenbeck type process is used in finance to capture important
distributional deviations from Gaussianity and to model dependence structures
[35, 36, 37, 38]. The extension of the Ornstein-Uhlenbeck processes is
possible by replacing the Brownian motion in (19) by the Lévy process
$\\{L_{t}\\}_{t\geq 0}$, called the background driving Lévy process (BDLP)
corresponding to the process $\\{X_{t}\\}_{t\geq 0}$. The SDE (19) becomes

$\displaystyle dX_{t}=-\lambda X_{t}dt+dL_{\lambda t} \quad\lambda>0\. $ (20)

$dL(\lambda t)$ is deliberately chosen [36] so that the marginal distribution
of X is unchanged, whatever the value of $\lambda$. The solution to the SDE
(20) is a stationary process $\\{X_{t}\\}_{t\geq 0}$ such that
$X_{t}\overset{d}{=}X$. $\\{X_{t}\\}_{t\geq 0}$ is a stationary process of
Ornstein-Uhlenbeck type developed by Barndorff-Nielsen and Shephard [39, 38,
40]. $\\{X_{t}\\}_{t\geq 0}$ moves up entirely by jumps and then tails off
exponentially [36].

######  Theorem 3.

Let $\phi$ be the characteristic function of a random variable X . If X is
self-decomposable then there exists a stationary stochastic process
$\left\\{X(t)\right\\}_{t>0}$ and a background driving Lévy process (BDLP)
$\left\\{L(t)\right\\}_{t>0}$ independent of $X(0)$ , such that
$X(t)\overset{d}{=}X$ and

$\displaystyle X(t)=e^{-\lambda
t}X(0)+\int_{0}^{t}e^{-\lambda(t-s)}dL_{\lambda s} \quad\lambda>0\. $ (21)

Conversely, if $\left\\{X(t)\right\\}_{t>0}$ is a stationary stochastic
process and $\left\\{L(t)\right\\}_{t>0}$ is a Lévy process, independent of
$X(0)$, such that $\left\\{X(t)\right\\}_{t>0}$ and
$\left\\{L(t)\right\\}_{t>0}$ satisfy the equation (20) for all $\lambda>0$
then $X(t)$ is self-decomposable.

For the theorem-proof, see [40]

##  3  Self-decomposability of GTS distribution

###  3.1  Background Driving Lévy Process (BDLP) of the GTS distribution

######  Theorem 4.

The Generalized Tempered Stable (GTS) distribution is self-decomposable, and
its background driving Lévy process $Y$ is the compound Poisson process with
Poisson exponent.

$\displaystyle
N(dx)=\left[\alpha_{+}\frac{\beta_{+}+\lambda_{+}x}{x^{1+\beta_{+}}}e^{-\lambda_{+}x}\boldsymbol{1}_{x>0}\right]dx+\left[\alpha_{-}\frac{\beta_{-}+\lambda_{-}|x|}{|x|^{1+\beta_{-}}}e^{-\lambda_{-}|x|}\boldsymbol{1}_{x<0}\right]dx$
(22)

###### Proof.

Let $\left[a_{\phi^{+}},\sigma^{2}_{\phi^{+}},M_{\phi^{+}}\right]$, the
Lévy–Khintchine triplet of the Tempered stable distribution TS($\beta_{+}$,
$\alpha_{+}$,$\lambda_{+}$) and
$\left[b_{\varphi^{+}},s^{2}_{\varphi^{+}},N_{\varphi^{+}}\right]$, the
Lévy–Khintchine triplet of the associated distribution.
The property (18) is applied by choosing $A=(y,+\infty)$

$\displaystyle
M_{\phi^{+}}((y,+\infty))=\int_{y}^{+\infty}\frac{\alpha_{+}}{x^{1+\beta_{+}}}e^{-\lambda_{+}x}dx=\int_{0}^{+\infty}N_{\varphi^{+}}((e^{t}y,+\infty))dt=\int_{y}^{+\infty}\frac{N_{\varphi^{+}}((u,+\infty))}{u}du$
(23)

In order to have the relation (23), we need to have:

$\displaystyle
N_{\varphi^{+}}((u,+\infty))=\frac{\alpha_{+}}{u^{\beta_{+}}}e^{-\lambda_{+}u}$

The Lévy density becomes

$\displaystyle
N_{\varphi^{+}}(du)=-\frac{dN_{\varphi^{+}}((u,+\infty))}{du}du=\alpha_{+}(\lambda_{+}+\beta_{+}u^{-1})u^{-\beta_{+}}e^{-\lambda_{+}u}
du$ (24)

The derivation of the relation (23) provides the relation between the Lévy
densities $M_{\phi^{+}}(dy)$ and $N_{\varphi^{+}}(dv)$ as follows.

$\displaystyle M_{\phi^{+}}(dy)$
$\displaystyle=\int_{0}^{+\infty}N_{\varphi^{+}}(e^{t}dy)dt$ (25)
$\displaystyle=\int_{0}^{+\infty}\alpha_{+}(\lambda_{+}+\beta_{+}e^{-t}y^{-1})e^{(1-\beta_{+})t}y^{-\beta_{+}}e^{-\lambda_{+}ye^{t}}dtdy$

We assume $\phi^{+}(\xi)$ is the characteristic function of the Tempered
stable distribution TS($\beta_{+}$, $\alpha_{+}$,$\lambda_{+}$).

$\displaystyle\log(\phi^{+}(\xi))$
$\displaystyle=\int_{0}^{+\infty}(e^{iy\xi}-1)M_{\phi^{+}}(dy)$
$\displaystyle=\int_{0}^{+\infty}\int_{0}^{+\infty}(e^{iy\xi}-1)\alpha_{+}(\lambda_{+}+\beta_{+}e^{-t}y^{-1})e^{(1-\beta_{+})t}y^{-\beta_{+}}e^{-\lambda_{+}ye^{t}}dtdy$
$\displaystyle=\int_{0}^{+\infty}\int_{0}^{+\infty}(e^{i\xi
ve^{-t}}-1)\alpha_{+}(\lambda_{+}+\beta_{+}v^{-1})v^{-\beta_{+}}e^{-\lambda_{+}v}dtdv\quad
v=ye^{t}$ $\displaystyle=\int_{0}^{1}\int_{0}^{+\infty}(e^{i\xi
vu}-1)\alpha_{+}(\lambda_{+}+\beta_{+}v^{-1})v^{-\beta_{+}}e^{-\lambda_{+}v}dv\frac{du}{u}\quad
u=e^{t}$ $\displaystyle=\int_{0}^{1}\log(\varphi^{+}(\xi
u))\frac{du}{u}=\int_{0}^{\xi}\log(\varphi^{+}(x))\frac{dx}{x}\quad\quad x=\xi
u$

We have

$\displaystyle\Psi^{+}(\xi)=\log(\phi^{+}(\xi))=\int_{0}^{\xi}\log(\varphi^{+}(x))\frac{dx}{x}$
(26)
$\displaystyle\log(\varphi^{+}(x))=\int_{0}^{+\infty}(e^{ivx}-1)N_{\varphi^{+}}(dv)\quad\quad
N_{\varphi^{+}}(dx)$
$\displaystyle=\alpha_{+}\frac{\beta_{+}+\lambda_{+}x}{x^{1+\beta_{+}}}e^{-\lambda_{+}x}dx$
(27)

Following the same procedure on the left side, the Tempered stable
distribution TS($\beta_{-}$, $\alpha_{-}$,$\lambda_{-}$), we have

$\displaystyle\Psi^{-}(\xi)=\log(\phi^{-}(\xi))=\int_{0}^{\xi}\log(\varphi^{-}(x))\frac{dx}{x}$
$\displaystyle\log(\varphi^{-}(x))=\int_{0}^{+\infty}(e^{ivx}-1)N_{\varphi^{-}}(dv)$
$\displaystyle\quad N_{\varphi^{-}}(dx)$
$\displaystyle=\alpha_{-}\frac{\beta_{-}+\lambda_{-}x}{x^{1+\beta_{-}}}e^{-\lambda_{-}x}dx$

By putting the left and the right sides of the Tempered stable distribution
together, we have

$\displaystyle\log(\varphi^{+}(x))+\log(\varphi^{-}(-x))$
$\displaystyle=\int_{-\infty}^{+\infty}(e^{ivx}-1)N_{\varphi}(dx)$ (28)
$\displaystyle\log(\phi^{+}(\xi))+\log(\phi^{-}(-\xi))$
$\displaystyle=\int_{0}^{\xi}(\log(\varphi^{+}(x))+\log(\varphi^{-}(-x)))\frac{dx}{x}$
$\displaystyle=\int_{0}^{\xi}\int_{-\infty}^{+\infty}(e^{ivx}-1)N_{\varphi}(dx)\frac{dx}{x}$

Where

$\displaystyle
N_{\varphi}(dx)=\left[\alpha_{+}\frac{\beta_{+}+\lambda_{+}x}{x^{1+\beta_{+}}}e^{-\lambda_{+}x}\boldsymbol{1}_{x>0}\right]dx+\left[\alpha_{-}\frac{\beta_{-}+\lambda_{-}|x|}{|x|^{1+\beta_{-}}}e^{-\lambda_{-}|x|}\boldsymbol{1}_{x<0}\right]dx$

We have the characteristic exponent (28) of the Generalized Tempered Stable
GTS($\mu$, $\beta_{+}$, $\beta_{-}$, $\alpha_{+}$,$\alpha_{-}$, $\lambda_{+}$,
$\lambda_{-}$)

$\displaystyle\Psi(\xi)=$ $\displaystyle Log\left(Ee^{iL\xi}\right)=\mu\xi
i+\log(\phi^{+}(\xi))+\log(\phi^{-}(-\xi))$ (29)
$\displaystyle=\int_{0}^{\xi}\left(\mu
xi+\int_{-\infty}^{+\infty}(e^{ivx}-1)N_{\varphi}(dx)\right)\frac{dx}{x}=\int_{0}^{\xi}\log(\varphi(x)\frac{dx}{x}$

Where

$\displaystyle\log(\varphi(x))=\mu
xi+\int_{-\infty}^{+\infty}(e^{ivx}-1)N_{\varphi}(dx)$

We conclude from (29) that the Generalized Tempered Stable GTS is self-
decomposable. ∎

######  Theorem 5.

(Characteristic exponents of BDL process of the GTS distribution)
The Generalized Tempered Stable (GTS) distribution is self-decomposable, and
its background driving Lévy process $Y$ has the following characteristic
exponent.

$\displaystyle\log(\varphi(y))$ $\displaystyle=\mu
iy+\alpha_{+}\Gamma(1-\beta_{+})\frac{iy}{(\lambda_{+}-iy)^{1-\beta_{+}}}+\alpha_{-}\Gamma(1-\beta_{-})\frac{-iy}{(\lambda_{-}+iy)^{1-\beta_{-}}}
$ (30)

###### Proof.

From (26), we have :

$\displaystyle\log(\varphi^{+}(x))$
$\displaystyle=\int_{0}^{+\infty}(e^{ivx}-1)\alpha_{+}(\lambda_{+}+\beta_{+}v^{-1})v^{-\beta_{+}}e^{-\lambda_{+}v}dv$
(31)
$\displaystyle=\sum_{k=1}^{\infty}\frac{(ix)^{k}}{k!}\alpha_{+}\left(\lambda_{+}\int_{0}^{+\infty}v^{k-\beta_{+}}e^{-\lambda_{+}v}dv+\beta_{+}\int_{0}^{+\infty}v^{-1-\beta_{+}}e^{-\lambda_{+}v}dv\right)$
$\displaystyle=\sum_{k=1}^{\infty}\frac{(ix)^{k}}{k!}\alpha_{+}\left(\lambda_{+}\frac{\Gamma(k+1-\beta_{+})}{\lambda_{+}^{k+1-\beta_{+}}}+\beta_{+}\frac{\Gamma(k-\beta_{+})}{\lambda_{+}^{k-\beta_{+}}}\right)$
$\displaystyle=\alpha_{+}\lambda_{+}^{\beta_{+}}\Gamma(1-\beta_{+})\left(\sum_{k=1}^{\infty}\frac{\Gamma(k+1-\beta_{+})}{\Gamma(1-\beta_{+})k!}(\frac{ix}{\lambda_{+}})^{k}-\sum_{k=1}^{\infty}\frac{\Gamma(k-\beta_{+})}{\Gamma(-\beta_{+})k!}(\frac{ix}{\lambda_{+}})^{k}\right)$
$\displaystyle=\alpha_{+}\lambda_{+}^{\beta_{+}}\Gamma(1-\beta_{+})\left(\sum_{k=1}^{\infty}{\beta_{+}-1\choose
k}(-\frac{ix}{\lambda_{+}})^{k}-\sum_{k=1}^{\infty}{\beta_{+}\choose
k}(\frac{ix}{\lambda_{+}})^{k}\right)$
$\displaystyle=\alpha_{+}\Gamma(1-\beta_{+})\left(\lambda_{+}(\lambda_{+}-ix)^{\beta_{+}-1}-(\lambda_{+}-ix)^{\beta_{+}}\right)$
$\displaystyle=\alpha_{+}\Gamma(1-\beta_{+})\frac{ix}{(\lambda_{+}-ix)^{1-\beta_{+}}}$

Similarly, we have

$\displaystyle\log(\varphi^{-}(x))$
$\displaystyle=\int_{0}^{+\infty}(e^{ivx}-1)\alpha_{-}(\lambda_{-}+\beta_{-}v^{-1})v^{-\beta_{-}}e^{-\lambda_{-}v}dv=\alpha_{-}\Gamma(1-\beta_{-})\frac{ix}{(\lambda_{-}-ix)^{1-\beta_{-}}}$

And

$\displaystyle\log(\varphi(x))$
$\displaystyle=\log(\varphi^{+}(x))+\log(\varphi^{-}(-x))$
$\displaystyle=\alpha_{+}\Gamma(1-\beta_{+})\frac{ix}{(\lambda_{+}-ix)^{1-\beta_{+}}}+\alpha_{-}\Gamma(1-\beta_{-})\frac{-ix}{(\lambda_{-}+ix)^{1-\beta_{-}}}$

∎

###  3.2  GTS distribution as Background Driving Lévy Process (BDLP)

######  Theorem 6.

The Generalized Tempered Stable (GTS) distribution is the Background Driving
Lévy Process (BDLP) of a self-decomposable random variable $X$. Then the self-
decomposable random variable $X$ has the following Lévy–Khintchine triplet
$\left[\mu,0,U\right]$.

$\displaystyle
U(dx)=\left[\alpha_{+}\lambda_{+}^{\beta_{+}}\frac{\Gamma(-\beta_{+},\lambda_{+}x)}{x}\boldsymbol{1}_{x>0}\right]dx+\left[\alpha_{-}\lambda_{-}^{\beta_{-}}\frac{\Gamma(-\beta_{-},\lambda_{-}x)}{x}\boldsymbol{1}_{x<0}\right]dx$
(32)

###### Proof.

The Lévy density of the Generalized Tempered Stable (GTS) distribution is
defined in (9) as follows.

$\displaystyle
W(dx)=\left(\frac{\alpha_{+}e^{-\lambda_{+}x}}{x^{1+\beta_{+}}}\boldsymbol{1}_{x>0}\right)dx+\left(\frac{\alpha_{-}e^{-\lambda_{-}|x|}}{|x|^{1+\beta_{-}}}\boldsymbol{1}_{x<0}\right)dx=W^{+}(dx)+W^{-}(dx)
$ (33)

Let $\left[a^{+},0,W^{+}\right]$, the Lévy–Khintchine triplet of the Tempered
stable distribution TS($\beta_{+}$, $\alpha_{+}$,$\lambda_{+}$) and
$\left[b^{+},0,U^{+}\right]$, the Lévy–Khintchine triplet of the self-
decomposable random variable $X$.

We apply the property (18) by choosing $A=(y,+\infty)$

$\displaystyle
U^{+}((y,+\infty))=\int_{y}^{+\infty}U^{+}(dx)=\int_{0}^{+\infty}W^{+}((e^{t}y,+\infty))dt=\int_{y}^{+\infty}\frac{W^{+}((u,+\infty))}{u}du$
(34)

In order to have the relation (34), we need to have:

$\displaystyle U^{+}(dx)=\frac{W^{+}((x,+\infty))}{x}dx$

The Lévy density becomes

$\displaystyle U^{+}(dx)=\frac{W^{+}((x,+\infty))}{x}dx$
$\displaystyle=\frac{1}{x}\int_{x}^{+\infty}W^{+}(dy)dx=\frac{1}{x}\int_{x}^{+\infty}\frac{\alpha_{+}e^{-\lambda_{+}y}}{y^{1+\beta_{+}}}dydx$
$\displaystyle=\frac{\alpha_{+}\lambda_{+}^{\beta_{+}}}{x}\int_{\lambda_{+}x}^{+\infty}z^{-\beta_{+}-1}e^{-z}dzdx\hskip
14.22636pt \hbox{$z=\lambda_{+}y$}$
$\displaystyle=\alpha_{+}\lambda_{+}^{\beta_{+}}\frac{\Gamma(-\beta_{+},\lambda_{+}x)}{x}dx$

We have:

$\displaystyle
U^{+}(dx)=\alpha_{+}\lambda_{+}^{\beta_{+}}\frac{\Gamma(-\beta_{+},\lambda_{+}x)}{x}dx$
(35)

We follow the same procedure on the left side of the Tempered stable
distribution TS($\beta_{-}$, $\alpha_{-}$,$\lambda_{-}$) and we have:

$\displaystyle
U^{-}(dx)=\alpha_{-}\lambda_{-}^{\beta_{-}}\frac{\Gamma(-\beta_{-},\lambda_{-}x)}{x}dx$
(36)

The Lévy density of the random variable $X$ has the following expression

$\displaystyle
U(dx)=U^{+}(dx)+U^{-}(dx)=\left[\alpha_{+}\lambda_{+}^{\beta_{+}}\frac{\Gamma(-\beta_{+},\lambda_{+}x)}{x}\boldsymbol{1}_{x>0}+\alpha_{-}\lambda_{-}^{\beta_{-}}\frac{\Gamma(-\beta_{-},\lambda_{-}|x|)}{|x|}\boldsymbol{1}_{x<0}\right]dx$

The relation between the Lévy densities $U^{+}(dx)$ and $W^{+}(dx)$ is
provided if we derive the relation (34).

$\displaystyle U^{+}(dy)$
$\displaystyle=\int_{0}^{+\infty}W^{+}(e^{t}dy)dt=\int_{0}^{+\infty}\alpha_{+}\frac{e^{-\lambda_{+}ye^{t}}}{(ye^{t})^{1+\beta_{+}}}e^{t}dtdy$
(37)

We assume $\phi^{+}(\xi)$ is the characteristic function of the random
variable $X^{+}$.

$\displaystyle\log(\phi^{+}(\xi))$
$\displaystyle=\int_{0}^{+\infty}(e^{iy\xi}-1)U^{+}(dy)=\int_{0}^{+\infty}\int_{0}^{+\infty}(e^{iy\xi}-1)\alpha_{+}\frac{e^{-\lambda_{+}ye^{t}}}{(ye^{t})^{1+\beta_{+}}}e^{t}dtdy$
$\displaystyle=\int_{0}^{+\infty}\int_{0}^{+\infty}(e^{i\xi
ve^{-t}}-1)\alpha_{+}\frac{e^{-\lambda_{+}v}}{v^{1+\beta_{+}}}dtdv=\int_{0}^{1}\int_{0}^{+\infty}(e^{i\xi
vu}-1)W^{+}(dv)dv\frac{du}{u}$ $\displaystyle=\int_{0}^{1}\Psi^{+}(\xi
u))\frac{du}{u}=\int_{0}^{\xi}\Psi^{+}(x))\frac{dx}{x}\quad\quad x=\xi u$

Similarly, we have

$\displaystyle\log(\phi^{-}(\xi))=\int_{0}^{\xi}\Psi^{-}(x))\frac{dx}{x}$

We assume $\phi(\xi)$ is the characteristic function of the random variable
$X$.

$\displaystyle\log(\phi(\xi))$ $\displaystyle=\mu
xi+\log(\phi^{+}(\xi))+\log(\phi^{-}(-\xi))$ (38)
$\displaystyle=\int_{0}^{\xi}(\mu
xi+\Psi^{+}(x)+\Psi^{-}(-x))\frac{dx}{x}=\int_{0}^{\xi}\Psi(x)\frac{dx}{x}$

The random variable $X$ is self-decomposable, and its background driving Lévy
process is the Generalized Tempered Stable (GTS) distribution. ∎

####  3.2.1  Asymptotic behavior of $U(dx)$

As $x\to 0$, we have

$\displaystyle\Gamma(-\beta_{+},\lambda_{+}x)\sim\frac{1}{\beta_{+}\lambda_{+}^{\beta_{+}}}\frac{1}{x^{\beta_{+}}}$
$\displaystyle
U^{+}(dx)\sim\frac{\alpha_{+}}{\beta_{+}}\frac{1}{x^{1+\beta_{+}}}dx $

The Lévy density of the self-decomposable random variable $X$ becomes the Lévy
density of a stable distribution [41].

$\displaystyle
U(dx)\sim\left[\frac{\alpha_{+}}{\beta_{+}}\frac{1}{x^{1+\beta_{+}}}\boldsymbol{1}_{x>0}\right]dx+\left[\frac{\alpha_{-}}{\beta_{-}}\frac{1}{x^{1+\beta_{-}}}\boldsymbol{1}_{x<0}\right]dx$
(39)

As $x\to+\infty$, we have

$\displaystyle\Gamma(-\beta_{+},\lambda_{+}x)\sim\frac{1}{\lambda_{+}^{1+\beta_{+}}}\frac{e^{-\lambda_{+}x}}{x^{1+\beta_{+}}}$
$\displaystyle
U^{+}(dx)\sim\frac{\alpha_{+}}{\lambda_{+}}\frac{e^{-\lambda_{+}x}}{x^{2+\beta_{+}}}dx
$

The Lévy density of the self-decomposable random variable $X$ becomes the
Generalized Tempered Stable (GTS) distribution.

$\displaystyle
U(dx)\sim\left[\frac{\alpha_{+}}{\lambda_{+}}\frac{e^{-\lambda_{+}x}}{x^{2+\beta_{+}}}\boldsymbol{1}_{x>0}\right]dx+\left[\frac{\alpha_{-}}{\lambda_{-}}\frac{e^{-\lambda_{-}x}}{x^{2+\beta_{-}}}\boldsymbol{1}_{x<0}\right]dx$
(40)

U(dx) behaves like a $\alpha$-stable distribution near zero, with a GTS
distribution at the tails.

######  Theorem 7.

(Characteristic Exponents)
The Generalized Tempered Stable (GTS) distribution is the Background Driving
Lévy Process (BDLP) of a self-decomposable random variable $X$. Then the
characteristic exponents of the self-decomposable random variable $X$ is:

$\displaystyle\gamma(\xi)=\mu\xi i+\gamma^{+}(\xi)+\gamma^{-}(-\xi)$ (41)

Where

$\displaystyle\gamma^{+}(\xi)$
$\displaystyle=\alpha_{+}\Gamma(-\beta_{+})\int_{0}^{1}\frac{(\lambda_{+}-i\xi
u)^{\beta_{+}}-\lambda_{+}^{\beta_{+}}}{u}du$ $\displaystyle\gamma^{-}(\xi)$
$\displaystyle=\alpha_{-}\Gamma(-\beta_{-})\int_{0}^{1}\frac{(\lambda_{-}-i\xi
u)^{\beta_{-}}-\lambda_{-}^{\beta_{-}}}{u}du$

###### Proof.

$\displaystyle\Gamma(-\beta_{+},\lambda_{+}y)$
$\displaystyle=\int_{\lambda_{+}y}^{+\infty}x^{-\beta_{+}-1}e^{-x}dx=\int_{1}^{+\infty}(\lambda_{+}y)^{-\beta_{+}}u^{-\beta_{+}-1}e^{-\lambda_{+}yu}du$
(42)

$\displaystyle\int_{0}^{+\infty}(e^{iy\xi}-1)\frac{\Gamma(-\beta_{+},\lambda_{+}y)}{y}dy$
$\displaystyle=\sum_{k=1}^{\infty}\frac{(i\xi)^{k}}{k!}\left[\int_{0}^{+\infty}\frac{y^{k}\Gamma(-\beta_{+},\lambda_{+}y)}{y}dy\right]\hskip
14.22636pt \hbox{recall (\ref{eq:l081a})}$ (43)
$\displaystyle=\sum_{k=1}^{\infty}\frac{(i\xi)^{k}}{k!}\left[\int_{0}^{+\infty}y^{k}\int_{1}^{+\infty}\lambda_{+}^{-\beta_{+}}(uy)^{-\beta_{+}-1}e^{-\lambda_{+}yu}dudy\right]$
$\displaystyle=\sum_{k=1}^{\infty}\frac{(i\xi)^{k}}{k!}\left[\int_{1}^{+\infty}\lambda_{+}^{-\beta_{+}}u^{-\beta_{+}-1}\int_{0}^{+\infty}y^{k-\beta_{+}-1}e^{-\lambda_{+}yu}dydu\right]$
$\displaystyle=\sum_{k=1}^{\infty}\frac{(i\xi)^{k}}{k!}\left[\frac{\Gamma(k-\beta_{+})}{\lambda_{+}^{k}}\int_{1}^{+\infty}u^{-k-1}du\right]$
$\displaystyle=\Gamma(-\beta_{+})\sum_{k=1}^{\infty}\frac{\Gamma(k-\beta_{+})}{k!\Gamma(-\beta_{+})k}(\frac{i\xi}{\lambda_{+}})^{k}$

$\displaystyle\frac{\Gamma(j-\beta)}{\Gamma(-\beta)\Gamma(j+1)}=(-1)^{j}{\beta\choose
j}\hskip 14.22636pt \hbox{ratio of falling factorials (Pochhammer symbol)}$

$\displaystyle\xi\frac{d\left(\sum_{k=1}^{\infty}\frac{\Gamma(k-\beta_{+})}{k!\Gamma(-\beta_{+})}\frac{1}{k}(\frac{i\xi}{\lambda_{+}})^{k}\right)}{d\xi}=\sum_{k=1}^{\infty}\frac{\Gamma(k-\beta_{+})}{k!\Gamma(-\beta_{+})}(\frac{i\xi}{\lambda_{+}})^{k}=\sum_{k=1}^{\infty}{{\beta_{+}}\choose
k}(\frac{-i\xi}{\lambda_{+}})^{k}$ (44)

Newton’s generalized binomial theorem can be applied, and (44) becomes

$\displaystyle\xi\frac{d\left(\sum_{k=1}^{\infty}\frac{\Gamma(k+1-\beta_{+})}{k!\Gamma(1-\beta_{+})}\frac{1}{k}(\frac{i\xi}{\lambda_{+}})^{k}\right)}{d\xi}=(1-\frac{i\xi}{\lambda_{+}})^{\beta_{+}}-1\hskip
14.22636pt \forall\xi\in\mathbb{R}$ (45)

The first-order differential equation can be solved as follows

$\displaystyle\sum_{k=1}^{\infty}\frac{\Gamma(k+1-\beta_{+})}{k!\Gamma(1-\beta_{+})}\frac{1}{k}(\frac{i\xi}{\lambda_{+}})^{k}$
$\displaystyle=\frac{1}{\lambda_{+}^{\beta_{+}}}\int_{0}^{\xi}\frac{(\lambda_{+}-iy)^{\beta_{+}}-\lambda_{+}^{\beta_{+}}}{y}dy$
$\displaystyle=\frac{1}{\lambda_{+}^{\beta_{+}}}\int_{0}^{1}\frac{(\lambda_{+}-i\xi
u)^{\beta_{+}}-\lambda_{+}^{\beta_{+}}}{u}du\quad\quad u=\frac{y}{\xi}$

From the previous development (43), we have :

$\displaystyle\int_{0}^{+\infty}(e^{iy\xi}-1)U^{+}(dx)=\alpha_{+}\Gamma(-\beta_{+})\int_{0}^{1}\frac{(\lambda_{+}-i\xi
u)^{\beta_{+}}-\lambda_{+}^{\beta_{+}}}{u}du$ (46)

We have the same results on $U^{-}(dx)$

$\displaystyle\int_{0}^{+\infty}(e^{iy\xi}-1)U^{-}(dx)=\alpha_{-}\Gamma(-\beta_{-})\int_{0}^{1}\frac{(\lambda_{-}-i\xi
u)^{\beta_{-}}-\lambda_{-}^{\beta_{-}}}{u}du$ (47)

∎

##  4 Fitting General Tempered Stable Distribution Results

###  4.1 Review of the Maximum Likelihood Method

From a probability density function $f(x,V)$ with parameter $V$ of size $p=7$
and the sample data $X$ of size $m$, we definite the likelihood function and
its derivatives as follows:

$\displaystyle L(x,V)$ $\displaystyle=\prod_{j=1}^{m}f(x_{j},V)\quad$
$\displaystyle l(x,V)$ $\displaystyle=\sum_{j=1}^{m}log(f(x_{j},V)) $ (48)
$\displaystyle \frac{dl(x,V)}{dV_{j}}$
$\displaystyle=\sum_{i=1}^{m}\frac{\frac{df(x_{i},V)}{dV_{j}}}{f(x_{i},V)}\quad\quad
\frac{d^{2}l(x,V)}{dV_{k}dV_{j}}$
$\displaystyle=\sum_{i=1}^{m}\left(\frac{\frac{d^{2}f(x_{i},V)}{dV_{k}dV_{j}}}{f(x_{i},V)}-\frac{\frac{df(x_{i},V)}{dV_{k}}}{f(x_{i},V)}\frac{\frac{df(x_{i},V)}{dV_{j}}}{f(x_{i},V)}\right).$
(49)

To perform the maximum of the likelihood function (48), the quantities
$\frac{df(x,V)}{dV_{j}}$ and $\frac{d^{2}f(x,V)}{dV_{k}dV_{j}}$ in (49) are
the first and second order derivatives of the probability density and should
be computed with high accuracy. The quantities
$\frac{d^{2}l(x,V)}{dV_{k}dV_{j}}$ are critical in computing the Hessian
matrix and the Fisher information matrix.

Given the parameter
$V=(\textbf{$\mu$},\textbf{$\beta_{+}$},\textbf{$\beta_{-}$},\textbf{$\alpha_{+}$},\textbf{$\alpha_{-}$},\textbf{$\lambda_{+}$},\textbf{$\lambda_{-}$})$
and the sample data set $X$, we derive from (49) the following vector and
matrix: (50) .

$\displaystyle I^{\prime}(X,V)=\left(\frac{dl(x,V)}{dV_{j}}\right)_{1\leq
j\leq p}\quad\quad
I^{\prime\prime}(X,V)=\left(\frac{d^{2}l(x,V)}{dV_{k}dV_{j}}\right)_{\begin{subarray}{c}{1\leq
k\leq p}\\\ {1\leq j\leq p}\end{subarray}}. $ (50)

The advanced fast FRFT scheme developed previously is used to compute the
likelihood function (48) and its derivatives (50) in the optimization process.

The local solution $V^{0}$ should meet the following requirements:

$\displaystyle I^{\prime}(x,V^{0})=0\quad\quad
U^{T}\mathbf{I^{\prime\prime}(X,V^{0})}U\leq 0\hskip 14.22636pt \hbox{
$\forall U\in\mathbb{R}^{p}$}. $ (51)

The inequality in (51) is met when $I^{\prime\prime}(x,V^{0})$ is a negative
semi-definite matrix. The Newton-Raphson iteration algorithm provides the
numerical solution (52):

$\displaystyle
V^{n+1}=V^{n}-{\left(I^{\prime\prime}(x,V^{n})\right)^{-1}}I^{\prime}(x,V^{n}).
$ (52)

See [42] for details on maximum likelihood and Newton-Raphson iteration
procedure.

###  4.2 GTS Parameter Estimation from S$\&$P 500 Index

The results of the GTS parameter estimation from S$\&$P 500 return data are
reported in Table 1. As expected, the stability indexes
($\beta_{-}$,$\beta_{+}$), the process intensities
($\alpha_{-}$,$\alpha_{+}$), and the decay rate ($\lambda_{-}$,$\lambda_{+}$)
are all positive. The results show that $0\leq\beta_{+}\leq 1$ and
$0\leq\beta_{-}\leq 1$, S$\&$P 500 return is of infinite activity process (10)
with infinite number of jumps in any given time interval; S$\&$P 500 is also a
finite variance process (11).

Table 1:  FRFT Maximum Likelihood GTS Parameter Estimation Model | $\mu$ | $\beta_{+}$ | $\beta_{-}$ | $\alpha_{+}$ | $\alpha_{-}$ | $\lambda_{+}$ | $\lambda_{-}$
---|---|---|---|---|---|---|---
GTS | -0.693477 | 0.682290 | 0.242579 | 0.458582 | 0.414443 | 0.822222 | 0.727607

As shown in Table 1, we have $\alpha_{-}$ $\leq$ $\alpha_{+}$ and the arrival
rate of the positive jump is more than that of the negative jump. In addition,
when $\lambda_{-}$ $\leq$ $\lambda_{+}$, the S$\&$P 500 return is a bit left-
skewed distribution. Based on $\alpha$ and $\lambda$, the tail distribution is
thicker on the negative side of the S$\&$P 500 return distribution ($X_{-}$)
than on the positive side of the distribution.

The Newton-Raphson iteration algorithm (52) was implemented, and the results
are reported in Table 2. Each raw has eleven columns made of the iteration
number, the seven parameters $\mu$, $\beta_{+}$, $\beta_{-}$, $\alpha_{+}$,
$\alpha_{-}$, $\lambda_{+}$, $\lambda_{-}$ and three statistical indicators:
the log-likelihood ($Log(ML)$), the norm of the partial derivatives
($||\frac{dLog(ML)}{dV}||$), the maximum value of the eigenvalues
($MaxEigenValue$). The statistical indicators aim at checking if the two
necessary and sufficient conditions described in equations(51) are all met.
$Log(ML)$ displays the value of the Naperian logarithm of the likelihood
function $L(x,V)$ as described in (48); $||\frac{dLog(ML)}{dV}||$ displays the
value of the norm of the first derivatives ($\frac{dl(x,V)}{dV_{j}}$)
described in equations (49); and $MaxEigenValue$ displays the maximum value of
the seven eigenvalues generated by the Hessian Matrix
($\frac{d^{2}l(x,V)}{dV_{k}dV_{j}}$) as described in equations (50).

Table 2: GTS Parameter Estimations from S$\&$P 500 Index

$Iterations$ | $\mu$ | $\beta_{+}$ | $\beta_{-}$ | $\alpha_{+}$ | $\alpha_{-}$ | $\lambda_{+}$ | $\lambda_{-}$ | $Log(ML)$ | $||\frac{dLog(ML)}{dV}||$ | $MaxEigenValue$
---|---|---|---|---|---|---|---|---|---|---
1 | -0.5266543 | 0.67666185 | 0.43662658 | 0.43109407 | 0.32587754 | 0.85012595 | 0.60145985 | -4664.7647 | 289.206866 | 48.3287566
2 | -0.5459715 | 0.67059496 | 0.42415971 | 0.44989796 | 0.34810329 | 0.80921924 | 0.60289955 | -4660.2156 | 35.9853332 | -6.0431046
3 | -0.7108484 | 0.66676659 | 0.20613817 | 0.46963149 | 0.41246296 | 0.8368645 | 0.73679498 | -4663.5777 | 1082.00304 | 449.252496
4 | -0.6704327 | 0.66891111 | 0.11250993 | 0.46528242 | 0.50028087 | 0.83422813 | 0.86330646 | -4660.5268 | 135.684998 | 15.1103621
5 | -0.739816 | 0.66779246 | 0.09097181 | 0.48302677 | 0.45922815 | 0.85551132 | 0.81320574 | -4660.0208 | 45.7082317 | 10.8419307
6 | -0.6517205 | 0.65580459 | 0.19673172 | 0.47996977 | 0.42743878 | 0.85248358 | 0.75350741 | -4659.833 | 46.3329967 | 11.8534517
7 | -0.8136505 | 0.71997129 | 0.21558026 | 0.44023152 | 0.41952419 | 0.79423596 | 0.74025892 | -4662.4815 | 1187.98211 | 166.006596
8 | -0.7805857 | 0.70635344 | 0.22952102 | 0.44666893 | 0.4166063 | 0.80360771 | 0.7334486 | -4659.7758 | 85.6584355 | -3.4305336
9 | -0.7543747 | 0.69912144 | 0.23468747 | 0.45029318 | 0.41542774 | 0.80935861 | 0.73077041 | -4659.1944 | 1.07446211 | -0.7993855
10 | -0.7533784 | 0.69885561 | 0.23480071 | 0.45042677 | 0.41541378 | 0.80956671 | 0.73072468 | -4659.1943 | 1.03706137 | -0.8136858
11 | -0.752414 | 0.69859801 | 0.2349108 | 0.45055614 | 0.41540021 | 0.8097682 | 0.73068024 | -4659.1942 | 1.00184387 | -0.8273569
12 | -0.7514794 | 0.69834808 | 0.23501796 | 0.45068156 | 0.41538701 | 0.80996353 | 0.73063702 | -4659.1942 | 0.96860649 | -0.8404525
13 | -0.7496917 | 0.69786927 | 0.23522417 | 0.4509216 | 0.41536164 | 0.81033731 | 0.73055392 | -4659.194 | 0.90739267 | -0.8650984
14 | -0.7471903 | 0.69719768 | 0.23551545 | 0.45125776 | 0.41532584 | 0.81086063 | 0.73043671 | -4659.1938 | 0.82669598 | -0.8987475
15 | -0.7463993 | 0.69698491 | 0.23560822 | 0.45136413 | 0.41531445 | 0.8110262 | 0.73039942 | -4659.1937 | 0.80232744 | -0.9091961
16 | -0.7456279 | 0.69677723 | 0.23569899 | 0.45146791 | 0.41530331 | 0.8111877 | 0.73036294 | -4659.1937 | 0.77907591 | -0.9193009
17 | -0.7434222 | 0.69618233 | 0.23596025 | 0.45176484 | 0.41527126 | 0.81164974 | 0.73025805 | -4659.1935 | 0.71530396 | -0.9477492
18 | -0.7427203 | 0.69599272 | 0.2360439 | 0.45185938 | 0.41526101 | 0.81179683 | 0.73022449 | -4659.1934 | 0.69583386 | -0.9566667
19 | -0.7376152 | 0.69460887 | 0.23665992 | 0.45254797 | 0.41518555 | 0.81286777 | 0.72997772 | -4659.193 | 0.56549166 | -1.0197019
20 | -0.7353516 | 0.69399264 | 0.23693733 | 0.45285383 | 0.41515159 | 0.81334323 | 0.72986678 | -4659.1929 | 0.51375646 | -1.0466686
21 | -0.7342715 | 0.69369811 | 0.23707047 | 0.45299987 | 0.41513529 | 0.81357023 | 0.72981356 | -4659.1928 | 0.47702329 | -1.0633442
22 | -0.727748 | 0.69191195 | 0.23788938 | 0.45388257 | 0.41503509 | 0.81494168 | 0.72948683 | -4659.1924 | 0.30254549 | -1.1564316
23 | -0.7269375 | 0.69168915 | 0.23799284 | 0.45399234 | 0.41502243 | 0.81511217 | 0.72944561 | -4659.1924 | 0.28659447 | -1.1671214
24 | -0.7261524 | 0.69147312 | 0.23809343 | 0.45409871 | 0.41501011 | 0.81527736 | 0.72940554 | -4659.1923 | 0.27235218 | -1.177299
25 | -0.7074205 | 0.68625723 | 0.24059125 | 0.45665159 | 0.41470173 | 0.81923664 | 0.72841073 | -4659.1916 | 0.15127988 | -1.3705213
26 | -0.7028887 | 0.68497607 | 0.24122265 | 0.45727561 | 0.4146222 | 0.82020261 | 0.72815815 | -4659.1915 | 0.12096032 | -1.4033087
27 | -0.6987925 | 0.68381109 | 0.24180316 | 0.45784257 | 0.4145479 | 0.82107955 | 0.72792474 | -4659.1914 | 0.07862954 | -1.4283679
28 | -0.6934505 | 0.68228736 | 0.24256739 | 0.45858391 | 0.41444901 | 0.82222574 | 0.72761631 | -4659.1914 | 0.75324649 | -1.6442249
29 | -0.6934765 | 0.68229002 | 0.24257963 | 0.45858236 | 0.41444404 | 0.82222285 | 0.72760762 | -4659.1914 | 9.93E-06 | -1.4542674
30 | -0.6934774 | 0.68229032 | 0.24257975 | 0.45858219 | 0.41444396 | 0.8222226 | 0.7276075 | -4659.1914 | 7.4193E-08 | -1.4542632

As shown in Table 2, the log-likelihood ($Log(ML)$) value starts at
$-4664.7647$ and increases to a limit of $-4659.1914$; the
$||\frac{dLog(ML)}{dV}||$ value starts at $289.206866$ and decreases to $0$;
and the maximum value of the eigenvalues ($MaxEigenValue$) starts at
$48.3287566$ and converges to $-1.4542632$, which is negative. The Hessian
matrix in (50) is a negative semi-definite matrix at a converged solution.
Therefore, both conditions in (51) are met, and we have a locally optimal
solution.

(a) Lévy densities

(b) Characteric exponents

Figure 1:  S&P500 index Data: $\mu=-0.693477$, $\beta_{+}=0.682290$,
$\beta_{-}=0.242579$, $\alpha_{+}=0.458582$, $\alpha_{-}=0.414443$,
$\lambda_{+}=0.822222$, $\lambda_{-}=0.727607$

###  4.3 GTS Parameter Estimation: Bitcoin

The results of the GTS parameter estimation from Bitcoin returns are reported
in Table 3. As expected, the stability indexes ($\beta_{-}$,$\beta_{+}$), the
process intensities ($\alpha_{-}$,$\alpha_{+}$), and the decay rate
($\lambda_{-}$,$\lambda_{+}$) are all positive. The results show that
$0\leq\beta_{+}\leq 1$ and $0\leq\beta_{-}\leq 1$, Bitcoin return is of
infinite activity process (10) with infinite number of jumps in any given time
interval; Bitcoin is also a finite variance process (11).

Table 3:  FRFT Maximum Likelihood GTS Parameter Estimation Model | $\mu$ | $\beta_{+}$ | $\beta_{-}$ | $\alpha_{+}$ | $\alpha_{-}$ | $\lambda_{+}$ | $\lambda_{-}$
---|---|---|---|---|---|---|---
GTS | -0.736924 | 0.461378 | 0.267178 | 0.810017 | 0.517347 | 0.215628 | 0.191937

As shown in Table 3, we have $\alpha_{-}$ $\leq$ $\alpha_{+}$ and the arrival
rate of the positive jump is significantly more than that of the negative
jump. In addition, when we have $\lambda_{-}$ $\leq$ $\lambda_{+}$ then
Bitcoin return is a left-skewed distribution. According to the values of the
process intensity ($\alpha$) and the decay rate($\lambda$), Bitcoin tail
distribution is thicker on the negative side .

The Newton-Raphson iteration algorithm (52) was implemented, and the results
of the iteration process are reported in Table 4. Table 4 and Table 2 have
similar structures. As shown in Table 4, the log-likelihood ($Log(ML)$) value
starts at $-10036.656$ and increases to a limit of $-9751.2193$; the
$||\frac{dLog(ML)}{dV}||$ value starts at $3188.1469$ and decreases to almost
$0$; and the maximum value of the eigenvalues ($MaxEigenValue$) starts at
$2854.2231$ and converges to $-0.000436$, which is negative. The Hessian
matrix in (50) is a negative semi-definite matrix at a converged solution.
Therefore, both conditions in (51) are met, and we have a locally optimal
solution.

Table 4: GTS Parameter Estimation from Bitcoin Returns

$Iterations$ | $\mu$ | $\beta_{+}$ | $\beta_{-}$ | $\alpha_{+}$ | $\alpha_{-}$ | $\lambda_{+}$ | $\lambda_{-}$ | $Log(ML)$ | $||\frac{dLog(ML)}{dV}||$ | $MaxEigenValue$
---|---|---|---|---|---|---|---|---|---|---
1 | -0.5973 | 0.3837 | 0.4206 | 1.872 | 0.694 | 0.5333 | 0.219 | -10036.656 | 3188.1469 | 2854.2231
2 | -1.2052311 | 0.35850753 | 0.44451399 | 2.00512495 | 0.62313779 | 0.53840744 | 0.20635235 | -9912.1541 | 1971.95236 | 1518.15345
3 | -1.624469 | 0.31755508 | 0.45798507 | 2.23748588 | 0.58475918 | 0.56388213 | 0.17998332 | -9865.5242 | 952.027546 | 716.347539
4 | -1.9959515 | 0.2615866 | 0.47359748 | 2.57864607 | 0.5494987 | 0.60990478 | 0.16377681 | -9862.2551 | 440.489423 | 295.233361
5 | -2.8340545 | -0.0033236 | 0.55268657 | 4.51611949 | 0.47359381 | 0.87264266 | 0.130233 | -9911.337 | 510.396179 | 60.6518499
6 | -2.701598 | 0.07378625 | 0.54127007 | 3.76512451 | 0.47819392 | 0.7743675 | 0.13294829 | -9896.0802 | 611.613266 | 117.692562
7 | -1.4130144 | 0.37715064 | 0.48510637 | 1.50024644 | 0.5523787 | 0.39998447 | 0.15474644 | -9804.2548 | 801.962681 | 307.039536
8 | -1.049984 | 0.45100489 | 0.4741659 | 1.13318569 | 0.56070768 | 0.30817133 | 0.15667085 | -9775.871 | 728.753228 | 295.768229
9 | -0.8211241 | 0.50542632 | 0.46591368 | 0.90824445 | 0.54487666 | 0.23555274 | 0.1533588 | -9756.7912 | 397.102674 | 133.652433
10 | -0.7319208 | 0.53998085 | 0.44372538 | 0.81377008 | 0.54717214 | 0.19930192 | 0.15902826 | -9752.0888 | 89.8942291 | 12.4808028
11 | -1.2317854 | 0.58475806 | 0.23780021 | 0.81260395 | 0.49719481 | 0.18597445 | 0.19343462 | -9751.3131 | 32.1533364 | -2.5188051
12 | -0.7150133 | 0.4721273 | 0.29761245 | 0.80521805 | 0.52356611 | 0.21204916 | 0.18644449 | -9750.6869 | 5.52013773 | 0.11490414
13 | -0.7271098 | 0.45992936 | 0.27054512 | 0.80932883 | 0.51829775 | 0.21585683 | 0.19139837 | -9751.2179 | 1.13726334 | 0.19172007
14 | -0.7750442 | 0.46778988 | 0.25646086 | 0.81245021 | 0.51411912 | 0.21450942 | 0.19360685 | -9751.2252 | 1.00910053 | -0.9423667
15 | -0.7090246 | 0.45678856 | 0.27535855 | 0.80819689 | 0.51977265 | 0.21641582 | 0.19065445 | -9751.2151 | 0.68576109 | 0.57268124
16 | -0.7494266 | 0.46338031 | 0.26341052 | 0.81084098 | 0.51624184 | 0.21529106 | 0.19253194 | -9751.2211 | 0.65747338 | -0.2888343
17 | -0.7429071 | 0.46233754 | 0.26537785 | 0.8104111 | 0.51681851 | 0.21546689 | 0.19222166 | -9751.2201 | 0.65058847 | -0.1361475
18 | -0.7401206 | 0.46189156 | 0.2662183 | 0.81022749 | 0.51706495 | 0.21554213 | 0.19208913 | -9751.2197 | 0.64914609 | -0.0724136
19 | -0.7369888 | 0.4613886 | 0.26715947 | 0.81002148 | 0.51734136 | 0.21562718 | 0.19194083 | -9751.2193 | 0.64887078 | -0.0018685
20 | -0.7369633 | 0.4613845 | 0.26716714 | 0.8100198 | 0.51734361 | 0.21562787 | 0.19193962 | -9751.2193 | 0.64887314 | -0.0012986
21 | -0.7369455 | 0.46138165 | 0.26717247 | 0.81001863 | 0.51734517 | 0.21562835 | 0.19193878 | -9751.2193 | 0.64887482 | -0.0009025
22 | -0.7369332 | 0.46137967 | 0.26717617 | 0.81001782 | 0.51734626 | 0.21562869 | 0.1919382 | -9751.2193 | 0.64887601 | -0.0006273
23 | -0.7369246 | 0.46137829 | 0.26717875 | 0.81001726 | 0.51734702 | 0.21562892 | 0.19193779 | -9751.2193 | 0.64887685 | -0.000436

See [27] for details on fitting GTS distribution with fast FRFT.

(a) Lévy densities

(b) Characteric exponents

Figure 2: Bitcoin Data: $\mu=-0.737459$, $\beta_{+}=0.461722$,
$\beta_{-}=0.267500$, $\alpha_{+}=0.810017$, $\alpha_{-}=0.517386$,
$\lambda_{+}=0.215545$, $\lambda_{-}=0.191874$

##  5  Applications and Simulation of S&P index and Bitcoin process

###  5.1  Stationary process Algorithm of Ornstein-Uhlenbeck type

We recall the stationary process of Ornstein-Uhlenbeck type
$\left\\{X(t)\right\\}_{t>0}$ and a background driving Lévy process (BDLP)
$\left\\{L(t)\right\\}_{t>0}$ independent of $X_{0}$ , such that
$X(t)\overset{d}{=}X$ as described in Theorem (5)

$\displaystyle X_{t}=e^{-\lambda
t}X_{0}+\int_{0}^{t}e^{-\lambda(t-s)}dL_{\lambda s}=e^{-\lambda
t}X_{0}+\int_{0}^{\lambda t}e^{-\lambda t+s}dL_{s} \quad\lambda>0\. $ (53)

######  Theorem 8.

Let $\phi$ be the characteristic exponent of a random variable X and $\Psi(x)$
be the characteristic exponent of a random variable $L_{1}$.
$\left\\{X(t)\right\\}_{t>0}$ is the stationary process of Ornstein-Uhlenbeck
type, such that $X(t)\overset{d}{=}X$.

$\displaystyle\lim_{t\to+\infty}Log(E\left[e^{i\xi
X_{t}}\right])=\phi(\xi)=\int_{0}^{\xi}\frac{\Psi(u)}{u}du $ (54)

###### Proof.

$X_{0}$ is an initial value, not a variable.

$\displaystyle Log(E\left[e^{i\xi X_{t}}\right])$ $\displaystyle=e^{-\lambda
t}X_{0}\xi i+\int_{0}^{\lambda t}Log(E\left[e^{i\xi e^{-\lambda
t+s}L_{1}}\right])ds=e^{-\lambda t}X_{0}\xi i+\int_{0}^{\lambda t}\Psi(\xi
e^{-\lambda t+s})ds$ $\displaystyle=e^{-\lambda t}X_{0}\xi i+\int_{\xi
e^{-\lambda t+s}}^{\xi}\frac{\Psi(u)}{u}du\quad\quad u=\xi e^{-\lambda t+s}$

We take the limit on both side

$\displaystyle\lim_{t\to+\infty}Log(E\left[e^{i\xi
X_{t}}\right])=\phi(\xi)=\int_{0}^{\xi}\frac{\Psi(u)}{u}du$

∎

######  Theorem 9.

Let us consider the stationary process of Ornstein-Uhlenbeck type
$\left\\{X(t)\right\\}_{t>0}$ with background driving Lévy process (BDLP)
$\left\\{L(t)\right\\}_{t>0}$, such that $X(t)\overset{d}{=}X$. The discrete-
time horizon is $t_{0}<t_{1}<t_{2}<t_{3},...,<t_{n-1}<t_{n}$ with $\Delta
t=t_{i}-t_{i-1}$. The discrete version of the stationary process of Ornstein-
Uhlenbeck type becomes

$\displaystyle X_{k\Delta t}=e^{-\lambda\Delta t}X_{(k-1)\Delta
t}+\int_{0}^{\lambda\Delta t}e^{-\lambda\Delta t+s}dL_{s} $ (55)

###### Proof.

For $k=1$, replace $t$ by $\Delta t$ in the Equation (5.1) and we have :

$\displaystyle X_{\Delta t}=e^{-\lambda\Delta t}X_{0}+\int_{0}^{\lambda\Delta
t}e^{-\lambda\Delta t+s}dL_{s}$

We assume the equation (55) holds for k. We have the following relation for
$k+1$.

$\displaystyle X_{(k+1)\Delta t}$ $\displaystyle=e^{-\lambda(k+1)\Delta
t}X_{0}+\int_{0}^{\lambda(k+1)\Delta t}e^{-\lambda(k+1)\Delta t+s}dL_{s}$
$\displaystyle=e^{-\lambda\Delta t}\left(e^{-\lambda k\Delta
t}X_{0}+\int_{0}^{\lambda k\Delta t}e^{-\lambda k\Delta
t+s}dL_{s}\right)+\int_{\lambda k\Delta t}^{\lambda(k+1)\Delta
t}e^{-\lambda(k+1)\Delta t+s}dL_{s}$ $\displaystyle=e^{-\lambda\Delta
t}X_{k\Delta t}+\int_{0}^{\lambda\Delta t}e^{-\lambda\Delta t+u}dL_{u+\lambda
k\Delta t}\quad\quad u=s-\lambda k\Delta t\quad dL_{u+\lambda k\Delta
t}=dL_{u}$ $\displaystyle=e^{-\lambda\Delta t}X_{k\Delta
t}+\int_{0}^{\lambda\Delta t}e^{-\lambda\Delta t+u}dL_{u} $

∎

The relation (5.1) can be used to simulate $Y=\int_{0}^{\lambda\Delta
t}e^{-\lambda\delta t+s}dL_{s}$, given that $X_{(k-1)\Delta t}\overset{d}{=}X$
and $X_{k\Delta t}\overset{d}{=}X$. Let $\phi$ be the characteristic exponent
of a random variable X. From (5.1), we have

$\displaystyle Log(E\left[e^{i\xi X_{\Delta
t}}\right])=Log(E\left[e^{i\xi{e^{-\lambda\Delta t}X_{(k-1)\Delta
t}}}\right])+Log(E\left[e^{i\xi Y}\right])$ (56)

It is

$\displaystyle\phi(\xi)=\phi(e^{-\lambda\Delta t}\xi)+Log(E\left[e^{i\xi
Y}\right])\quad\quad Log(E\left[e^{i\xi
Y}\right])=\phi(\xi)-\phi({e^{-\lambda\Delta t}\xi})$ (57)

The Fourier transform generated by the characteristic exponent of
$Y=\int_{0}^{\lambda\Delta t}e^{-\lambda\delta t+s}dL_{s}$ in (57) and the
probability density function (PDF) have the following expression.

$F_{Y}[f](\xi)=e^{\phi(-\xi)-\phi(-{e^{-\lambda\Delta t}\xi})}\quad\quad
f_{Y}(y)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\\!F[f](\xi)e^{iy\xi}\,\mathrm{d}\xi.$
(58)

The composite of a FRFT of a 15-long weighted sequence and a FRFT of an N-long
sequence was used to compute the probability density functions 58. See [43,
44, 18] for details on composite FRFTs and applications.

(a) S&P500 index Daily Data

(b) Bitcoin Daily Data

Figure 3:  probability Density Function (PDF) of $Y=\int_{0}^{\lambda\Delta
t}e^{-\lambda\delta t+s}dL_{s}$ Algorithm 1 Simulating the stationary process
of Ornstein-Uhlenbeck type

1:GTS($\beta_{+}$, $\beta_{-}$, $\alpha_{+}$,$\alpha_{-}$, $\lambda_{+}$,
$\lambda_{-}$); discrete-time $t_{0}<t_{1}<t_{2},...,<t_{n-1}<t_{n}$

2:Initialization: $X_{0}\leftarrow x$.

3:For $i=1:N$

4:$a\leftarrow e^{-\lambda\Delta t_{i}}$ $\triangleright$ $\Delta
t_{i}=t_{i}-t_{i-1}$ and $\lambda=1$

5:$u_{i}\sim\mathscr{U}(0,1)$ $\triangleright$ $\mathscr{U}(0,1)$ is is the
Uniform distribution

6:$y_{i}\sim F_{Y}^{-1}(u_{i})$ $\triangleright$ $F_{Y}(y)$ is a CDF
associated to the PDF $f_{Y}(y)$ in (58)

7:$X_{i\Delta t_{i}}\leftarrow a*X_{(i-1)\Delta t_{i-1}}+y_{i}$

8:EndFor

###  5.2 Applications and Simulation results

The stationary process of Ornstein-Uhlenbeck type is performed using the
previously developed Algorithm to simulate the S&P index and Bitcoin process.
The simulations are carried out on the GTS - OU stationary process of the
Ornstein-Uhlenbeck type and the OU - GTS stationary process of the Ornstein-
Uhlenbeck type. The theoretical values generated by the the cumulant
generating function $(\ref{eq:l62})$ and the simulation values of means,
variance, skewness, and kurtosis are compared; the relative error (Error %)
for each statistical indicator is also provided.
The relation (54) can be used as a cumulant function of the stationary process
of Ornstein-Uhlenbeck type $\left\\{X(t)\right\\}_{t>0}$. The cumulants
($\kappa_{k}$) and the cumulant generating function of the GTS($\mu$,
$\beta_{+}$, $\beta_{-}$, $\alpha_{+}$,$\alpha_{-}$, $\lambda_{+}$,
$\lambda_{-}$) distribution was developed in Theorem 2.3 [18].

$\displaystyle\Psi(\xi)=Log\left(Ee^{iY\xi}\right)$
$\displaystyle=\sum_{j=1}^{+\infty}\kappa_{j}\frac{(i\xi)^{j}}{j!}
\quad\kappa_{1}=\mu+\alpha_{+}{\frac{\Gamma(1-\beta_{+})}{\lambda_{+}^{1-\beta_{+}}}}-\alpha_{-}{\frac{\Gamma(1-\beta_{-})}{\lambda_{-}^{1-\beta_{-}}}}$
(59) $\displaystyle\kappa_{k}$
$\displaystyle=\alpha_{+}{\frac{\Gamma(k-\beta_{+})}{\lambda_{+}^{k-\beta_{+}}}}+(-1)^{k}\alpha_{-}{\frac{\Gamma(k-\beta_{-})}{\lambda_{-}^{k-\beta_{-}}}}$

where $\kappa_{k}$ is the $k^{th}$ cumulant of the GTS distribution.
The cumulant generating function of the stationary process of Ornstein-
Uhlenbeck type $\left\\{X(t)\right\\}_{t>0}$ is deduced from relation (59)

$\displaystyle\phi(\xi)=\int_{0}^{\xi}\frac{\Psi(u)}{u}du=\sum_{k=1}^{+\infty}\frac{\kappa_{k}}{k}\frac{(i\xi)^{k}}{k!}\quad
m_{1}=\kappa_{1}\quad m_{k}=\frac{\kappa_{k}}{k}$ (60)

We have the following relation for the mean, Variance, skewness, and Kurtosis
for the stationary process of Ornstein-Uhlenbeck type
$\left\\{X(t)\right\\}_{t>0}$.

$\displaystyle m_{1}$
$\displaystyle=\kappa_{1}=\mu+\alpha_{+}{\frac{\Gamma(1-\beta_{+})}{\lambda_{+}^{1-\beta_{+}}}}-\alpha_{-}{\frac{\Gamma(1-\beta_{-})}{\lambda_{-}^{1-\beta_{-}}}}$
(61) $\displaystyle m_{2}$
$\displaystyle=\frac{\kappa_{2}}{2}=\frac{1}{2}\left[\alpha_{+}{\frac{\Gamma(k-\beta_{+})}{\lambda_{+}^{k-\beta_{+}}}}+(-1)^{k}\alpha_{-}{\frac{\Gamma(k-\beta_{-})}{\lambda_{-}^{k-\beta_{-}}}}\right]$
$\displaystyle\frac{m_{3}}{m_{2}^{\frac{3}{2}}}$
$\displaystyle=\frac{2^{\frac{3}{2}}}{3}\frac{\kappa_{3}}{\kappa_{2}^{\frac{3}{2}}}\quad\quad\left(3+\frac{m_{4}}{m_{2}^{2}}\right)=\left(3+\frac{\kappa_{4}}{\kappa_{2}^{2}}\right)$

####  5.2.1  Results for OU- GTS Process

The daily cumulative returns for the S&P500 index and Bitcoin are simulated
under the assumption that the daily return follows the stationary distribution
defined by the Lévy measure (32). The simulations are provided by the
stationary process of Ornstein-Uhlenbeck type (55). The sample path shows in
Fig 4(a) for the S&P500 daily cumulative returns and in Fig 4(b) for the
Bitcoin daily cumulative returns. The sample path in red is compared to the
actual values of daily cumulative returns in black.

(a) S&P 500 index daily returns

(b) Bitcoin daily returns

Figure 4: Simulation OU- GTS Process: daily cumulative returns ($\%$)

The mean ($m_{1}$), Variance ($m_{2}$), skewness and Kurtosis developed in
(61) for the stationary distribution were computed and summarised in Table
(5), on a raw labelled exact value. These theoretical values are compared with
the empirical values from sample paths generated by the stationary process of
Ornstein-Uhlenbeck type (55). The sample size of the sample paths increases
from 1000 to 5000. As shown in Table (5), the empirical value converges to the
exact value when the sample size increases for each statistical indicator.

Table 5: S$\&$P 500 return theoretical values versus the empirical values of the OU-GTS stationary process | $\displaystyle{\lim_{t\to\infty}}E\left[X_{t}|X_{0}\right]$ | $\displaystyle{\lim_{t\to\infty}}Var\left[X^{2}_{t}|X_{0}\right]$ | $\displaystyle{\lim_{t\to\infty}}Skew\left[X_{t}|X_{0}\right]$ | $\displaystyle{\lim_{t\to\infty}}Kurtosis\left[X_{t}|X_{0}\right]$
---|---|---|---|---
Exact value | 0.04013 | 0.77410 | -0.54649 | 8.92320
Sample Size | Empirical Statistics from Simulations
| Error(%) |  | Error(%) |  | Error(%) |  | Error(%)
1000 | 0.03835 | -4.45524 | 0.76772 | -0.82393 | -0.44509 | -18.55559 | 7.53707 | -15.53397
1500 | 0.03488 | -13.09001 | 0.77405 | -0.00658 | -0.57732 | 5.64163 | 8.98251 | 0.66473
2500 | 0.03956 | -1.41919 | 0.77140 | -0.34865 | -0.53762 | -1.62352 | 8.52938 | -4.41345
5000 | 0.04039 | 0.64644 | 0.77349 | -0.07943 | -0.54792 | 0.26251 | 8.87311 | -0.56136

Table (5) shows the results for S$\&$P 500 returns, whereas the results for
Bitcoin returns are displayed in Table 7 in Appendice A. Both tables show that
the simulation process is effective, and each sample path comes from the OU-
GTS stationary process.

####  5.2.2  Results for GTS - OU Process

The daily cumulative returns for S&P500 index and Bitcoin are simulated under
the assumption that the daily return follows the GTS distribution. The sample
path shows in Fig 5(a) for S&P500 daily cumulative returns and in Fig 5(b) for
Bitcoin daily cumulative returns. The sample path in red is compared to the
actual values of daily cumulative returns in black.

(a) S&P 500 index daily returns

(b) Bitcoin daily returns

Figure 5: Simulation GTS - OU Process: Daily cumulative returns ($\%$)

The mean ($m_{1}$) , Variance ($m_{2}$), skewness and Kurtosis developed in
(59) for GTS distribution were computed and summarised in Table (6). The
empirical value converges to the exact value for each statistical indicator
when the sample size increases.

Table 6:  S$\&$P 500 return theoretical values versus empirical values of the GTS - OU stationary process | $\displaystyle{\lim_{t\to\infty}}E\left[X_{t}|X_{0}\right]$ | $\displaystyle{\lim_{t\to\infty}}Var\left[X^{2}_{t}|X_{0}\right]$ | $\displaystyle{\lim_{t\to\infty}}Skew\left[X_{t}|X_{0}\right]$ | $\displaystyle{\lim_{t\to\infty}}Kurtosis\left[X_{t}|X_{0}\right]$
---|---|---|---|---
Exact value | 0.04013 | 1.09475 | -0.57964 | 8.92320
Sample Size | Empirical Statistics from Simulations
| Error(%) |  | Error(%) |  | Error(%) |  | Error(%)
1000 | 0.03682 | -8.26002 | 1.08798 | -0.61770 | -0.56865 | -1.89692 | 8.83489 | -0.98962
1500 | 0.04204 | 4.75799 | 1.10069 | 0.54257 | -0.57351 | -1.05806 | 8.56435 | -4.02145
2500 | 0.04224 | 5.23894 | 1.09164 | -0.28379 | -0.53099 | -8.39354 | 8.64516 | -3.11584
5000 | 0.04178 | 4.11213 | 1.09141 | -0.30449 | -0.56619 | -2.32057 | 8.70841 | -2.40708

Table (6) shows the results for S$\&$P 500 Returns, whereas the results for
Bitcoin returns are displayed in Table 8 in Appendice A. Both tables show that
the simulation process is effective, and each sample path comes from the GTS-
OU stationary process.

##  Appendix A Bitcoin: Theoretical versus Empirical values

Table 7: Bitcoin return theoretical values versus empirical values of the OU-GTS stationary process | $\displaystyle{\lim_{t\to\infty}}E\left[X_{t}|X_{0}\right]$ | $\displaystyle{\lim_{t\to\infty}}Var\left[X^{2}_{t}|X_{0}\right]$ | $\displaystyle{\lim_{t\to\infty}}Skew\left[X_{t}|X_{0}\right]$ | $\displaystyle{\lim_{t\to\infty}}Kurtosis\left[X_{t}|X_{0}\right]$
---|---|---|---|---
Exact value | 0.14890 | 2.81898 | -0.30158 | 9.74634
Sample Size | Empirical Statistics from Simulations
| Error(%) |  | Error(%) |  | Error(%) |  | Error(%)
1000 | 0.13623 | -8.51088 | 2.81760 | -0.04900 | -0.30070 | -0.29135 | 8.23142 | -15.54341
1500 | 0.13806 | -7.28426 | 2.80073 | -0.64725 | -0.32346 | 7.25614 | 9.37698 | -3.78972
2500 | 0.15173 | 1.89851 | 2.81712 | -0.06598 | -0.29138 | -3.38203 | 9.57965 | -1.71027
5000 | 0.14987 | 0.64693 | 2.81808 | -0.03186 | -0.29731 | -1.41514 | 9.66934 | -0.79007
Table 8: Bitcoin return theoretical values versus empirical values of the GTS - OU stationary process | $\displaystyle{\lim_{t\to\infty}}E\left[X_{t}|X_{0}\right]$ | $\displaystyle{\lim_{t\to\infty}}Var\left[X^{2}_{t}|X_{0}\right]$ | $\displaystyle{\lim_{t\to\infty}}Skew\left[X_{t}|X_{0}\right]$ | $\displaystyle{\lim_{t\to\infty}}Kurtosis\left[X_{t}|X_{0}\right]$
---|---|---|---|---
Exact value | 0.14890 | 3.98664 | -0.31987 | 9.74634
Sample Size | Empirical Statistics from Simulations
| Error(%) |  | Error(%) |  | Error(%) |  | Error(%)
1000 | 0.15521 | 4.23755 | 4.00641 | 0.49589 | -0.24091 | -24.68461 | 9.49887 | -2.53906
1500 | 0.11793 | -20.80070 | 3.98615 | -0.01217 | -0.36310 | 13.51377 | 9.56633 | -1.84691
2500 | 0.14404 | -3.26698 | 3.96137 | -0.63372 | -0.33935 | 6.08765 | 9.22822 | -5.31603
5000 | 0.15275 | 2.58583 | 3.97762 | -0.22608 | -0.32867 | 2.75017 | 9.42721 | -3.27430

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