A Note on the Background Driving Process Associated with Generalized Tempered Stable Distributions

1. Introduction

The modeling of financial asset returns has long challenged the classical assumptions of normality and geometric Brownian motion (GBM). Empirical evidence overwhelmingly demonstrates that asset returns exhibit heavy tails, excess kurtosis, asymmetry, and volatility clustering; stylized facts that the normal distribution fails to capture [1,2,3,4,5]. Tempered stable distributions were introduced to overcome the limitations of both the stable distribution, which lacks finite variance and higher moments, and the normal distribution, which underestimates tail risk [6,7,8,9].

The Generalized Tempered Stable (GTS) class is a flexible framework constructed by applying exponential tempering to the Lévy measure of a stable distribution [10,11]. The resulting six-parameter Lévy measure is given by:

$$M\left(d\xi \right)=\left[{\displaystyle \frac{{\alpha }_{+}}{{\xi }^{1+{\beta }_{+}}}}{e}^{-{\lambda }_{+}\xi }{\mathbf{1}}_{(0,\infty )}\left(\xi \right)+{\displaystyle \frac{{\alpha }_{-}}{{\left|\xi \right|}^{1+{\beta }_{-}}}}{e}^{-{\lambda }_{-}\left|\xi \right|}{\mathbf{1}}_{(-\infty ,0)}\left(\xi \right)\right]d\xi ,$$

(1)

where $\xi \in \mathbb{R}$, $0\le {\beta }_{\pm }\le 1$, ${\alpha }_{\pm }\ge 0$, and ${\lambda }_{\pm }\ge 0$. This specification nests the KoBoL, bilateral Gamma, Variance Gamma, and CGMY models as special cases [11,12,13,14,15,16,17,18,19,20].

Despite its theoretical flexibility and empirical success, the Background Driving Lévy Process (BDLP) underlying the GTS distribution remains unexplored [21,22]. The BDLP is central to the theory of self-decomposable distributions [23,24,25,26] and to the construction of stationary Ornstein–Uhlenbeck processes widely used in stochastic volatility modeling [27,28,29,30].

This paper fills this gap by investigating the self-decomposable structure of the GTS distribution. Our main contributions are as follows:

• Prove that the GTS distribution is self-decomposable by deriving the explicit form of its Background Driving Lévy Density (BDLD).
• Establish that the associated BDLP is a finite-variation, infinite-activity Type B Lévy process.
• And derive closed-form expressions for the cumulants of the GTS background driving distribution.

The paper is organized as follows. Section 2 provides an overview of the GTS distribution and its key properties. Section 3 establishes the main theoretical results, deriving the explicit BDLD. Section 4 analyzes the structure of the GTS background driving process and presents its cumulants. Section 5 concludes.

2. Generalized Tempered Stable (GTS) Distribution: Overview

The Generalized Tempered Stable (GTS) distribution is a family of infinitely divisible distributions that generalizes the classical stable laws by introducing exponential tempering into the Lévy measure. This modification preserves the power-law behavior in the central part of the distribution while damping the tails, ensuring the existence of moments and improving the tractability of the model in applications.

Formally, a random variable $Z\sim GTS({\beta }_{+},{\beta }_{-},{\alpha }_{+},{\alpha }_{-},{\lambda }_{+},{\lambda }_{-})$ has a Lévy measure $V\left(dx\right)$ given by Eq (1), where

• ${\beta }_{+},{\beta }_{-}\in (0,1)$ are stability index parameters, controlling the heaviness of the tails on the positive and negative axes, respectively;
• ${\alpha }_{+},{\alpha }_{-}>0$ are scale parameters, determining the overall intensity of jumps;
• ${\lambda }_{+},{\lambda }_{-}>0$ are tempering parameters, governing the exponential decay of large jumps in either direction.

The importance of the Lévy measure is shown by the Lévy–Khintchine representation [31,32], which states that the characteristics of any Lévy process are uniquely defined by a triplet $(a,{\sigma }^2,V)$, consisting of

• A drift term (a);
• A diffusion or variance coefficient (${\sigma }^2$), which controls the continuous, Gaussian-motion component;
• The Lévy measure (V), which precisely quantifies the frequency and size of the jumps.

As shown in [26,33,34], GTS distribution is a finite variation process, and generates a type B Lévy process [35], which is a purely non-Gaussian infinite activity Lévy process of finite variation whose sample paths have an infinite number of small jumps and a finite number of large jumps in any finite time interval. In particular, being of bounded variation shows that $Z\sim GTS({\beta }_{+},{\beta }_{-},{\alpha }_{+},{\alpha }_{-},{\lambda }_{+},{\lambda }_{-})$ can be written as the difference of two independent subordinators [36,37]:

$$Z=Z^{+}-Z^{-}\mathrm{with}{Z}^{+}\sim TS({\beta }_{+},{\alpha }_{+},{\lambda }_{+}),Z^{-}\sim TS({\beta }_{-},{\alpha }_{-},{\lambda }_{-}),$$

where $Z_{+}\sim TS({\beta }_{+},{\alpha }_{+},{\lambda }_{+})$ and $Z_{-}\sim TS({\beta }_{-},{\alpha }_{-},{\lambda }_{-})$ are subordinators.

By adding a drift parameter, we have the following expression:

$$\begin{array}{c}\hfill V=\mu +Z\sim GTS(\mu ,{\beta }_{+},{\beta }_{-},{\alpha }_{+},{\alpha }_{-},{\lambda }_{+},{\lambda }_{-}).\end{array}$$

(2)

Theorem 1.

Consider a variable $V\sim GTS(\mu ,{\beta }_{+},{\beta }_{-},{\alpha }_{+},{\alpha }_{-},{\lambda }_{+},{\lambda }_{-})$, the characteristic exponents can be written as

$$\begin{array}{cc}\hfill \psi \left(\xi \right)& =Log\left(E{e}^{iV\xi }\right)=\mu \xi i+{\psi }^{+}\left(\xi \right)+{\psi }^{-}(-\xi )\hfill \end{array}$$

(3)

Where

$$\begin{array}{cc}\hfill {\psi }^{+}\left(\xi \right)& =Log\left(E{e}^{i{Z}^{+}\xi }\right)={\alpha }_{+}\Gamma (-{\beta }_{+})\left({({\lambda }_{+}-i\xi )}^{{\beta }_{+}}-{{\lambda }_{+}}^{{\beta }_{+}}\right)\hfill \\ \hfill {\psi }^{-}\left(\xi \right)& =Log\left(E{e}^{i{Z}^{-}\xi }\right)={\alpha }_{-}\Gamma (-{\beta }_{-})\left({({\lambda }_{-}-i\xi )}^{{\beta }_{-}}-{{\lambda }_{-}}^{{\beta }_{-}}\right)\hfill \end{array}$$

See [33,38] for theorem 1 proof.

3. GTS Background Driving Lévy Density (GTSBDLD)

Theorem 2.

The Generalized Tempered Stable (GTS) distribution is self-decomposable. Its background driving Lévy process Y is a compound Poisson process with Lévy exponent given by the Lévy density

$$N\left(dx\right)=\left[{\alpha }_{+}{\displaystyle \frac{{\beta }_{+}+{\lambda }_{+}x}{x^{1+{\beta }_{+}}}}{e}^{-{\lambda }_{+}x}{\mathbf{1}}_{x>0}\right]dx+\left[{\alpha }_{-}{\displaystyle \frac{{\beta }_{-}+{\lambda }_{-}\left|x\right|}{{\left|x\right|}^{1+{\beta }_{-}}}}{e}^{-{\lambda }_{-}\left|x\right|}{\mathbf{1}}_{x<0}\right]dx.$$

Proof:

Let $\left[a_{{\psi }^{+}},{\sigma }_{{\psi }^{+}}^2,M_{{\psi }^{+}}\right]$ be the Lévy–Khintchine triplet of the tempered stable distribution $\mathrm{TS}({\beta }_{+},{\alpha }_{+},{\lambda }_{+})$, and let $\left[b_{{\phi }^{+}},s_{{\phi }^{+}}^2,N_{{\phi }^{+}}\right]$ be the Lévy–Khintchine triplet of the associated distribution.

For $A=(y,+\infty )$, we have

$$M_{{\psi }^{+}}\left((y,+\infty )\right)={\int }_y^{+\infty }{\displaystyle \frac{{\alpha }_{+}}{x^{1+{\beta }_{+}}}}{e}^{-{\lambda }_{+}x}dx={\int }_0^{+\infty }{N}_{{\phi }^{+}}\left((e^{t}y,+\infty )\right)dt={\int }_y^{+\infty }{\displaystyle \frac{N_{{\phi }^{+}}\left((u,+\infty )\right)}{u}}du.$$

(4)

For (1) to hold, we require

$$N_{{\phi }^{+}}\left((u,+\infty )\right)={\displaystyle \frac{{\alpha }_{+}}{u^{{\beta }_{+}}}}{e}^{-{\lambda }_{+}u}.$$

Differentiating with respect to u yields and we have the Lévy density:

$$N_{{\phi }^{+}}\left(du\right)=-{\displaystyle \frac{d}{du}}{N}_{{\phi }^{+}}\left((u,+\infty )\right)du={\alpha }_{+}\left({\lambda }_{+}+{\beta }_{+}{u}^{-1}\right)u^{-{\beta }_{+}}{e}^{-{\lambda }_{+}u}du.$$

From (1), the relationship between the Lévy densities $M_{{\psi }^{+}}\left(dy\right)$ and $N_{{\phi }^{+}}\left(dv\right)$ follows:

$$M_{{\psi }^{+}}\left(dy\right)=-{\displaystyle \frac{d{M}_{{\psi }^{+}}\left((y,+\infty )\right)}{dy}}dy={\int }_0^{+\infty }{e}^t{N}_{{\phi }^{+}}\left(e^{t}dy\right)dt={\int }_0^{+\infty }{\alpha }_{+}\left({\lambda }_{+}+{\beta }_{+}{e}^{-t}{y}^{-1}\right)e^{(1-{\beta }_{+})t}{y}^{-{\beta }_{+}}{e}^{-{\lambda }_{+}y{e}^t}dtdy.$$

Let ${\psi }^{+}\left(\xi \right)$ be the characteristic function of $\mathrm{TS}({\beta }_{+},{\alpha }_{+},{\lambda }_{+})$. Then

$$log{\psi }^{+}\left(\xi \right)={\int }_0^{+\infty }(e^{iy\xi }-1)M_{{\psi }^{+}}\left(dy\right).$$

Substituting the expression for $M_{{\psi }^{+}}\left(dy\right)$ and changing variables $v=y{e}^t$, $u=e^{-t}$, we obtain

$$log{\psi }^{+}\left(\xi \right)={\int }_0^1{\int }_0^{+\infty }(e^{i\xi vu}-1){\alpha }_{+}\left({\lambda }_{+}+{\beta }_{+}{v}^{-1}\right)v^{-{\beta }_{+}}{e}^{-{\lambda }_{+}v}dv{\displaystyle \frac{du}{u}}.$$

Thus,

$$log{\psi }^{+}\left(\xi \right)={\int }_0^{1}log{\phi }^{+}\left(\xi u\right){\displaystyle \frac{du}{u}}={\int }_0^{\xi }log{\phi }^{+}\left(x\right){\displaystyle \frac{dx}{x}},x=\xi u.$$

(5)

Here

$$log{\phi }^{+}\left(x\right)={\int }_0^{+\infty }(e^{ivx}-1)N_{{\phi }^{+}}\left(dv\right),N_{{\phi }^{+}}\left(dx\right)={\alpha }_{+}{\displaystyle \frac{{\beta }_{+}+{\lambda }_{+}x}{x^{1+{\beta }_{+}}}}{e}^{-{\lambda }_{+}x}dx.$$

The same reasoning applied to the negative side (with parameters ${\beta }_{-},{\alpha }_{-},{\lambda }_{-}$) gives

$$log{\psi }^{-}\left(\xi \right)={\int }_0^{\xi }log{\phi }^{-}\left(x\right){\displaystyle \frac{dx}{x}},$$

$$log{\phi }^{-}\left(x\right)={\int }_0^{+\infty }(e^{ivx}-1)N_{{\phi }^{-}}\left(dv\right),N_{{\phi }^{-}}\left(dx\right)={\alpha }_{-}{\displaystyle \frac{{\beta }_{-}+{\lambda }_{-}x}{x^{1+{\beta }_{-}}}}{e}^{-{\lambda }_{-}x}dx.$$

Combining both sides, we define the Lévy density for the GTS distribution:

$$N_{\phi }\left(dx\right)=\left\{\begin{array}{cc}{\alpha }_{+}{\displaystyle \frac{{\beta }_{+}+{\lambda }_{+}x}{x^{1+{\beta }_{+}}}}{e}^{-{\lambda }_{+}x}dx,\hfill & x>0,\hfill \\ {\alpha }_{-}{\displaystyle \frac{{\beta }_{-}+{\lambda }_{-}\left|x\right|}{{\left|x\right|}^{1+{\beta }_{-}}}}{e}^{-{\lambda }_{-}\left|x\right|}dx,\hfill & x<0.\hfill \end{array}\right.$$

Then

$$log{\phi }^{+}\left(x\right)+log{\phi }^{-}(-x)={\int }_{-\infty }^{+\infty }(e^{ivx}-1)N_{\phi }\left(dv\right).$$

(6)

Using (2) and its negative-side counterpart,

$$log{\psi }^{+}\left(\xi \right)+log{\psi }^{-}(-\xi )={\int }_0^{\xi }\left({\int }_{-\infty }^{+\infty }(e^{ivx}-1)N_{\phi }\left(dv\right)\right){\displaystyle \frac{dx}{x}}.$$

(3)

The characteristic exponent of the GTS distribution with parameters $(\mu ,{\beta }_{+},{\beta }_{-},{\alpha }_{+},{\alpha }_{-},{\lambda }_{+},{\lambda }_{-})$ is therefore

$$log\psi \left(\xi \right)=log\mathbb{E}{e}^{iX\xi }=i\mu \xi +log{\psi }^{+}\left(\xi \right)+log{\psi }^{-}(-\xi ).$$

From (3),

$$log\psi \left(\xi \right)={\int }_0^{\xi }\left(i\mu x+{\int }_{-\infty }^{+\infty }(e^{ivx}-1)N_{\phi }\left(dv\right)\right){\displaystyle \frac{dx}{x}}={\int }_0^{\xi }log\phi \left(x\right){\displaystyle \frac{dx}{x}},$$

where

$$log\phi \left(\xi \right)=i\mu x+{\int }_{-\infty }^{+\infty }(e^{ivx}-1)N_{\phi }\left(dv\right).$$

(7)

The representation $log\psi \left(\xi \right)={\int }_0^{\xi }log\phi \left(x\right){\displaystyle \frac{dx}{x}}$ is exactly the condition for self-decomposability. Hence, the GTS distribution is self-decomposable.□

4. Theoretical Properties of the Background Driving Lévy Process

As shown in Theorem 2, the Background Driving Lévy Density (BDLD) associated with the Generalized Tempered Stable (GTS) distribution differs from the GTS Lévy measure (1). Specifically, the BDLD is expressed as the product of a tempering (or tilting) function $q\left(y\right)$ and a rational function $Q\left(y\right)$, defined as follows:

$$\begin{array}{cc}\hfill q\left(y\right)& =exp(-{\lambda }_{+}y){\mathbf{1}}_{(0,\infty )}\left(y\right)+exp(-{\lambda }_{-}\left|y\right|){\mathbf{1}}_{(-\infty ,0)}\left(y\right),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill Q\left(y\right)& ={\alpha }_{+}{\displaystyle \frac{{\beta }_{+}+{\lambda }_{+}y}{y^{1+{\beta }_{+}}}}{\mathbf{1}}_{(0,\infty )}\left(y\right)+{\alpha }_{-}{\displaystyle \frac{{\beta }_{-}+{\lambda }_{-}\left|y\right|}{{\left|y\right|}^{1+{\beta }_{-}}}}{\mathbf{1}}_{(-\infty ,0)}\left(y\right).\hfill \end{array}$$

(9)

Consider a background driving random variable (BDRV) Y associated with the GTS distribution. Then Y follows a background driving distribution, denoted GTSBD$({\beta }_{+},{\beta }_{-},{\alpha }_{+},{\alpha }_{-},{\lambda }_{+},{\lambda }_{-})$, defined by the Lévy measure $N\left(dy\right)$ given by:

$$N\left(dy\right)=\left[{\alpha }_{+}{\displaystyle \frac{{\beta }_{+}+{\lambda }_{+}y}{y^{1+{\beta }_{+}}}}{e}^{-{\lambda }_{+}y}{\mathbf{1}}_{y>0}\right]dy+\left[{\alpha }_{-}{\displaystyle \frac{{\beta }_{-}+{\lambda }_{-}\left|y\right|}{{\left|y\right|}^{1+{\beta }_{-}}}}{e}^{-{\lambda }_{-}\left|y\right|}{\mathbf{1}}_{y<0}\right]dy,$$

(10)

The activity process of the GTSBD distribution can be studied via the integral of the Lévy measure in (10):

$${\int }_{-\infty }^{+\infty }N\left(dy\right)=+\infty \mathrm{for}{\beta }_{+},{\beta }_{-}\in (0,1).$$

(11)

As shown in (11), when ${\beta }_{+},{\beta }_{-}\in (0,1)$, the Lévy measure $N\left(dy\right)$ is not integrable [38]; it diverges as $y\to 0$ due to an accumulation of very small jumps [39]. Consequently, the GTSBD distribution exhibits infinite activity, meaning that its sample paths have an infinite number of very small jumps within any finite time interval.

The variation of the background driving process can be assessed using the following integral condition [36]:

$${\int }_{-\infty }^{+\infty }min(1,|x\left|\right)N\left(dx\right)<+\infty \mathrm{for}{\beta }_{+},{\beta }_{-}\in (0,1).$$

(12)

Condition (12) implies that the GTSBD distribution is a finite-variation process (see [26,38] for further developments). It generates a Type B L’evy process [35,40]: a purely non-Gaussian, infinite-activity L’evy process of finite variation whose sample paths contain infinitely many small jumps but only finitely many large jumps in any finite time interval.

Because the process has bounded variation, $Y\sim \mathrm{GTSBD}({\beta }_{+},{\beta }_{-},{\alpha }_{+},{\alpha }_{-},{\lambda }_{+},{\lambda }_{-})$ can be written as the difference of two independent subordinators [36,37]:

$$Y=Y^{+}-Y^{-},$$

(13)

Adding a drift parameter $\mu$ yields the GTSBD distribution:

$$\mu +Y\sim \mathrm{GTSBD}(\mu ,{\beta }_{+},{\beta }_{-},{\alpha }_{+},{\alpha }_{-},{\lambda }_{+},{\lambda }_{-}).$$

(14)

Theorem 3.

(GTSBD Cumulants ${\kappa }_k$)

Consider a variable $Y\sim GTS(\mu ,{\beta }_{+},{\beta }_{-},{\alpha }_{+},{\alpha }_{-},{\lambda }_{+},{\lambda }_{-})$. The cumulants ${\kappa }_k$ of the GTS distribution are defined as follows:

$$\begin{array}{cc}\hfill {\kappa }_1& =\mu +{\alpha }_{+}{\displaystyle \frac{\Gamma (1-{\beta }_{+})}{{\lambda }_{+}^{1-{\beta }_{+}}}}-{\alpha }_{-}{\displaystyle \frac{\Gamma (1-{\beta }_{-})}{{\lambda }_{-}^{1-{\beta }_{-}}}}{\kappa }_2=2\left({\alpha }_{+}{\displaystyle \frac{\Gamma (2-{\beta }_{+})}{{\lambda }_{+}^{2-{\beta }_{+}}}}+{\alpha }_{-}{\displaystyle \frac{\Gamma (2-{\beta }_{-})}{{\lambda }_{-}^{2-{\beta }_{-}}}}\right)\hfill \\ \hfill {\kappa }_k& =k\left({\alpha }_{+}{\displaystyle \frac{\Gamma (k-{\beta }_{+})}{{\lambda }_{+}^{k-{\beta }_{+}}}}+{\alpha }_{-}{\displaystyle \frac{\Gamma (k-{\beta }_{-})}{{\lambda }_{-}^{k-{\beta }_{-}}}}{(-1)}^k\right)\forall k\in \mathbb{N}\setminus \{0,1\}.\hfill \end{array}$$

(15)

Proof:

We reconsider the characteristic exponents $log{\phi }^{+}\left(\xi \right)$, $log{\phi }^{-}\left(\xi \right)$ and $log{\phi }^{(}\xi )$ in (6) and (7).

$$\begin{array}{cc}\hfill log\left({\phi }^{+}\left(x\right)\right)& ={\int }_0^{+\infty }(e^{ivx}-1){\alpha }_{+}({\lambda }_{+}+{\beta }_{+}{v}^{-1})v^{-{\beta }_{+}}{e}^{-{\lambda }_{+}v}dv\hfill \\ \hfill & ={\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{{\left(ix\right)}^k}{k!}}{\alpha }_{+}\left({\lambda }_{+}{\int }_0^{+\infty }{v}^{k-{\beta }_{+}}{e}^{-{\lambda }_{+}v}dv+{\beta }_{+}{\int }_0^{+\infty }{v}^{-1-{\beta }_{+}}{e}^{-{\lambda }_{+}v}dv\right)\hfill \\ \hfill & ={\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{{\left(ix\right)}^k}{k!}}{\alpha }_{+}\left({\lambda }_{+}{\displaystyle \frac{\Gamma (k+1-{\beta }_{+})}{{\lambda }_{+}^{k+1-{\beta }_{+}}}}+{\beta }_{+}{\displaystyle \frac{\Gamma (k-{\beta }_{+})}{{\lambda }_{+}^{k-{\beta }_{+}}}}\right)\hfill \\ \hfill & ={\alpha }_{+}{\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{k\Gamma (k-{\beta }_{+})}{{\lambda }_{+}^{k-{\beta }_{+}}}}{\displaystyle \frac{{\left(ix\right)}^k}{k!}}\hfill \end{array}$$

(16)

Similarly, we have

$$\begin{array}{c}\hfill log\left({\phi }^{-}\left(x\right)\right)={\alpha }_{-}{\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{k\Gamma (k-{\beta }_{-})}{{\lambda }_{-}^{k-{\beta }_{-}}}}{\displaystyle \frac{{(-ix)}^k}{k!}}\end{array}$$

We have

$$\begin{array}{c}\hfill log\left(\phi \left(x\right)\right)=\mu xi+log\left({\phi }^{+}\left(x\right)\right)+log\left({\phi }^{-}(-x)\right)\end{array}$$

(17)

$$\begin{array}{cc}\hfill log\left(\phi \left(x\right)\right)& =i\mu x+log\left({\phi }^{+}\left(x\right)\right)+log\left({\phi }^{-}(-x)\right)\hfill \\ \hfill & =i\mu x+{\displaystyle \sum _{k=1}^{\infty }}\left[{\alpha }_{+}{\displaystyle \frac{\Gamma (k-{\beta }_{+})}{{\lambda }_{+}^{k-{\beta }_{+}}}}+{\alpha }_{-}{\displaystyle \frac{\Gamma (k-{\beta }_{-})}{{\lambda }_{-}^{k-{\beta }_{-}}}}{(-1)}^k\right]{\displaystyle \frac{{\left(ix\right)}^k}{(k-1)!}}\hfill \\ \hfill & ={\displaystyle \sum _{k=1}^{\infty }}{\kappa }_k{\displaystyle \frac{{\left(ix\right)}^k}{k!}}.\hfill \end{array}$$

(18)

Hence, the k-th order cumulant ${\kappa }_k$ is given by comparing the coefficients of both polynomial functions in $ix$. For more details on the relationship between the characteristic exponent and cumulant functions, refer to [41,42].□

5. Conclusion

This paper has identified and characterized the Background Driving Lévy Process (BDLP) associated with the Generalized Tempered Stable (GTS) distribution, which is a contribution to the theory of self-decomposable processes and their financial applications. The main theoretical result is the explicit derivation of the Lévy density (BDLD) for the GTS background driving process, establishing that the GTS distribution is self-decomposable. Specifically, we demonstrated that the BDLD differs from the GTS Lévy measure but conserves the same Lévy process properties, such as infinite activity and finite variation. We derived closed-form expressions for the cumulants of the GTS background driving distribution (GTSBD).