Variance-Gamma Scaled Self-Decomposable Process

• The Variance-Gamma Scaled Self-Decomposable Process is a Lévy process built via self-decomposability using Thorin-subordinated Brownian motion or the gamma law's remainder.
• It employs weak subordination to extend classical variance-gamma models to multivariate settings, enabling both common and idiosyncratic jump structures.
• Deterministic scaling preserves its structural form, with self-decomposability ensured under zero drift and explicitly broken in dimensions n≥2 with nonzero drift.

A variance-gamma scaled self-decomposable process is a variance-gamma-type Lévy process whose construction uses self-decomposability either through a Thorin-subordinated Brownian motion or through the self-decomposability remainder of the gamma law. In the framework of Buchmann–Lu–Madan, the classical variance-gamma process $V_t=B(G(t))$ arises by subordinating an $n$-dimensional Brownian motion $B\sim BM^n(\mu,\Sigma)$ with an independent gamma subordinator $G\sim Ts(b)$, while weak subordination extends this construction to arbitrary multivariate subordinators and dependent Brownian components (Buchmann et al., 2016). In the framework of Gardini–Sabino–Sasso, the Variance Gamma++ process is obtained by replacing the classical gamma subordinator with the $a$-remainder from the self-decomposability of the gamma law, introducing an additional scaling parameter $\kappa=a\in(0,1)$ (Gardini et al., 2021). Within the broader weak variance generalised gamma convolution class, self-decomposability is guaranteed for a driftless Brownian subordinate, and under explicit moment conditions on the Thorin measure this condition is also necessary in dimensions $n\ge 2$ (Buchmann et al., 2017).

1. Classical variance-gamma structure and weak subordination

The classical variance-gamma process is defined by

$V_t=B(G(t)), \qquad t\ge 0,$

where $B\sim BM^n(\mu,\Sigma)$ and $G\sim Ts(b)$ are independent. Its characteristic exponent at $n$0 is

$n$1

The associated Lévy density can be written explicitly in terms of the modified Bessel function $n$2 (Buchmann et al., 2016).

Weak subordination generalizes both univariate and multivariate subordination. If $n$3 is an arbitrary $n$4-dimensional Lévy process and $n$5 is any $n$6-dimensional subordinator, then there exists a $n$7-dimensional Lévy process $n$8 whose projection $n$9 is written

$B\sim BM^n(\mu,\Sigma)$0

Its characteristic exponent satisfies

$B\sim BM^n(\mu,\Sigma)$1

and its marginal Lévy measure is

$B\sim BM^n(\mu,\Sigma)$2

This construction allows arbitrary dependent-component Lévy processes and arbitrary multivariate subordinators, so it supports both common jumps and idiosyncratic jumps in multivariate settings (Buchmann et al., 2016).

2. Weak variance generalised gamma convolutions

The weak variance generalised gamma convolution class is formed by weakly subordinating Brownian motion with a Thorin subordinator. Let

$B\sim BM^n(\mu,\Sigma)$3

with $B\sim BM^n(\mu,\Sigma)$4 and $B\sim BM^n(\mu,\Sigma)$5 independent, $B\sim BM^n(\mu,\Sigma)$6, $B\sim BM^n(\mu,\Sigma)$7, $B\sim BM^n(\mu,\Sigma)$8, and Thorin measure $B\sim BM^n(\mu,\Sigma)$9 on $G\sim Ts(b)$0 satisfying

$G\sim Ts(b)$1

The weakly subordinated process

$G\sim Ts(b)$2

is called a weak variance generalised gamma convolution process and is denoted

$G\sim Ts(b)$3

Its characteristic exponent is

$G\sim Ts(b)$4

In the related $G\sim Ts(b)$5 notation, the classical variance-gamma model is recovered by taking $G\sim Ts(b)$6 and $G\sim Ts(b)$7, where $G\sim Ts(b)$8. Then $G\sim Ts(b)$9 is the gamma subordinator on the diagonal ray and

$a$0

This identifies variance-gamma as a special case of the broader variance generalised gamma convolution class (Buchmann et al., 2017, Buchmann et al., 2016).

3. Self-decomposability criteria and the role of drift

The central sufficient condition is Theorem 3.1 of Buchmann–Lu–Madan: if

$a$1

then $a$2 is self-decomposable in $a$3. When $a$4, the characteristic exponent reduces to

$a$5

and the proof proceeds by approximating this law by finite superpositions of ordinary variance-gamma laws, using the facts that ordinary variance-gamma laws are self-decomposable and that the class of self-decomposable laws is closed under convolution and weak limits (Buchmann et al., 2017).

The corresponding necessity statement is formulated in Theorem 3.2. Let $a$6, $a$7, and $a$8. If there exists a Borel subset $a$9 of positive surface measure such that, for all directions $\kappa=a\in(0,1)$0, the two integrals

$\kappa=a\in(0,1)$1

are finite and strictly positive, then $\kappa=a\in(0,1)$2 is not self-decomposable. A stronger sufficient integrability condition is

$\kappa=a\in(0,1)$3

and this guarantees the preceding two conditions, hence non-self-decomposability whenever $\kappa=a\in(0,1)$4 (Buchmann et al., 2017).

A common misconception is that self-decomposability is automatic for every variance-gamma-type weakly subordinated process. The one-dimensional gamma case shows a more specific picture. For a standard gamma subordinator $\kappa=a\in(0,1)$5, the Thorin measure is

$\kappa=a\in(0,1)$6

and

$\kappa=a\in(0,1)$7

Hence the classical variance-gamma process with no drift on $\kappa=a\in(0,1)$8 is self-decomposable. When a nonzero Gaussian drift $\kappa=a\in(0,1)$9 is introduced, the same condition shows that non-zero drift breaks self-decomposability in all dimensions $n\ge 2$0 (Buchmann et al., 2017).

4. Deterministic scaling and scaled self-decomposable variance-gamma laws

Deterministic scaling preserves the variance generalised gamma convolution form. If

$n\ge 2$1

define

$n\ge 2$2

Then

$n\ge 2$3

and

$n\ge 2$4

The same summary states that the resulting law is again self-decomposable for every $n\ge 2$5, with Lévy–Khintchine data obtained by the same substitution in the formula of Theorem 4.1 (Buchmann et al., 2016).

This scaling rule isolates a precise sense in which a variance-gamma-type self-decomposable law remains within the same structural family after deterministic amplitude rescaling. A plausible implication is that scaling can be handled at the level of the Brownian drift, Brownian covariance, and Thorin measure without leaving the $n\ge 2$6 class, which is useful when comparing parametrizations or normalizations across applications.

5. Gamma self-decomposability and the Variance Gamma++ process

Gardini–Sabino–Sasso introduce the Variance Gamma++ process by exploiting the self-decomposability of the gamma law. A distribution is self-decomposable if, for every $n\ge 2$7,

$n\ge 2$8

with $n\ge 2$9 independent of the $V_t=B(G(t)), \qquad t\ge 0,$0-remainder $V_t=B(G(t)), \qquad t\ge 0,$1. For $V_t=B(G(t)), \qquad t\ge 0,$2, the remainder satisfies

$V_t=B(G(t)), \qquad t\ge 0,$3

The process $V_t=B(G(t)), \qquad t\ge 0,$4 is constructed so that

$V_t=B(G(t)), \qquad t\ge 0,$5

and equivalently its Lévy density is

$V_t=B(G(t)), \qquad t\ge 0,$6

Because $V_t=B(G(t)), \qquad t\ge 0,$7, $V_t=B(G(t)), \qquad t\ge 0,$8 is a compound-Poisson subordinator of finite activity (Gardini et al., 2021).

The Variance Gamma++ process is then

$V_t=B(G(t)), \qquad t\ge 0,$9

where $B\sim BM^n(\mu,\Sigma)$0 is a Brownian motion with drift $B\sim BM^n(\mu,\Sigma)$1 and volatility $B\sim BM^n(\mu,\Sigma)$2, independent of $B\sim BM^n(\mu,\Sigma)$3. An equivalent integral form is

$B\sim BM^n(\mu,\Sigma)$4

Since $B\sim BM^n(\mu,\Sigma)$5 is nondecreasing, $B\sim BM^n(\mu,\Sigma)$6 is a pure-jump process of finite variation and finite activity. In this model one sets $B\sim BM^n(\mu,\Sigma)$7. As $B\sim BM^n(\mu,\Sigma)$8, the remainder $B\sim BM^n(\mu,\Sigma)$9 collapses to the classical gamma subordinator and one recovers the infinite-activity variance-gamma model. In practice, one may fix a variance-per-unit-time parameter $G\sim Ts(b)$0 and choose

$G\sim Ts(b)$1

so that the classical variance-gamma subordinator has $G\sim Ts(b)$2 law and $G\sim Ts(b)$3 has the same first moment $G\sim Ts(b)$4 (Gardini et al., 2021).

6. Analytic representation, computation, and multivariate extension

For the Variance Gamma++ process, the characteristic triplet is $G\sim Ts(b)$5, with no continuous part,

$G\sim Ts(b)$6

Writing $G\sim Ts(b)$7 as the difference of two independent Gamma++ subordinators with parameters $G\sim Ts(b)$8 and $G\sim Ts(b)$9, where

$n$00

the Lévy density is

$n$01

and

$n$02

The drift coefficient can be written in closed form but is most often absorbed into the martingale adjustment (Gardini et al., 2021).

Its characteristic function is obtained from subordination. With

$n$03

one has

$n$04

and therefore

$n$05

Because $n$06 is compound-Poisson with an atom at zero, $n$07 also has an atom at zero, and the transition density is

$n$08

where $n$09 is the classical variance-gamma density with integer shape $n$10 and rate $n$11 (Gardini et al., 2021).

Simulation is available both forward and backward in time. On a time grid $n$12, forward simulation first samples $n$13 either through a Polya-sum method or through a compound-Poisson representation with random rate, and then samples

$n$14

Backward simulation uses the Polya bridge for $n$15, the gamma bridge for $n$16, and the Gaussian bridge for $n$17. In particular,

$n$18

$n$19

and

$n$20

(Gardini et al., 2021).

Under the risk-neutral measure, one sets

$n$21

with

$n$22

so that $n$23. A European call with strike $n$24 and maturity $n$25 has price

$n$26

and Proposition 4.1 gives the integral-free representation

$n$27

where

$n$28

Alternatively, the characteristic function may be inserted into a Carr–Madan or Lewis-type FFT inversion (Gardini et al., 2021).

Parameter estimation is described through maximum likelihood and generalized method of moments. For log-returns $n$29, the log-likelihood is

$n$30

to be maximized numerically with respect to $n$31. The generalized method of moments matches cumulants $n$32, using

$n$33

A multivariate extension is obtained by taking independent Gamma++ processes $n$34, an independent common factor $n$35, and defining

$n$36

Then each $n$37, while dependence is induced through the common subordinator $n$38. Time-changing independent Brownian motions by $n$39 yields a multivariate Variance Gamma++ process with the same marginal laws and a parsimonious dependence structure (Gardini et al., 2021).