--- title: Variance-Gamma Scaled Self-Decomposable Process url: https://www.emergentmind.com/topics/variance-gamma-scaled-self-decomposable-process type: topic --- # Variance-Gamma Scaled Self-Decomposable Process 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 [1609.04481]. 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 \(\kappa=a\in(0,1)\) [2106.15452]. 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 \(n\ge 2\) [1712.03640]. ## 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 \(t=1\) is \[ \Psi_{VG}(\theta)= -\,b\;\ln\!\frac{b - i\langle\theta,\mu\rangle +\tfrac12\langle\theta,\Sigma\theta\rangle}{b}, \qquad \theta\in\mathbb R^n. \] The associated Lévy density can be written explicitly in terms of the modified Bessel function \(K_\nu\) [1609.04481]. Weak subordination generalizes both univariate and multivariate subordination. If \(X\sim L^n(p,\Sigma,\mathcal X)\) is an arbitrary \(n\)-dimensional Lévy process and \(T\sim S^n(d,\mathcal T)\) is any \(n\)-dimensional subordinator, then there exists a \(2n\)-dimensional Lévy process \(Z=(T,Y)\) whose projection \(Y\) is written \[ Y=X\odot T. \] Its characteristic exponent satisfies \[ \Psi_Y(\theta)= i\langle d,\theta\rangle + \tfrac12\langle d\Sigma\theta,\theta\rangle + \int_{[0,\infty)^n}\bigl(e^{\Psi_X(t;\theta)}-1\bigr)\,\mathcal T(dt), \] and its marginal Lévy measure is \[ \nu_Y(dx)= d\,\mathcal X(dx)+\int_{[0,\infty)^n}\mathbb P\{X(t)\in dx\}\,\mathcal T(dt). \] This construction allows arbitrary dependent-component Lévy processes and arbitrary multivariate subordinators, so it supports both common jumps and idiosyncratic jumps in multivariate settings [1609.04481]. ## 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 \[ B=(B(t))_{t\ge0}\sim BM^n(\mu,\Sigma), \qquad S=(S(t))_{t\ge0}\sim GGC^n_S(d,\nu), \] with \(B\) and \(S\) independent, \(\mu\in\mathbb R^n\), \(\Sigma\succ 0\), \(d\ge 0\), and Thorin measure \(\nu\) on \([0,\infty)^n_*\) satisfying \[ \int_{[0,\infty)_*^n}\Bigl[(1+\ln^-\|u\|)\wedge(1/\|u\|)\Bigr]\,\nu(du)<\infty. \] The weakly subordinated process \[ X_t = B\odot S := (B_1(S_1(t)),\dots,B_n(S_n(t))) \] is called a weak variance generalised gamma convolution process and is denoted \[ X\sim WVGG^n(d,\mu,\Sigma,\nu). \] Its characteristic exponent is \[ \Psi(\theta) = i\langle\mu,\theta\rangle -\tfrac12\|\theta\|^2_\Sigma -\int_{[0,\infty)^n_*} \ln\!\Bigl(\tfrac{\|u\|^2+ \tfrac12\|\theta\|^2_\Sigma}{\|u\|^2}\Bigr)\,\nu(du), \qquad \theta\in\mathbb R^n. \] In the related \(VGG^n(d,p,\Sigma,U)\) notation, the classical variance-gamma model is recovered by taking \(d=0\) and \(U=b\delta_e\), where \(e=(1,\dots,1)\). Then \(GGC^n(0,b\delta_e)\) is the gamma subordinator on the diagonal ray and \[ VGG^n(0,p,\Sigma,b\delta_e)=VG_n(b,p,\Sigma). \] This identifies variance-gamma as a special case of the broader variance generalised gamma convolution class [1712.03640][1609.04481]. ## 3. Self-decomposability criteria and the role of drift The central sufficient condition is Theorem 3.1 of Buchmann–Lu–Madan: if \[ X\sim WVGG^n(d,\mu,\Sigma,\nu) \quad\text{and}\quad \mu=\mathbf 0, \] then \(X\) is self-decomposable in \(\mathbb R^n\). When \(\mu=0\), the characteristic exponent reduces to \[ \Psi(\theta) = -\int_{[0,\infty)^n_*} \ln\!\Bigl(\tfrac{\|u\|^2+\tfrac12\|\theta\|^2_\Sigma}{\|u\|^2}\Bigr)\,\nu(du), \] 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 [1712.03640]. The corresponding necessity statement is formulated in Theorem 3.2. Let \(n\ge 2\), \(\Sigma\succ 0\), and \(\mu\neq 0\). If there exists a Borel subset \(B\subset S_{**}\subset S\) of positive surface measure such that, for all directions \(\theta\in B\), the two integrals \[ \int_{(0,\infty)^n}A(\theta,u)\,\frac{\nu(du)}{D(\theta,u)}<\infty, \qquad \int_{(0,\infty)^n}E(\theta,u)\,\frac{\nu(du)}{D(\theta,u)}>0 \] are finite and strictly positive, then \(X\) is not self-decomposable. A stronger sufficient integrability condition is \[ \int_{(0,\infty)^n}\bigl(1+\|u\|^{1/2}\bigr) \Bigl(\tfrac{\|u\|^n}{\prod_{k=1}^n u_k}\Bigr)^{1/2}\,\nu(du)<\infty, \] and this guarantees the preceding two conditions, hence non-self-decomposability whenever \(\mu\neq 0\) [1712.03640]. 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 \(G\sim\Gamma_S(a,b)\), the Thorin measure is \[ \nu(du)=\frac{a}{u}e^{-bu}\,du,\qquad u>0, \] and \[ \int_0^\infty u^{1/2}\,\nu(du) = a\int_0^\infty u^{-1/2}e^{-bu}\,du = a\,\Gamma\!\bigl(\tfrac12\bigr)\,b^{-1/2} <\infty. \] Hence the classical variance-gamma process with no drift on \(B\) is self-decomposable. When a nonzero Gaussian drift \(\mu\neq 0\) is introduced, the same condition shows that non-zero drift breaks self-decomposability in all dimensions \(n\ge 2\) [1712.03640]. ## 4. Deterministic scaling and scaled self-decomposable variance-gamma laws Deterministic scaling preserves the variance generalised gamma convolution form. If \[ Y\sim VGG^n(d,p,\Sigma,U) \quad\text{and}\quad c>0, \] define \[ Y'_t=c\,Y_t. \] Then \[ \Psi_{Y'}(\theta)=\Psi_Y(c\,\theta), \] and \[ Y'\sim VGG^n\bigl(d,\;c\,p,\;c^2\,\Sigma,\;U\circ(u\mapsto u/c)\bigr). \] The same summary states that the resulting law is again self-decomposable for every \(c>0\), with Lévy–Khintchine data obtained by the same substitution in the formula of Theorem 4.1 [1609.04481]. 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 \(VGG^n\) 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 \(a\in(0,1)\), \[ X\stackrel d= aY+Z_a, \] with \(Y\stackrel d=X\) independent of the \(a\)-remainder \(Z_a\). For \(X\sim\Gamma(\alpha,\beta)\), the remainder satisfies \[ \phi_{Z_a}(u)=\Bigl(\tfrac{\beta-i\,a\,u}{\beta-i\,u}\Bigr)^{\alpha}. \] The process \(\{Z_a(t);t\ge 0\}\) is constructed so that \[ Z_a(t)\stackrel d= \begin{cases} \sum_{i=1}^{S(t)}X_i, & S(t)\sim \mathrm{Polya}(\alpha t,1-a),\; X_i\sim\mathrm{Exp}(\beta/a),\\[0.6em] 0, & S(t)=0, \end{cases} \] and equivalently its Lévy density is \[ \nu_{Z_a}(x)=\frac{\alpha}{x}\bigl(e^{-\beta x}-e^{-\beta x/a}\bigr)\,\mathbf 1_{x>0}(x). \] Because \(\int_0^\infty \nu_{Z_a}(x)\,dx<\infty\), \(Z_a\) is a compound-Poisson subordinator of finite activity [2106.15452]. The Variance Gamma++ process is then \[ X(t)=\theta\,Z_a(t)+\sigma\,W(Z_a(t)),\qquad t\ge 0, \] where \(W_t\) is a Brownian motion with drift \(\theta\in\mathbb R\) and volatility \(\sigma>0\), independent of \(Z_a\). An equivalent integral form is \[ X(t)=\int_0^{Z_a(t)}\theta\,du+\int_0^{Z_a(t)}\sigma\,dW(u). \] Since \(Z_a\) is nondecreasing, \(X\) is a pure-jump process of finite variation and finite activity. In this model one sets \(\kappa=a\in(0,1)\). As \(\kappa\to 0\), the remainder \(Z_a\) 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 \(\nu_0>0\) and choose \[ \alpha=\frac1{\nu_0},\qquad \beta=\frac1{\nu_0},\qquad \kappa=a\in(0,1), \] so that the classical variance-gamma subordinator has \(\Gamma(t/\nu_0,\nu_0)\) law and \(Z_a(t)\) has the same first moment \(\mathbb E[Z_a(t)]=t\) [2106.15452]. ## 6. Analytic representation, computation, and multivariate extension For the Variance Gamma++ process, the characteristic triplet is \((0,b,\nu)\), with no continuous part, \[ \nu(dx)=\nu(x)\,dx, \qquad b=\int_{|x|\le 1}x\,\nu(x)\,dx. \] Writing \(X=X^+-X^-\) as the difference of two independent Gamma++ subordinators with parameters \((a,\alpha,\beta_p)\) and \((a,\alpha,\beta_n)\), where \[ \beta_{p,n}=\frac{\sqrt{\theta^2+2\sigma^2\beta}\,\mp\,\theta}{\sigma^2}, \] the Lévy density is \[ \nu(x)=\frac{\alpha}{x}\bigl(e^{-\beta_p x}-e^{-\beta_p x/a}\bigr),\qquad x>0, \] and \[ \nu(x)=\frac{\alpha}{|x|}\bigl(e^{-\beta_n|x|}-e^{-\beta_n|x|/a}\bigr),\qquad x<0. \] The drift coefficient can be written in closed form but is most often absorbed into the martingale adjustment [2106.15452]. Its characteristic function is obtained from subordination. With \[ \omega(u)=\theta\,u+i\,\tfrac{\sigma^2}{2}u^2, \] one has \[ \phi_{X(t)}(u)=\mathbb E[e^{i\,u\,X(t)}] = \phi_{Z_a(t)}(\omega(u)) = \Bigl(\frac{\beta-i\,a\,\omega(u)}{\beta-i\,\omega(u)}\Bigr)^{\alpha t}, \] and therefore \[ \psi_{X(t)}(u)=\log \phi_{X(t)}(u) = \alpha t\,\ln\!\Bigl(\tfrac{\beta-i\,a\,\omega(u)}{\beta-i\,\omega(u)}\Bigr). \] Because \(Z_a(t)\) is compound-Poisson with an atom at zero, \(X(t)\) also has an atom at zero, and the transition density is \[ p_{X(t)}(x) = a^{\alpha t}\,\delta_0(x) + \sum_{k=1}^\infty \binom{\alpha t+k-1}{k}\,a^{\alpha t}(1-a)^k\,f^{\mathrm{VG}_{k,\beta/a}}(x), \] where \(f^{\mathrm{VG}_{k,\beta/a}}(x)\) is the classical variance-gamma density with integer shape \(k\) and rate \(\beta/a\) [2106.15452]. Simulation is available both forward and backward in time. On a time grid \(0=t_0