196 E. Luciano and P. Semeraro
It depends on three marginal parameters(μj,σj,αj)and an additional common
parametera. Its characteristic function is the following
ψX(t)(u)=∏nj= 1(
1 −αj(
iμjuj−1
2
σ^2 ju^2 j))−t(
α^1
j−a)⎛
⎝ 1 −
∑nj= 1αj(
iμjuj−1
2
σ^2 ju^2 j)
⎞
⎠
−ta. (4)
The usual multivariate VG obtains forαj=α,j= 1 ,...,nanda=^1 α.
For the sake of simplicity, from now on we consider the bivariate case.
Since the marginal processes are VG, the corresponding distributions at timet,
Ft^1 andFt^2 can be obtained in a standard way, i.e., conditioning with respect to the
marginal time change:
Fti(xi)=∫+∞
0(
xi−μi(wi+αiz)
σi√
wi+αiz)
fGi(t)(z)dz, (5)where is a standard normal distribution function andfGi(t)is the density of a gamma
distribution with parameters
(
t
αi,t
αi)
. The expression for the joint distribution at time
t,Ft=FX(t),is:
Ft(x 1 ,x 2 )=∫∞
0∫∞
0∫∞
0(
x 1 −μ 1 (w 1 +αz)
σ 1√
w 1 +α 1 z)
(
x 2 −μ 2 (w 2 +βz)
σ 2√
w 2 +α 2 z)
(6)
·fY 1 (t)(w 1 )fY 2 (t)(w 2 )fZ(t)(z)dw 1 dw 2 dz, (7)wherefY 1 (t), fY 2 (t), fZ(t)are densities of gamma distributions with parameters re-
spectively:
(
t(
1
α 1 −a)
,α^11)
,
(
t(
1
α 2 −a)
,α^12)
and(ta, 1 )[11].2.1 Dependence structure
In this section, we investigate the dependence or association structure of theα-VG
process.
We know from Sklar’s Theorem that there exists a copula such that any joint
distribution can be written in terms of the marginal ones:
Ft(x 1 ,x 2 )=Ct(Ft^1 (x 1 ),Ft^2 (x 2 )). (8)The copulaCtsatisfies:Ct(u 1 ,u 2 )=Ft((Ft^1 )−^1 (u 1 ),(Ft^2 )−^1 (u 2 )), (9)where(Fti)−^1 is the generalised inverse ofFti,i= 1 ,2.