In this section, X := {Xi
t}i=1,...,dt≥0 denotes a L´evy process on Rd with characteristic triplet
(A, ν, γ). For I ⊂ {1, . . . , d} we define Ic := {1, . . . , d} \ I and |I| denotes the number of elements in I. We start by defining the margins of a L´evy process.
Definition 4.1. Let I ⊂ {1, . . . , d} nonempty. The I-margin of X is the L´evy process XI := {Xi
t}i∈It≥0.
The following lemma explains that the L´evy measure of XI only depends on the L´evy measure of X and shows how it can be computed.
Lemma 4.1 (Marginal L´evy measures). Let I ⊂ {1, . . . , d} nonempty. Then the L´evy process XI has L´evy measure νI given by
νI(B) = ν({x ∈ Rd: (xi)i∈I ∈ B}), ∀B ∈ B(R|I|\ {0}). (4.1)
In view of the above lemma, for a given L´evy measure ν we will refer to the L´evy measure νI defined by Equation (4.5) as the I-margin of ν. To simplify notation, when I = {k} for some k, the I-margin of ν will be denoted by νk and called simply k-th margin of ν.
Next we would like to characterize the independence of L´evy processes in terms of their characteristic triplets.
Lemma 4.2. The components X1, . . . , Xd of an Rd-valued L´evy process X are independent if and only if their continuous martingale parts are independent and the L´evy measure ν is supported by the coordinate axes. ν is then given by
ν(B) =
d
X
i=1
νi(Bi) ∀B ∈ B(Rd\ {0}), (4.2)
where for every i, νi denotes the i-th margin of ν and
Bi= {x ∈ R : ( 0, . . . , 0
| {z } i − 1 times
, x, 0, . . . , 0) ∈ B}.
Proof. Since the continuous martingale part and the jump part of X are independent, we can assume without loss of generality that X has no continuous martingale part, that is, its characteristic triplet is given by (0, ν, γ).
The “if” part. Suppose ν is supported by the coordinate axes. Then necessarily for every B ∈ B(Rd\ {0}), ν(B) =Pdi=1ν˜i(Bi) with some measures ˜νi, and Lemma 4.1 show that these
measures coincide with the margins of ν: ˜νi = νi ∀i. Using the L´evy-Khintchine formula for
the process X, we obtain: E[eihu,Xti] = exp t{ihγ, ui +
Z
Rd\{0}
(eihu,xi− 1 − ihu, xi1|x|≤1)ν(dx)}
= exp t d X k=1 {iγkuk+ Z R\{0} (eiukxk − 1 − iu kxk1|xk|≤1)νk(dxk)} = d Y k=1 E[eiukXtk],
which shows that the components of X are independent L´evy processes.
The “only if” part. Define a measure ˜ν on Rd\ {0} by ˜ν(B) =Pdi=1νi(Bi), where νi is the
i-th marginal L´evy measure of X and Bi is as above. It is straightforward to check that ˜ν is
a L´evy measure. Since the components of X are independent, applying the L´evy-Khintchine formula to each component of X, we find:
E[eihu,Xti] = exp t{ihγ, ui +
Z
Rd\{0}
Now from the uniqueness of L´evy-Khintchine representation we conclude that ˜ν is the L´evy measure of X.
The complete dependence of L´evy processes is a new notion that is worth being discussed in detail. First, the following definition is in order.
Definition 4.2. A subset S of Rd is called ordered if, for any two vectors v, u ∈ S, either
vk ≤ uk, k = 1, . . . , d or vk ≥ uk, k = 1, . . . , d. S is called strictly ordered if, for any two
different vectors v, u ∈ S, either vk< uk, k = 1, . . . , d or vk> uk, k = 1, . . . , d.
We recall that random variables Y1, . . . , Yd are said to be completely dependent or comono-
tonic if there exists a strictly ordered set S ⊂ Rdsuch that (Y1, . . . , Yd) ∈ S with probability 1.
However, saying that the components of a L´evy process are completely dependent only if they are completely dependent for every fixed time is too restrictive; the components of a L´evy pro- cess can be completely dependent as processes without being completely dependent as random variables for every fixed time. The following example clarifies this point.
Example 4.2 (Dynamic complete dependence for L´evy processes). Let {Xt}t≥0be a L´evy process
with characteristic triplet (A, ν, γ) such that A = 0 and γ = 0 and let {Yt}t≥0be a L´evy process,
constructed from the jumps of X: Yt=Ps≤t∆Xs3. From the dynamic point of view X and Y
are completely dependent, because the trajectory of any one of them can be reconstructed from the trajectory of the other. However, the copula of Xtand Ytis not that of complete dependence
because Yt is not a deterministic function of Xt. Indeed, if X is a compound Poisson process
having jumps of size 1 and 2 and Xt = 3 for some t, this may either mean that X has three
jumps of size 1 in the interval [0, t], and then Yt= 3, or that X has one jump of size 1 and one
jump of size 2, and then Yt= 9.
This example motivates the following definition. In this definition and below,
K := {x ∈ Rd: sgn x1 = · · · = sgn xd}. (4.3)
Definition 4.3. Let X be a Rd-valued L´evy process. Its jumps are said to be completely dependent or comonotonic if there exists a strictly ordered subset S ⊂ K such that ∆Xt :=
Clearly, an element of a strictly ordered set is completely determined by one coordinate only. Therefore, if the jumps of a L´evy process are completely dependent, the jumps of all components can be determined from the jumps of any single component. If the L´evy process has no continuous martingale part, then the trajectories of all components can be determined from the trajectory of any one component, which indicates that Definition 4.3 is a reasonable dynamic notion of complete dependence for L´evy processes. The condition ∆Xt∈ K means that
if the components of a L´evy process are comonotonic, they always jump in the same direction. For any Rd-valued L´evy process X with L´evy measure ν and for any B ∈ B(Rd\ {0}) the
number of jumps in the time interval [0, t] with sizes in B is a Poisson random variable with parameter tν(B). Therefore, Definition 4.3 can be equivalently restated in terms of the L´evy measure ν of X as follows:
Definition 4.4. Let X be a Rd-valued L´evy process with L´evy measure ν. Its jumps are said
to be completely dependent or comonotonic if there exists a strictly ordered subset S of K such that ν(Rd\ S) = 0.