Processus de Lévy en Finance - Laboratoire de Probabilités et ...

4.1. INTRODUCTION 137
distribution at time t for any fixed t > 0, the dependence structure of a multidimensional
Lévy process X ≡ {X i t} i=1...d
t≥0
can be parametrized by the copula C t of the random vector
{X i t} i=1...d for some t > 0. However, this approach has a number of drawbacks. First, for given
infinitely divisible one-dimensional laws µ 1 t , . . . , µ d t , it is unclear, which copulas C t will yield a
d-dimensional infinitely divisible law. Second, the copula C t may depend on t and C s for some
s ≠ t cannot in general be computed from C t alone; to compute it one also needs to know the
marginal distributions at time t and at time s. We will now construct an explicit example of a
Lévy process with nonconstant copula.
Example 4.1. Let Z := {Z t } t≥0 be a 2-dimensional Cauchy process, that is, a Lévy process with
characteristic triplet (0, ν Z , 0), where ν Z has a density, also denoted by ν Z , given by
ν Z (x, y) =
1
(x 2 + y 2 ) 3/2 .
The probability distribution of Z t for every t > 0 has a density
p Z t (x, y) = 1 t
2π {(x 2 + y 2 ) 2 + t 2 } 3/2 .
The copula of Z t does not depend on time and can be computed explicitly:
⎧ ⎫
C Z (u, v) = − 1 4 + u 2 + v 2 + 1 ⎨
2π arctan tan π(u − 1 2 ) tan π(v − 1 2 ) ⎬
⎩
√1 + tan 2 π(u − 1 2 ) + tan2 π(v − 1 2 ) ⎭ ,
which is clearly different from the independence copula C ⊥ (u, v) = uv.
Let W := {W t } t≥0 be a standard planar Brownian motion, independent from Z. Since the
components of W are independent, the copula of W t for each t > 0 is the independence copula
C ⊥ (u, v) = uv. For every t, let X t = Z t + W t . {X t } t≥0 is a Lévy process with characteristic
triplet (Id 2 , ν Z , 0).
X t √t
Since Z is a 1-stable process and W is 1 2 -stable, X t
t
d = Z √
t
+ W 1 . Therefore, the random variable Xt
t
d
= Z 1 + W 1/t and
is infinitely divisible with characteristic
triplet ( 1 t Id 2, 0, ν Z ) and the random variable Xt √
t
is infinitely divisible with characteristic triplet
(Id 2 , 0, √ tν Z ). From Proposition 1.7 it follows that Xt d
t
−−−→ Z 1 and Xt d
√
t→∞
t
−−→ W 1 . Since the
t→0
copula is invariant with respect to transformations of margins by strictly increasing functions,
X t , Xt √
t
and Xt
t
have the same copula. Therefore, Theorem 2.1 in [64] implies that the copula
C t of X t has the following properties
∀u, v, C t (u, v) → C Z (u, v) as t → ∞,
∀u, v, C t (u, v) → C ⊥ (u, v) as t → 0.