Multidimensional Lévy Models - Computational Methods for Quantitative Finance - page 195

Chapter 14
Multidimensional Lévy Models

In this chapter, we extend the one-dimensional Lévy models described in Chap. 10
to multidimensional Lévy models. Since the law of a Lévy process X is time-
homogeneous, it is completely characterized by its characteristic triplet ( ,ν,γ) .

The drift γ has no effect on the dependence structure between the components of X .
The dependence structure of the Brownian motion part of X is given by its covari-
ance matrix

Q

. For purposes of financial modeling, it remains to specify a paramet-
ric dependence structure of the purely discontinuous part of X which can be done
by using Lévy copulas.

Q

14.1 Lévy Processes

We associate to the multidimensional Lévy process X ={ X t : t ∈[ 0 ,T ]} the Lévy

measure

= E # } ,B

d ).

ν(B) { t ∈[ 0 , 1 ]: X t = 0 ,X t ∈ B ∈ B ( R

The Lévy measure satisfies
R

2 ν( d z) < 2

=
i = 1 z i . Using the Lévy-Khinchine representation, we see that every Lévy process
is uniquely defined by a drift vector γ , a positive definite matrix

d 1

∧ | z | ∞ , where we denote by | z |

d × d
sym

Q ∈ R and the

Lévy measure ν .

Theorem 14.1.1 (Lévy-Khinchine representation) Let X be a Lévy process with
state space ,ν,γ) . Assume

d

R Q

> 1 | z | ν( d z) <

∞

and characteristic triplet ( .

| z |

Then , for t 0,

E e i ξ,X t =

≥

e − tψ(ξ) ,ξ

d ,

∈ R

1

ν( d z).

(14.1)

1
2

e i ξ,z +

with ψ(ξ) =− i γ,ξ + ξ, Q ξ + − i ξ,z

R

d