Multidimensional Lévy Models - Computational Methods for Quantitative Finance - page 196

In what follows, we denote by X i , i

=

1 ,...,d , the coordinate projection of

(X 1 ,...,X d ) ∈ R

d . Coordinate projections of

=

the multidimensional process X

Lévy processes are again Lévy processes.

(X 1 ,...,X d )

=

Proposition 14.1.2 Let X be a Lévy process with state space

,ν,γ) . Assume
|

d

R Q

> 1 | z | ν( d z) <

∞

and characteristic triplet ( . Then , the

z |

marginal processes X j , j

= 1 ,...,d , are again Lévy processes with the charac-

teristic triplet ( Q jj ,ν j ,γ j ) where the marginal Lévy measures are defined as

ν j (B) := ν( { x ∈ R

d

: x j ∈ B \{ 0 }} ), ∀ B ∈ B ( R ), j = 1 ,...,d.

If all the marginal Lévy processes X i

satisfy the martingale condition as

in Lemma 10.1.5,i.e. e X i

are martingales with respect to the filtration of X i ,

i = 1 ,...,d , they are also martingales with respect to the filtration F of X =

(X 1 ,...,X d ) .

d

Lemma 14.1.3 Let X be a Lévy process with state space R and characteris-

,ν,γ) . Assume , and

| z | > 1 e z j ν j ( d z) <

tic triplet ( Q

| z | > 1 |

z | ν( d z) < ∞ ∞ , j =

1 ,...,d . Then , e X j , j

= 1 ,...,d , are martingales with respect to the filtration F of

X if and only if
Q jj
2

e z j

z j ν j ( d z)

+ γ j + − 1 − = 0 ,j = 1 ,...,d.

R

Proof As in the proof of Lemma 10.1.5,wehave
E e X s | F t = e X t e (t − s)ψ( − ie j ) .

Therefore, setting ψ( − ie j ) = 0, j = 1 ,...,d , the Lévy-Khinchine formula ( 14.1 )

yields

e z j

− z j ν( d z) =

Q jj
2

+ γ j + − 1 0 ,j = 1 ,...,d.

R

d

The result follows with the definition of the marginal Lévy density ν j given in
Proposition 14.1.2 .

14.2 Lévy Copulas

d

We first gi ve so m e definitions. The F - volume of (a, b ] , a,b ∈ R for a function

d

F : S → R , S ⊂ R is defined by

1 ) N(u) F(u), V F ( (a, b ] := ( −

)

u ∈{ a 1 ,b 1 }×···×{ a d ,b d }

where N(u) = |{ k : u k = a k }| .