Multidimensional Lévy Models - Computational Methods for Quantitative Finance - page 201

sgn (z j )ν I
j

I(z j ) d z I

d − 1

+ z I )
j

∂ I f( 0 +

R

i

i = 2 | I |= i

I 1 < ··· < I i

∈ I ∈ I

+ ∂ d f( 0 ,..., 0 ,z d ) sgn (z d )ν ( R ,..., R ,I(z d ))

R

d − 1

+ ∂ i ∂ d f( 0 ,...,z i ,..., 0 ,z d )

R R

i = 1

×

sgn (z i ) sgn (z d )ν i,d (I(z i ), I (z d )) d z i d z d

d − 1

∂ I ∂ d f(z { I ,d } )

+

i

R R

i = 2 | I |= i

I 1 < ··· < I i

sgn (z j )ν { I ,d } I(z j ) d z I d z d ,

×

∈{ I ,d } ∈{ I ,d } j j

which is the claimed result.

Using Lemma 14.2.7 , we immediately obtain

(X 1 ,...,X d ) be a d-dimensional square integrable
Lévy process with characteristic triplet ( 0 ,ν,γ) . Then ,

Corollary 14.2.8 Let X =

t t

Cov (X t ,X t )

F { i,j } (U i (z i ), U j (z j )) d z i d z j , = z i z j ν( d z) = ∀ i = j,

d 2

R R

where F is the Lévy copula from Theorem 14.2.6 .

We conclude with examples of Lévy copulas.

Example 14.2.9 Examples of Lévy copulas are:

(i) Independence Lévy copula

d

u i
j

F(u 1 ,...,u d ) = 1 (u j ). (14.4)

{∞}

i = 1 = i

(ii) Complete dependence Lévy copula

d

F(u 1 ,...,u d ) = min {| u 1 | ,..., | u d |} 1 K (u 1 ,...,u d ) sgn u j , (14.5)

j = 1

d

where K := { x ∈ R : sgn (x 1 ) =···= sgn (x d ) } .