Multidimensional Lévy Models - Computational Methods for Quantitative Finance - page 222

Fig. 14.9 Convergence rates
with respect to ε for Y 1 ( top )
and Y 2 ( bottom )in d = 1

ε z j ν j ( d z j )<

= ε
−

Now with c j ∞ , j = 1 ,...,d and Jensen's inequality, we

have
ε

2 ν j ( d z j ) 2

c j ε
−

2

c j ε
−

z j z j | z j

z j | ν j ( d z j )

c j

| z j | ν j ( d z j )

c j

2

= ≤

− ε ε ε

c j ε
−

z j

3 ν j ( d z j ),

= j = 1 ,...,d.

ε

Using Propositions 14.7.3 and 14.7.4 , we immediately obtain

Corollary 14.7.5 Assume the Lévy measure ν satisfies (10.12) with α = (α 1 ,...,

C 4 (

d )

α d ) . Then , for g ∈ R