Lévy Models - Computational Methods for Quantitative Finance - page 131

The removal of drift is important for stability of the numerical algorithm. For nota-
tional simplicity, we also removed the interest rate r .
Remark 10.3.4 If the Lévy measure satisfies

1 | z | ν( d z) < ∞

, we remove the

| z |≤

drift γ 0 which is given, due to the martingale condition, by

σ 2
2 −

(e z

γ 0 =− − 1 )ν( d z),

R

and obtain the generator

1
2 σ 2 ∂ xx f(x)

J f )(x)

( A = + (f (x + z) − f (x))ν( d z).

R

10.4 Variational Formulation

For the variational formulation, we need Sobolev spaces of fractional order, i.e.
H s ( ) , s> 0. For any integer m , we defined the H m -norm using the derivative ∂ m

(see Sect. 3.1). Equivalently, we can also define the norm using the Fourier transfor-
mation. In particular, for ∂ m u ∈ L 2 ( R ) we have, by using Plancherel's theorem,

R

2 π 2 π

∂ m u(ξ)

2 d ξ

∂ m u(x)

2 d x 2 m 2 d ξ,

| | = = | ξ | | u(ξ ) |

R R R

( 2 π) − 1 e − iξz u(z) d z denotes the Fourier transform of u . Therefore,
we can define an equivalent H s -norm for s> 0via

where u(ξ) =

2

+| ξ | ) 2 s

2 d ξ,

u

H s ( R ) :=

( 1 | u(ξ ) |

R

and set H s ( L 2 ( R ) := { u ∈ R ) : u H s ( R ) < ∞} .Wealsoset

1 if σ> 0 ,
α/ 2 f σ

ρ =

0 ,
with α giveninAssumption 10.2.3 . The variational formulation of the PIDE ( 10.19 )
reads

=

L 2 (J H ρ ( H 1 (J L 2 (

; R ∩ ; R

Find u ∈ )) )) such that

a J (u, v) H ρ (

(∂ t u, v) + = 0 , ∀ v ∈ R ), a.e. in J, (10.21)

u( 0 ) = u 0 ,

g(e x ) and the bilinear form a J ( H ρ ( H ρ (

where u 0 (x) := · , · ) : R ) × R ) → R is given

by

1
2 σ 2 (ϕ ,φ )

a J (ϕ, φ) zϕ (x))φ(x)ν( d z) d x.

(10.22)

:= − (ϕ(x + z) − ϕ(x) −

R R