Lévy Models - Computational Methods for Quantitative Finance - page 133

1
2 σ 2 ξ 2

1
2 σ 2 ξ 2 ,

ψ(ξ) = + ( 1 − cos (ξ z))ν( d z) ≥

R

1
2 σ 2 ξ 2

e iξz C 7 ξ 2 | ψ(ξ) |≤ + | − 1 − iξz | ν( d z) ≤ + C 8 .

R

Lemma 10.4.2 shows that ψ satisfies the so-called sector condition | ψ(ξ) |≤

C ψ(ξ) , for all ξ ∈ R . Using this, we can show

Theorem 10.4.3 Let X be a Lévy process with characteristic triplet (σ 2 ,ν, 0 )
where the Lévy measure satisfies ( 10.12 ), ( 10.13 ). Then , there exist constants
C i > 0, i = 1 , 2 , 3, such that for all ϕ,φ ∈ H ρ ( R ) one has

| a J (ϕ, φ) |≤ C 1 ϕ H ρ ( R ) φ H ρ ( R ) ,

J (ϕ, ϕ) ≥ C 2 ϕ 2 2

H ρ ( R ) − C 3 ϕ

L 2 ( R ) .

Proof Using Lemma 10.4.2 , there exist positive constants C 1 ,C 2 > 0 such that

| ψ(ξ) |≤ C 2 | ξ | 1 .

2 ρ , 2 ρ

ψ(ξ) ≥ C 1 | ξ | +

Then, we have, using Hölder's inequality and Remark 10.4.1 ,

ϕ(ξ)φ(ξ) d ξ

a J (ϕ, φ)

| |=

ψ(ξ)

R

≤ C 2

2 ρ ) | ϕ(ξ)φ(ξ) |

( 1 +| ξ | d ξ

R

≤ C 2

) 2 ρ ϕ(ξ)φ(ξ)

+| | | |

( 1 ξ d ξ

R

≤ C 2

2 d ξ 1 / 2 2 d ξ 1 / 2

) 2 ρ ) 2 ρ | φ(ξ)

( 1 +| ξ | | ϕ(ξ) | ( 1 +| ξ | |

R R

= C 2 ϕ H ρ ( R ) φ H ρ ( R ) ,

where we used the fact that there exists a c> 0 such that

) 2 ρ

( 1 +| ξ | 1

c <

0 <c ≤ ≤ ∞ , ∀ ξ ∈ R .

1 +| ξ |

2 ρ

Furthermore, to prove the Gårding inequality, one finds

C 1 a J (ϕ, ϕ)

2 d ξ 2 d ξ 2 d ξ,

= ψ(ξ) | ϕ(ξ) | = (C 1 + ψ(ξ)) | ϕ(ξ) | − | ϕ(ξ) |

R R R

and

C 1

2 d ξ 2 ρ ) 2 d ξ

(C 1 +

| | ≥ +| | | |

ψ(ξ)) ϕ(ξ) ( 1 ξ ϕ(ξ)

R R

≥ C 1 ) 2 ρ

2 d ξ.

( 1 +| ξ | | ϕ(ξ) |

R