Processus de Lévy en Finance - Laboratoire de Probabilités et ...

54 CHAPTER 1. LEVY PROCESSES AND EXP-LEVY MODELS
From the martingale condition,
Γ T (v) = Φ T (v − i) − Φ T (−i)
v
=
∫ 1
0
Φ ′ T (vt − i)dt.
Applying the dominated convergence theorem once again, we conclude that
and therefore |Γ (n)
T (v)| ≤ C n+1
n+1
Γ (n)
T (v) = ∫ 1
0
for all n ≥ 0.
t n Φ (n+1)
T
(vt − i)dt,
Proposition 1.13. Suppose that the integral in (1.28) is approximated using the trapezoidal
rule. Then the discretization error ε D satisfies
{
2∑
|ε D | ≤ ∆2 C 3−l + C3−l
Σ (
∆ + π 6π (3 − l)! 2
l=0
)
e |k−rT | + log
(
L
2 + √
L 2
4 + 1 )
where C Σ k is computed as in (1.31) for the characteristic function ΦΣ T .
}
|k − rT | l
, (1.32)
l!
If this integral is approximated using the Simpson rule, ε D satisfies
{ (
4∑
|ε D | ≤ ∆4 C 5−l + C5−l
Σ (
2∆ + π √ ) }
)
e |k−rT | L L
+ log
5π (5 − l)!
2
2 + 2
4 + 1 |k − rT | l
, (1.33)
l!
l=0
Proof. The trapezoidal rule (cf. [32]) is defined by
∫ x+h
x
f(ξ)dξ = 1 2 h(f(x) + f(x + h)) + R with R = − 1 12 h3 f ′′ (x ∗ )
for some x ∗ ∈ [x, x + h]. The sampling error in (1.28) therefore satisfies:
|ε D | ≤ ∆3 N−2
∑
24π
= ∆3
24π
m=0 v∈[−L/2+m∆,−L/2+(m+1)∆]
N−2
∑
m=0
sup
∂ 2
∣∂v 2 e−ivk ˜ζT (v)
∣
{
sup
∂ 2
∣ ∣∣∣
∣∂v 2 e−ivk ζ T (v)
∣ + sup ∂ 2
}
∂v 2 e−ivk ζT Σ (v)
∣ , (1.34)
where ζT Σ (v) := e ivrT ΦΣ T
(v − i) − 1
.
iv(1 + iv)
Observe that e −ivk ζ T (v) = e−iv(k−rT )
i(1+iv)
Γ T (v), and therefore
∂ n
∣∂v n (eivk ζ T (v))
∣
≤
n∑
l=0
≤ n!
|Γ (n−l)
T
n∑
l=0
n!
(v)|
(n − l)!
C n−l+1
(n − l + 1)!
l∑ |k − rT | j (
1
√
j! 1 + v 2
j=0
) l−j+1
l∑ |k − rT | j ( )
1 l−j+1
√ := ˜g n (v).
j! 1 + v 2
j=0