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

52 CHAPTER 1. LEVY PROCESSES AND EXP-LEVY MODELS
1. ∀ v ∈ R, |Φ T (v − i)| ≤ exp(− T Av2
2
).
2. Suppose that ν has a density of the form ν(x) = e−x
|x|
f(x), where f is increasing on (−∞, 0)
and decreasing on (0, ∞). Then |Φ T (v − i)| is increasing on v ∈ (−∞, 0) and decreasing
on v ∈ (0, ∞).
Proof. The martingale condition implies that ∀ v ∈ R,
|Φ T (v − i)| = exp
{− T ∫ ∞
}
Av2 − e x (1 − cos(vx))ν(dx) . (1.29)
2
This immediately entails the first part of the lemma. For the second part, let 0 > v 1 > v 2 (the
case v 1 < v 2 < 0 can be shown in the same way). Then, for all t ∈ R,
Therefore,
∫ ∞
−∞
e x (1 − cos v 1 x)ν(dx) =
∫ ∞
−∞
e t/v 1
ν(t/v 1 )
v 1
≤ et/v 2
ν(t/v 2 )
v 2
.
(1 − cos t) et/v 1
ν(t/v 1 )
dt
−∞
∫ ∞
≤
−∞
v 1
(1 − cos t) et/v 2
ν(t/v 2 )
v 2
dt =
and it follows from Equation (1.29) that |Φ T (v 1 − i)| ≥ |Φ T (v 2 − i)|.
∫ ∞
−∞
e x (1 − cos v 2 x)ν(dx),
Proposition 1.11. Let {X t } t≥0 be a real-valued Lévy process with triplet (A, ν, γ) and characteristic
function Φ T and let Σ > 0.
1. Suppose A > 0. Then the truncation error in Equation (1.28) satisfies:
|ε T | ≤
8
πT Σ 2 L 3 e− T L2 Σ 2 8
8 +
πT AL 3 e− T L2 A
8 . (1.30)
2. Suppose that ν has a density of the form ν(x) = e−x
|x|
f(x), where f is increasing on (−∞, 0)
and decreasing on (0, ∞). Then the truncation error in Equation (1.28) satisfies:
Proof. From Equation, (1.28),
|ε T | ≤ 1 ∫ ∞
|˜ζ T (v)|dv + 1
2π L/2
2π
≤ 1 ∫
2π
8
|ε T | ≤
πT Σ 2 L 3 e− T L2 Σ 2
8 + |Φ T (L/2 − i)| + |Φ T (−L/2 − i)|
.
πL
∫ −L/2
−∞
(−∞,−L/2)∪(L/2,∞)
|˜ζ T (v)|dv
|Φ T (v − i)|dv
v 2 + 1 ∫
|Φ Σ T
(v − i)|dv
2π (−∞,−L/2)∪(L/2,∞) v 2 .