Spectral calibration of exponential Levy models ??
15
> 2 exp(-τσ2axv2 - 4TR´ =: κ(v). (27)
The only reason for the factor 1/2 is the mathematical tractability giving
later the bound of Lemma 1.
Concerning the choice of the weight functions, we take advantage of the
smoothness s of μ by taking functions w such that Fw has s vanishing mo-
ments. Equivalently expressed in the spectral domain, the weight functions
w(u) grow with frequencies ∣u∣ like ∣u∣s to profit from the decay of ∣Fμ(u) |.
Hence, we define for all U > 0 families of weight functions by rescaling:
wσU (u) = U -3wσ1 (u/U), wγU (u) = U -2wγ1 (u/U), wλU (u) = U-1wλ1 (u/U),
(28)
where the functions wσ1 , wγ1 , wλ1 satisfy conditions (17) and (18) as well as
F(wσ1 (u)/us), F(wγ1 (u)/us), F(wλ1 (u)/us) ∈ L1(R). (29)
In addition the support of the weight functions wσ1 , wγ1 , wλ1 is assumed to
be contained in [-1, 1]. Note that the property F (w(u)/us) ∈ L1(R) means
in particular that w(u)/us is continuous and bounded such that
∣wσU(u)∣ .U-(s+3)∣u∣s, ∣wγU(u)∣ .U-(s+2)∣u∣s, ∣wλU(u)∣ .U-(s+1)∣u∣s.
(30)
6.2. A numerical approximation result
Proposition 2. Under the hypothesis e-A . ∆2 we obtain uniformly over
all Lévy triplets satisfying Assumption 1
sup ∣E[FO(u) - FO(u)] ∣ = sup ∣FOl (u) -FO(u) ∣ . ∆2. (31)
u∈R u∈R
Proof. By standard Fourier estimates the assertion follows once we have
proved kOl - OkL1 . ∆2 .
Note that O - β0 is twice differentiable except at the points xj0-1 , 0, xj0
and possibly γT by Proposition 1(d). Moreover, O - β0 has a derivative
near zero which is uniformly bounded by a constant C0 , which follows from
the L1-estimate for O00 in Proposition 1(d).
Starting with the case σ> 0, we obtain the classical quadrature estimate
for the trapezoidal rule using the mean value theorem:
xN
x1
∣Oι(x) -O(x)∣
dx 6 kO00kL1∆2 +2C0∆2.
By Assumption 1 and Proposition 1(b,c) the extrapolation error is bounded
by
/
[x0,x1]∪[xN,xN +1]
∣E[O(x) - O(x)] ∣ dx 6 4C2∆e-(a-δ).