Spectral calibration of exponential Lévy Models [1]



Spectral calibration of exponential Levy models ??

23


-2δ2 (1

-∞


- e-v2)2U-2v-2U dv


ε-2δ2U-1 - ε-2δ(2s+5)/(s+2).

Thus, the Kullback-Leibler divergence remains small for δ ε(2s+4)/(2s+5)
with a small constant, which gives the asymptotic lower bound for γ .

For the lower bound of λ we perturb the triplet T0 leaving γ0 and σ0 = 0
fixed and putting

1(u) := 0(u) + δe-u(u-i)/U2.

By similar estimates as for γ, when choosing U δ-1 / (s+1) with a suffi-
ciently small constant, the perturbation
T1 lies in Gs (R, 0) and the Kullback-
Leibler divergence is asymptotically bounded by

KL(TιT)) . ε-2δ2U-3 ε-2δ(2s+5)/(s + 1).

The basic lower bound results yields the asserted lower bound for λ.

7.3. Lower bound for μ in the case σ > 0

The interesting deviation from standard proofs of lower bounds (see e.g.
Butucea and Matias (2005)) for severely ill-posed problems is that we face
the restriction that
is analytic in a strip parallel to the real line and is
uniquely identifiable from its values on any open set. So, let
T0 = (σ20, μ0)
with
σ0 > 0 be a Levy triplet from the interior of Gs (R,σmax ). Consider
the perturbation
T1 = (σ2 0 1) with

1(u) := 0(u) + δm 1 /4e-(Tσ2u2/m)m/2(2m)mum(u - i)m, u R .

for m N, δ > 0. Then we have uniformly for m → ∞ and δ → 0

∣∣μ 1 - μо k2 2 = p2∏τ=∣ fo e-vv(1+2m)/2m (1 + m- 1 v- 1 /m)m dvδ2

Similarly, for k = 1, . . . , s we derive uniformly in m and δ

∣∣μ(k ) - μ0k)∣∣l2 = 2∣∣ukF(μ 1 - μθ)(u)∣∣l2 δmk/2,
∣∣μ 1s) - μ0s)∣∣ 6 ∣∣usF(μ 1 - μо)(u)∣∣li 6 δms/2- 1 /4.

Therefore choosing δm-s/2 with a small constant yields T1 Gs (R, σmax)
because we then also have that
μ 1 is real-valued and T1 satisfies the mar-
tingale condition and Assumption 1.

By the same arguments as before and by Stirling’s formula to estimate
the Gamma function, the Kullback-Leibler divergence between the observa-
tions under
T0 and under T1 is asymptotically bounded by

4ε


-2


I1P о ,t ( u


- i ) 12 T2 IF ( μ 1


- μ0)(u)12(u4 + u2) 1 du




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