Spectral calibration of exponential Lévy Models [1]

Spectral calibration of exponential Levy models ??

ε^-2δ²U^-1 - ε^-2δ(2^s+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

Fμ ₁(u) := Fμ₀(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(Tι∣T)) . 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 Fμ is analytic in a strip parallel to the real line and is
uniquely identifiable from its values on any open set. So, let T₀ = (σ²,γ₀, μ₀)
with σ₀ > 0 be a Levy triplet from the interior of G_s (R,σ_max ). Consider
the perturbation T1 = (σ² ,γ₀ ,μ 1) with

Fμ 1(u) := Fμ₀(u) + δm 1 /⁴e^-(T^σ2u2/^m)m/²(Tσ2∣m)^mu^m(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=∣ f_o e-v^v(1⁺2m)/2m (1 + m- 1 ^v- 1 /m)m d^v — δ2 ■

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

∣∣μ(k ) - μ0k)∣∣l2 = √2∏∣∣u^kF(μ 1 - μθ)(u)∣∣l2 — δm^k/²,
∣∣μ 1s) - μ0s)∣∣∞ 6 ∣∣u^sF(μ 1 - μо)(u)∣∣li 6 δm^s/^2- 1 /⁴.

Therefore choosing δ — m^-s/2 with a small constant yields T₁ ∈ G_s (R, σ_max)
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

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