Spectral calibration of exponential Lévy Models [1]

Denis Belomestny and Markus Reiβ

Hence, for 2^{j (2s+5)} ~ ε² with a sufficiently large constant the Kullback-
Leibler divergence remains bounded and the asymptotic lower bound for μ
follows.

7.2. Lower bound for γ and λ in the case σ = 0

Let us start with the lower bound for γ . We proceed as before by perturbing
a triplet T₀ = (0, γ₀, μ₀) from the interior of Gs (R, 0), but this time we only
consider one alternative T1 = (0, γ ₁, μ ₁) and choose the perturbation in such
a way that the characteristic function φ_τ (u — i) does not change for small
values of |u|. For any δ > 0 and U > 0 put

γ1 := Y0 + δ,   Fμι(u) := Fμo(u) — δi(u — i)e^-u /U , u ∈ R .

Then the function μ ₁ is real-valued. Moreover, the martingale condition (4)
is satisfied:

γ 1 + F μ 1(0) — F μ 1( i ) = Y 0 + δ + F μ 0(0) — δ — F μ 0( i ) + 0 = 0.

Because of

μ(₀s)∣∣∞ 6 2π f ∣u∣^s∣F(μ 1 — μ0)(u)∣ du < δ f |u|s⁺¹ e ^u2/U2 du
-∞                                   -∞

we get kμ1s) — μ0s) ∣∣∞ < δUs+2 and even better bounds for ∖∖μ1k) — μ0k) ∣∣^2,
k = 0,...,s. It suffices to choose U ~ δ¹ /(s⁺2) small enough to ensure
that T1 still lies in our nonparametric class Gs (R, 0). The basic lower bound
result (Korostelev and Tsybakov 1993, Prop. 2.2.2) then yields

in^{f      su}P      E γ,μ [^{|Y —} Y| 2] & ^δ 2^,

^γ (^0,γ^,μ)∈Gs(^r,0)

provided the Kullback-Leibler divergence between T1 and T0 remains asymp-
totically bounded. As in the lower bound proof for μ, in particular using
F(μ ₁ — μ₀)(i) = 0, we obtain

KL(T1|T0)

4 f∞ I^0,t(u — i)12T2|i(γ 1 — γ0)(u — i)+ F(μ 1 — μ0)(u)12 d

ε2 L₀₀                      (u⁴ + u2)                       u

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