Spectral calibration of exponential Lévy Models [1]

Spectral calibration of exponential Levy models ??

7.1. Lower bound for μ in the case σ = 0

Fix a positive integer j. Let ψ^(j) ∈ C^∞ (R) be some function with
support in [0, 1] satisfying kψ^(j) k_L2 = 1, ^R ψ ^(j) (x)e^{-2-j x} dx = 0 and
R ∖F ψ (^j ) ( u ) u^-21² du < ∞. Certainly, there are infinitely many functions ψ^{(j )}
fulfilling these requirements; the last property follows for instance if ψ is the
second derivative of an L²-function. Introduce the wavelet-like notation

ψ_jk ( x ) := 2^j/² ψ (^j )(2^j x - k ), j > 0, k = 0,..., 2^j - 1.

Consider for any r = (r_k ) ∈ {- 1, +1}²j and some β > 0 the perturbed Levy
triplets T_r = (0,γ₀, μ_r ) with

2^j

μr (x) = μo(x) + β2^-j(^s+1 /²) X rkψjk(x), x ∈ R .

k=1

We note that due to Fψjk (0) = 0 and e^-xψjk (x) dx = 0 the triplet
Tr satisfies the martingale condition such that Tr ∈ Gs (R, 0) holds for a
sufficiently small choice of the constant β > 0.

The Gaussian likelihood ratio of the observations under the probabilities
corresponding to T_r0 and T_r under the law of T_r for some r, r⁰ with r_k = r_k⁰
for all k except one k0 is given by

Λ(r⁰,r) = expf [ (O_ro-O_r)(x)ε^-1 dW(x)-- I ∣O_ro-O_r)(x)|²ε^-2 dx´ .
-∞                  ² -∞

Hence, the Kullback-Leibler divergence (relative entropy) between the two
observation models equals

KL(Tro ∖T_r) = 1 ∞ |(Oro
² -∞

The standard Assouad Lemma (Korostelev and Tsybakov 1993, Thm.

2.6.4) now yields the lower bound for the risk of any estimator μ of μ

inf sup Eτ h ∕∣^(x) - μ(x) |² dxi & 2^j ∖∖μ_r - μro ∣∣² 2 ~ 2^-2js,
^μ T =(0,γ,μ)∈Gs(R,0)     '-J                       ^j

provided the Kullback-Leibler divergence K L(T_ro |T_r) stays uniformly
bounded by a small constant. It remains to determine a minimal rate for
2^j → ∞ such that this holds when the noise level tends to zero.

Arguing in the spectral domain and using the general estimate |e^z - 1| 6
2∖z∖, for ∖z∖ 6 δ and some small δ > 0, together with ∣∣φτ,_r0∣ψτ,r∣∣∞ → 1 for
2^j → ∞, we obtain for all sufficiently large j

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