Spectral calibration of exponential Lévy Models [1]



Spectral calibration of exponential Levy models ??

21


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 (j) kL2 = 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

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

Consider for any r = (rk ) {- 1, +1}2j and some β > 0 the perturbed Levy
triplets
Tr = (00, μr ) with

2j

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

k=1

We note that due to jk (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
Tr0 and Tr under the law of Tr for some r, r0 with rk = rk0
for all k except one k0 is given by

Λ(r0,r) = expf [ (Oro-Or)(x)ε-1 dW(x)-- I Oro-Or)(x)|2ε-2 dx´ .
-∞                  2 -∞

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

KL(Tro Tr) = 1 |(Oro
2 -∞

- Or)(x)|2 ε-2 dx.


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 2j ∖∖μr - μro ∣∣2 2 ~ 2-2js,
μ T =(0,γ,μ)Gs(R,0)     '-J                       j

provided the Kullback-Leibler divergence K L(Tro |Tr) 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 |ez - 1| 6
2z, for z 6 δ and some small δ > 0, together with ∣∣φτ,r0ψτ,r∣∣ 1 for
2
j → ∞, we obtain for all sufficiently large j

KL(Tro |Tr)


1

= ∏2< J∞F( Or


- Or)(u)|2 du


6ε


-2


Γ∣

-∞


φτ,r (u - i) - Ψτ,rθ (u - i )
u(u - i)


2

∣ du




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