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




More intriguing information

1. The name is absent
2. Monetary Policy News and Exchange Rate Responses: Do Only Surprises Matter?
3. AGRICULTURAL TRADE LIBERALIZATION UNDER NAFTA: REPORTING ON THE REPORT CARD
4. TLRP: academic challenges for moral purposes
5. The name is absent
6. Beyond Networks? A brief response to ‘Which networks matter in education governance?’
7. The Impact of Minimum Wages on Wage Inequality and Employment in the Formal and Informal Sector in Costa Rica
8. Making International Human Rights Protection More Effective: A Rational-Choice Approach to the Effectiveness of Ius Standi Provisions
9. The name is absent
10. The name is absent