Spectral calibration of exponential Lévy Models [1]

20

6.5. Upper bound for μ

The assertion follows as soon as the following Gs (R, σmax)-uniform risk
bound for general U holds:

E h ∞ ∣μ(x) - μ(x)∣² dxi . U^-2s + E(Tσ²U²)U⁵ε² + E(2Tσ²_maxU²)U⁹ε⁴.

-∞

(44)

The bias in estimating μ due to the cutoff at U can be estimated by

¹  F μ ( u )(1

-∞

-1[_-U,U])|² du 6 U^-2s

∞ |u|^2s∣Fμ(u)∣² du = U^-2skμ(^s)k² 2.

-∞

(45)

The variance term can be split up according to the different risk contribu-
tions. For u ∈ [-U, U] we obtain

E[IF(μ - μ)(u)∣²]

6 4 E[∣ψ(u) — ψ(u)) 1²] + 4(u² + 1)² E[∣σ² — σ²1²]

+ 4( u² + 1) E[ ∣γ — γ∣² ] + 4 E[ ∣λ — λ∣² ]

. E[∣L(u) 1²] + E[∣R(u) 1²] + U⁴ E[∣σ² — σ²1²] + U² E[∣^ — γ∣²] + E[∣λ — λ∣²]

. E[|L(u)|²] +E[|R(u)|²]+U^-2(s+1) + E(Tσ²U²)U³ε² +E(Tσm²axU²)²U⁸ε⁴.

In analogy to the previous estimates for σ² we find

E[∣L(u)1²] 6∖φ_τ(u
_{. e}Tσ²u

2_u

i ) ∣^-2( u⁴ + u ²)( kF ( O — Oi ) k ∞ + Var[ FO⁾( u )])
^( ∆⁴ + ∆² kJk2₂ ´ .

With a look at Lemma 1 we estimate the remainder by

E[∣R(u)1²] 6 16κ(u)^-4(u⁴ + u²)² E[∣F(Oi — O)(u)∣⁴ + ∣F(O) — Oi)(u)1⁴]
. e²T^σm^axu u⁸(∆⁸ + ∆⁴∣∣tf∣∣⁴₂´ .

The Plancherel identity and these estimates yield together (44) via

E E[∣^(χ) — μ(χ) 1²] dχ . U

-∞

^2s +E(Tσ²U²)U⁵ε² +E(2Tσ_m²_axU²)U⁹ε⁴

+ E(Tσ²U²)U⁴ε² +E(Tσm²axU²)²U⁹ε⁴

^2s +E(σ²U²)U⁵ε² +E(2Tσ_m²_axU²)U⁹ε⁴.

7. Proof of the lower bounds

We follow the usual Bayes prior technique, see e.g. Korostelev and Tsybakov
(1993), and perturb a fixed Levy triplet T₀ = (0,γ₀,ν₀) in the interior of
Gs (R, σmax) such that the perturbations remain in Gs(R, σmax).