Spectral calibration of exponential Lévy Models [1]

6 ∣(O - β0)⁰∣L∞ (xj_- - xj_--1)²

We infer that this error term is also of order ∆² and thus does not enlarge
the convergence rate.

6.3. Upper bound for σ²

The rate for σ² follows once the general risk estimate

E[∣σ² - σ²1²] . U^-2(^s+3) + E(Tσ²U²)U^-1 ε² + E(Tσ²_maxU²)²U⁴ε⁴ (32)
has been shown for U . ∆^-1 uniformly over Gs (R, σmax ), since the explicit
choice of U renders the second and third term asymptotically negligible.

Consider in the definition (12) of ψ separately the linearisation L, ne-
glecting the stabilisation by κ, and the remainder term R:

L(u) := T^-1 φ_τ(u - i)^-1(u - i)uF(O) - O)(u),          (33)

R(u) := ψ⁾(u) - ψ(u) - L(u).                              (34)

When neglecting the remainder term, we may view ψ⁾(u) as observation
of ψ(u) in additive noise, whose intensity grows like ∣φT(u-i) ∣^-11(u-i)u∣ ~
u² e^{Tσ2 u2} for IuI → ∞. This heteroskedasticity reflects the degree of ill-
posedness of the estimation problem.

Lemma 1. For all u ∈ R the remainder term satisfies

∣R(u)∣ 6 T^-1κ(u)^-2(u⁴ +u²)∣F(O⁾ - O)(u)∣².

Proof. Let us set ⅛)T(u - i) := 1 - u(u - i)FO(u) which equals eT^ψ(^u) if
-⅛

∣⅛)T(u - i) I > κ(u). Using ∣eT^ψ(^u) ∣ > κ(u), u ∈ R, we obtain by a second-
order expansion of the logarithm

∣Tψ(u) - log(φT(u - i))) - φT(u - i)^-1(eT^ψ(^u) - φT(u - i))∣

6 1 κ(u)^-2 ∣eT^ψ(^u) - φT(u - i) ∣².

This gives the result whenever ∣⅛)T(u - i)∣ > κ(u). For the other values u
the inequalities ∣⅛)T(u - i)∣ < κ(u) 6 ∣φT(u - i)∣/2 imply 1 6 ∣⅛)T(u - i) -
φT(u - i) ∣κ(u)^-1 and hence

∣φT(u - i)^-1(eT^ψ(^u) - ⅛)T(u - i))∣ 6 2κ(u)^-21⅛)T(u - i) - φT(u - i)∣²

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