Spectral calibration of exponential Lévy Models [1]



16


Denis Belomestny and Markus Reiβ


An application of Proposition 1(d) therefore shows for σ > 0


∞  I E[ O)( x ) -O ( x )] I d x 6 e-A

-∞


+ 32 + 2C02 + 4C2∆e-(A-)


2.


In the case σ = 0 we consider the index jt with Xj-- 1 6 γT < xj- and
face an additional error estimated by


χxi*

xxi--1


IE[O)(x) - O(x)]I dx 6


xi-

k(O

xi--1


β0),∣∣L∞ I


2(x - xj--1)(xj- - x)
xj- - xj--1


dx


6 (O - β0)0L (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(2U2)U-1 ε2 + E(2maxU2)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 φτ(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 e2 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 eTψ(u) if
-⅛

⅛)T(u - i) Iκ(u). Using eTψ(u) κ(u), u R, we obtain by a second-
order expansion of the logarithm

(u) - log(φT(u - i))) - φT(u - i)-1(eTψ(u) - φT(u - i))

6 1 κ(u)-2 eTψ(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) -
φ
T(u - i) κ(u)-1 and hence

φT(u - i)-1(eTψ(u) - ⅛)T(u - i)) 6 2κ(u)-21⅛)T(u - i) - φT(u - i)2



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