24
Denis Belomestny and Markus Reiβ
e-Tσ2u2m1 /2e-(Tσ2u2/m)m (Tσ2/m)2mu2m-2 |u - i|2m-2 du
= ε-2 δ 2 m-7 / /
(T σ02 m
)-1/2
0
-mv1 /mlv2m- (i + m- 1 v- 1 /m)m- 1 dv
=ε
2δ2m-4
2δ2m-4
e-mv1/m
dv
e-zzm-1m1-m
dz
= ε-2δ2m-m-3Γ(m) < ε-2δ2m-m-3(m - 1)m-1 /2e1 -m ~ ε-2m-3-se-m
Consequently, the Kullback-Leibler divergence remains small when choosing
m > 2log(ε-1), but m < log(ε- 1), which gives δ ~ log(ε- 1)-s/2. From the
basic general lower bound result we therefore obtain the asymptotic lower
bound for μ.
7.4. Lower bound for σ2, γ and λ in the case σ > 0
Since the proof is very similar to the preceding calculations, we only give
the perturbations of the basic triplet T0 = (σ0, γ0, μ0) with σ0 > 0 which are
least favourable. More details can be found in Belomestny and Reiβ (2005).
For γ we leave σ0 fixed and use a perturbation of the form
u2m/U2m
γ 1 := Y0 + δ, Fμ 1(u) := Fμ0(u) — δi(u — i)e
For λ we keep σ0 and γ0 fixed and consider
i)m/U2m
Fμ 1(u) := Fμo(u) + δe--um(u
For σ2 we leave γ0 invariant and consider the perturbation
2m 2m
σ2 := σ2 + 2δ, Fμ 1(u):= Fμ0(u)+ δ(u — i)2e u /U
Each time m is chosen to be of order log(ε-1) and the value of δ > 0 results
from the smoothness class considered.
Acknowledgements
We thank Peter Tankov for intense discussions on different calibration ap-
proaches and for sharing his experience in assessing their practical perfor-
mances. We are grateful for the comments and questions by two anonymous
referees which have led to considerable improvements.