Spectral calibration of exponential Lévy Models [1]



18


Denis Belomestny and Markus Reiβ

Making use of R0U 2uecu2 du =    —1 = E (cU2) U2 for any c > 0, we esti-

mate the last integral by

U2

e eTσru +2TЫь 1 us+2 du 6 e2TkμkL1 Us+3E(T'.U2)

U

and derive from ∣∣μ∣∣^ 1 = (0) 6 2R for the bias part in the linear term

L> . 2E(T2U2).

(38)


For the variance part of the linear error term we use the support proper-
ties supp(
wσU) [-U, U] and supp(bk) = [xk1, xk+1]. Several applications
of the Plancherel identity, the Cauchy-Schwarz inequality and estimate (35)
then yield

=

U


Cov


ʌ'

(u — i)uFO(u) (v — i)vFO(v)
φT ( u — i )   ’   φT ( v — i )


wσU (u)wσU (v)


du dv


x I Γ
=X δ2
k=1   


ψτ ( u


2

— i)1(u — i)uF bk (u)wσU (u) duI

N I

2π     δk2 II

k=1    -  '


F 1Tτ ( u


— i)1(u — i)uwσU (u) (x)bk (—x) dx


6 2π


N       xk+1

Xδk2       II

k=1     xk-1


F


φτ ( u


i)1(u


i)uw


dx bk 2L2


δl2    III

J-∞a


F


φτ ( u


i)1(u


i)uw


dx


U
^HI2~ /  ψτ(u

U


i) 2(u4 + u2)wσ(u)2 du


∆U-1


E(2U2


)δl2.


Altogether we obtain for the linear error term

E


2

L(u)wσ(u)du∣ ] . E(2U2) (4 + U-1 ∣∣δ∣∣2^).

(39)


It remains to estimate the quadratic remainder term. Due to Lemma 1
and Proposition 2 we have

E

2

R(u)wσU (u) duII


(40)

U U I                            I2 u4wU (u)v4wU (v)

< E E F ( O-O )( u ) F ( O-O )( v )∣ ---σ. ( ) , σ2 ( ) d u d v.

-U -U                                       κ(u)2κ(v)2

The independence of (εk ) and the finiteness of their fourth order moments
entail the inequality



More intriguing information

1. Inhimillinen pääoma ja palkat Suomessa: Paluu perusmalliin
2. The name is absent
3. NEW DEVELOPMENTS IN FARM PRICE AND INCOME POLICY PROGRAMS: PART I. SITUATION AND PROBLEM
4. Ex post analysis of the regional impacts of major infrastructure: the Channel Tunnel 10 years on.
5. CAN CREDIT DEFAULT SWAPS PREDICT FINANCIAL CRISES? EMPIRICAL STUDY ON EMERGING MARKETS
6. Ruptures in the probability scale. Calculation of ruptures’ values
7. The name is absent
8. The name is absent
9. A Classical Probabilistic Computer Model of Consciousness
10. The name is absent
11. QUEST II. A Multi-Country Business Cycle and Growth Model
12. EXECUTIVE SUMMARIES
13. SLA RESEARCH ON SELF-DIRECTION: THEORETICAL AND PRACTICAL ISSUES
14. The name is absent
15. Text of a letter
16. Land Police in Mozambique: Future Perspectives
17. The name is absent
18. The name is absent
19. Neural Network Modelling of Constrained Spatial Interaction Flows
20. Spectral calibration of exponential Lévy Models [1]