Page 201 - Special Topic Session (STS) - Volume 1

STS425 Arifah B. et al.

1
2̂ − 2
̂
= ( ) (17)

2
Γ(2 + 1

Where  is the empirical moment of order 2 that can be given as :

1
2
̂ = ∑ (18)

=1
The asymptotic distribution of Least square estimator is given by (Chen et
al., 2017)
2.4 The estimation of Diffusion Coefficient on the Volatility
Process by using Quadratic Generalized Variations
The quadratic generalized variations (QGV) can be employed to
estimate the Diffusion coefficient  in the discretely fOU process. The

Hurst exponent H and the Diffusion coefficient  can be estimated
simultaneously. Let a  ( ,..., )a 0 a k be a discrete filter of +1, K N ,
with order L  , K  L . Then
1
K
K
 a k  0 for 0   L  1 and  a k  0 (19)
L
k
k
k 0 k 0
K
which can be normalised as  ( 1) 1 k a  k 1. The filter a is expanded to
k 0
a by using the following relation:
2
a  if k  2k
a   k for 0 k  2K (20)
2
k
 0 otherwise
2K K
2
Since  a k  2 r  k a k , then a and a produce same filtration. The
r
r
2
k
k 0 k 0
QGV associated to the filter a is presented as (Istas and Lang, 1997):
N K  K  2
V N ,a    a X i k  . (21)
k
i 0  k 0 
If we denote
mK nK
  a a mk n i  2H
m
n
k
 a m ,a n ( )i  k 0  0 , (22)
H
(mn ) H  a a k  2H
k
, k
then, the values of H and  can be estimated as follows:
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