Page 198 - Special Topic Session (STS) - Volume 1

STS425 Arifah B. et al.

t
Var [B t H  B s H ]   s 2H . (3)
The relation in Equation (3) defines its covariance structure as follows:
1 2H 2H 2H
Cov (B H , B H )  (t  s  t s ), (4)
s
t
2
Thus, for H  1 , the fOU process is Gaussian and ergodic but it is neither
2
1
Semimartingal nor Markovian. For H  , the fOU process shows a long
2
memory property. From Equation(2) we get:
t
Y  y e  t     e   (t s ) dB s H , t  0, X  x 0 . (5)
t
0
0
0
Let y  0 L 0 ( ),     and  ,  0. Then for all s  the
a
integration  a t e dB s H , t  , a exist as a Riemann-Stieltjes pathwise integral

s
which is continuous in t, and the unique almost surely continuous solution
of Equation (5) is

s
Z t H , 0 y  e  t  (y  0   0 t e dB s H ), t  0, (6)
Particularly, the restriction to t  0 of the following almost surely
continuous process
,
H
Z    0 t e   (t s ) dB s H , t R (7)
t

which can solve Equation (5) with initial condition y  Z 0 H . The Gaussian
0
process (Z t H ) tR is stationary since it follows the stationarity of the

increments of {B t H }. Additionally, for each initial condition y  0 L 0 ( ) ,
Z  Z t H ,x 0  e  t  (Z  y 0 )  0, as t  .  (8)
H
H
0
t

This implies that every stationary solution of Equation (5) has the same
distribution as (Z t H ) . The (Z t H ,x 0 ) t 0 is defined as a FOU process with
t
0
initial condition y and (Z t H ) tR is the stationary FOU process that is
0
1
ergodic and for H  it exhibits as long range dependence. Since the
2
values of H and  can be estimated without having the  value, then
the estimation for the unknown parameter   ( , , )H  will be carried
out in several steps. The covariance theorem of the fOU is shown in Chen
et al. (2017).

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