Information Technology Reference
In-Depth Information
so that
∞
(α
+
)
n
z
n
θ(
z
)
=
(4.53)
(
+
)(α)
n
1
n
=
0
for
|
z
| ≤
1
.
Therefore the coefficients in the solution (
4.51
)aregivenby
(α
+
k
)
θ
k
=
)(α)
,
(4.54)
(
k
+
1
from which the influence of the distant past is determined by the asymptotic form
of the coupling coefficient. Using the analytic properties of gamma functions it is
straightforward to show that for
k
→∞
k
α
−
1
(α)
,
θ
k
≈
(4.55)
so the strength of the contribution to the solution decreases with increasing time lag as
an inverse power law asymptotically as long as
α<
1
/
2
.
Financial time series
The fractional walk was originally developed for the study of long-time memory in
financial time series. The observable most often used to characterize the memory is
the slope of the spectrum obtained from the average square of the Fourier amplitude
of the time series. Given a set of measurements defined by (
4.51
) the discrete Fourier
amplitudes of the observed time series can be written as the product
Q
ω
=
θ
ω
ξ
ω
(4.56)
due to the convolution form of the solution to the fractional difference equation. The
spectrum of the solution is
Q
2
S
(ω)
=
ξ
,
(4.57)
ω
where the subscripted angle brackets denote an average over an ensemble of realizations
of the fluctuations driving the financial network. Inserting (
4.56
)into(
4.57
) yields
2
ξ
ω
2
θ
ω
S
(ω)
=
ξ
,
(4.58)
where for white-noise fluctuations the noise spectrum is the constant
D
ξ
.
The Fourier coefficient
θ
ω
is calculated by taking the discrete Fourier transform of
θ
k
.
Thus, for 0
<ω
≤
π
,
∞
θ
ω
=
0
θ
k
e
−
ik
ω
k
=
e
−
i
ω
k
∞
(
k
+
α
−
1
)
!
1
=
=
1
e
−
i
ω
α
,
(4.59)
k
!
(α
−
1
)
!
−
k
=
0
