Chemistry Reference
In-Depth Information
.N
C
1/A
Z
t
"
l
.s/ds
"
k
.t/
!
X
N
a
k
N
C
1
"
k
.t/
D
w
.t
s;T
k
;m
k
/
0
l
D
1
(5.126)
since at any time period t,firmk uses the current estimate "
k
.t/ in computing p
k
.t/
by (5.117), however it uses delayed actual price data, and therefore the computed
discrepancy p
k
is based on continuously distributed lagged values of
P
lD1
"
l
.
Equation (5.126) constitutes a system of linear Volterra-type integro-differential
equations. In order to compute the eigenvalues we seek the solution of the corre-
sponding homogenous equations in the exponential form "
k
.t/
D
v
k
e
t
.k
D
1;2;:::;N/. Substituting these into the homogenous equations implies that
C
v
k
e
t
Z
t
X
N
a
k
N
C
1
a
k
N
C
1
v
l
e
s
ds
D
0:
C
w
.t
s;T
k
;m
k
/
0
lD1
Letting t
!1
and using the limiting property of integral (D.3) we have
C
v
k
C
1
C
.m
k
C1/
X
N
a
k
N
C
1
a
k
N
C
1
T
k
p
k
v
l
D
0
lD1
or
N
C
1
a
k
C
1
1
C
m
k
C1
X
N
T
k
p
k
v
k
C
v
l
D
0;
(5.127)
lD1
where
(
1 if m
k
D
0;
m
k
if m
k
>0;
as before. Non-trivial solutions exist if and only if the determinant of the coefficient
matrix is zero. This determinantal equation has the special structure
p
k
D
0
1
A
1
./ B
1
./ ::: B
1
./
B
2
./ A
2
./ ::: B
2
./
:
:
:
:
:
:
:
:
:
:
:
:
B
N
./ B
N
./:::A
N
./
@
A
det
;
with
N
C
1
a
k
C
1
1
C
m
k
C1
T
k
p
k
A
k
./
D
C
1;
and
B
k
./
D
1;
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