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Lemma 13.4.1
Given d
=
d(ε)
,
there holds
d
d
2
λ
i
−
λ
i
|≤
s
j
,i
|
=
1
,...,d.
=
d
j
+
1
Proof
From the definitions of
η
and
λ
, we have that
η
k
)
≤
d
d
d
1
2
λ
i
−
λ
i
|=
1
|
Q
kk
−
Q
kk
|
|
U
ik
(η
k
−
1
|
η
k
−
η
k
|=
k
=
1
k
=
k
=
d
d
d
d
1
2
1
2
U
jk
(D
jj
−
D
jj
)
s
j
=
≤
k
=
1
j
=
1
k
=
1
=
d
j
+
1
d
d
2
s
j
,i
=
=
1
,...,d.
=
d
j
+
1
Remark 13.4.2
From (
13.29
) and Lemma
13.4.1
, we conclude that the fluctuation
components
T
ε,i
,
i
1
,...,d(ε)
, are pure drifts of order
ε
. Furthermore, note that,
=
upon setting
→
d
−
d(ε),
s
2
d(ε)
→
t
d
=
d
t
=
1
t,
+
the fluctuation components
T
ε,i
,
i
1
,...,d
, again define a
d
-dimensional
full-rank market of type (
13.19
)-(
13.21
) with timescale
t
, allowing, in principle, for
recursive
ε
-rank aggregation.
=
d(ε)
+
We again consider the
d
-dimensional Black-Scholes market model with log-
price process
X
and its
ε
-aggregate rank
d
process
X
ε
of the previous section, and
x
ε
)
in (
13.17
).
we estimate the error of approximating
u(t, x)
by
u(t,
ˆ
Theorem 13.4.3
Assume that the payoff g is Lipschitz
.
Then
,
there exists a constant
C(x) independent of ε such that
d
x
ε
)
s
i
.
|
u(t, x)
−
u(t,
ˆ
|≤
C(x)
=
d(ε)
i
+
1
Proof
We introduce the artificial process
Y
ε
with dynamics
d
Y
ε,i
t
s
i
d
W
t
,i
=
λ
i
d
t
+
1
=
1
,...,d.
≤
i
≤
d(ε)
}
{
1
Under the change of basis induced by
U
,wehave
x
ε
)
y
ε
)
|
−
ˆ
|=|
−
ˆ
|
u(t, x)
u(t,
v(t,y)
v(t,
y
ε
)
≤ |
v(t,y)
−
v(t,
y)
˜
| +|
v(t,
y)
˜
−
v(t,
ˆ
|
.
(13.30)
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