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Thetwotermsin(
13.30
) are estimated separately. Since
g
is globally Lipschitz, we
have for the first term, where
f(e
y
)
g(e
x
)
and constants may change between
=
lines,
E
f(e
y
+
Y
T
−
t
)
−
f(e
y
+
Y
T
−
t
)
|
v(t,y)
−
v(t, y)
| =
T
−
t
d
e
y
i
+
Y
ε,i
e
y
i
+
Y
T
−
t
≤
C
1
E
−
i
=
1
d
e
s
i
W
T
−
t
e
y
i
+
λ
i
(T
−
t)
=
C
E
−
i
=
d(ε)
+
1
e
y
i
+
λ
i
(T
−
t)
d
e
−
z
2
/(
2
s
i
)
d
z
e
z
=
C
|
−
1
|
R
=
d(ε)
i
+
1
d
d
e
y
i
s
i
s
i
.
≤
C
≤
C(y)
=
d(ε)
=
d(ε)
i
+
1
i
+
1
Similarly, using Lemma (
13.4.1
), we have for the second term
E
f(e
y
+
Y
T
−
t
)
f(e
y
+
Y
T
−
t
)
y
ε
)
=
v(t,
y)
˜
−
v(t,
ˆ
−
t
d
e
y
i
+
Y
ε,i
−
e
y
i
+
Y
ε,i
≤
C
1
E
T
−
t
T
−
i
=
e
1
{
1
≤
i
≤
d(ε)
}
s
i
W
T
−
t
e
λ
i
(T
−
t)
d
e
y
i
e
λ
i
(T
−
t)
=
C
E
−
i
=
1
d
d
d
1
2
s
i
(T
e
y
i
+
−
t)
λ
i
−
λ
i
|≤
e
y
i
(
1
s
i
)
s
j
|
+
≤
C(y)
C(y)
i
=
1
i
=
1
j
=
d(ε)
+
1
d
s
j
.
≤
C(y)
=
d(ε)
j
+
1
C
(x)
, which completes the proof.
Since
y
=
U
x
,
C(y)
=
13.4.2 Stochastic Volatility Models
Similarly to the one-dimensional case, multivariate stochastic volatility models re-
place the constant volatilities
Σ
ij
in the Black-Scholes model (
13.19
) by stochastic
processes
Σ
ij
=
f
ij
(Y )
, where
f
ij
are non-negative functions and
Y
is an additional
source of randomness, which is modeled by an Itô diffusion in
d
.
R
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