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⎛
⎞
10
···
0
⎝
⎠
ρ
1
.
L
=
,
L
ρ
n
v
where
L
n
v
×
n
v
∈ R
is defined as
⎧
⎨
0
if
i<j,
1
1
ρ
ki
1
/
2
−
j
−
1
k
L
ij
:=
−
ρ
i
if
i
=
j,
i, j
=
1
,...,n
v
.
=
⎩
ρ
ji
if
i>j,
The
constants
ρ
i
,
i
=
1
,...,n
v
, and
ρ
ij
,
i, j
=
1
,...,n
v
,
satisfy
|
ρ
i
|≤
1,
+
j
−
1
k
ρ
i
ρ
ki
≤
1
1.
=
1 is introduced in [68]. The process
Y
t
Example 9.1.1
The case
n
v
=
Y
t
follows
a mean-reverting Ornstein-Uhlenbeck (OU) process, i.e. the coefficients in (
9.6
)are
given by
=
=
−
=
c
1
(y)
α(m
y),
g
1
(y)
β,
with
α>
0 the rate of mean-re
version,
m>
0 the long-run mean level of volatility
and
β
1
. Here,
L
=
L
11
=
∈ R
−
ρ
2
, with
|
|≤
1. Note that this model includes
in particular the model of Stein-Stein [153](
ξ(y)
ρ
=|
|
=
y
,
ρ
0) and the model of
e
y
,
ρ
Scott [150](
ξ(y)
=
=
0).
Example 9.1.2
The case
n
v
=
2 is studied in [67]. The first component of
Y
t
is
assumed to follow a mean-reverting OU process modeling a fast scale volatility
factor, while the second component is a diffusion process, modeling a slow scale
volatility factor. In particular, the coefficients in (
9.6
)are
c
1
(y
1
)
=
α
1
(m
−
y
1
),
g
1
(y
1
)
=
α
2
,
c
2
(y
2
)
=
α
3
c(y
2
),
g
2
(y
2
)
=
α
4
g(y
2
),
(
√
α
1
)
as well as with
α
3
>
0 “small”, and
α
4
=
with
α
1
>
0 “large”,
α
2
=
O
(
√
α
3
)
. The functions
c, g
O
: R → R
are assumed to be smooth and at most lin-
early growing.
Introducing random volatility renders the market model incomplete so that there
is—unlike in the constant volatility case—no unique measure equivalent to the phys-
ical measure under which the discounted stock price is a martingale. We consider
the
(n
v
+
1
)
-dimensional standard Brownian motion under the risk-neutral measure
W
t
(W
t
, W
t
,...,W
n
v
+
1
)
=
t
:=
(W
t
,W
t
,...,W
n
v
+
1
)
t
t
ξ(Y
s
)
d
s,
t
γ
1
(Y
s
)
d
s,...,
t
0
γ
n
v
(Y
s
)
d
s
,
μ
−
r
+
0
0
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