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We consider the stochastic volatility extension of the Black-Scholes model as
described in [68, Chap. 10.6]. We set
Z
(X, Y )
, where
X
describes again the
log-price dynamics of
n>
1 assets and
Y
is an
:=
n
-valued Itô diffusion describing
R
f
ij
(Y )
. In particular, we assume that each
Y
i
the stochastic volatility
Σ
ij
=
evolves
according to the SDE
d
Y
t
c
i
(Y
t
)
d
t
b
i
(Y
t
)
d
W
t
,Y
0
=
y
i
,i
=
+
=
1
,...,n.
We pose the following assumptions: the state space domain of
Y
is
G
Y
n
, and
⊆ R
the coefficients
c
k
,b
k
:
G
Y
→ R
are globally Lipschitz-continuous and at most lin-
n
-valued standard Brownian motion
(W
t
)
t
≥
0
is
early growing. Furthermore, the
R
n
-valued standard Brownian motion
(W
t
)
t
≥
0
that drives the pro-
correlated to the
R
=
j
=
1
ρ
jk
W
j
cess
X
by
W
k
+
ρ
∗
W
k
, where
(W, W)
is a standard Brownian
−
j
=
1
ρ
jk
)
1
/
2
.
d
2
n
, and
ρ
k
:=
R
=
motion in
with
d
(
1
(x
1
,...,x
n
,y
1
,...,y
n
)
, the coefficients
μ
,
Σ
in
(
13.18
) under a non-unique EMM are given by
Denoting by
z
:=
(x, y)
=
:=
r
1
/
2
f
nn
(y), c
1
(y
1
),...,c
n
(y
n
)
∈ R
1
/
2
f
11
(y),...,r
d
,
μ(z)
−
−
(13.31)
Σ
X
(z)
0
d
×
d
,
Σ(z)
:=
∈ R
(13.32)
Σ
Y
(z)
D(z)
where the matrices
Σ
X
,Σ
Y
,D
n
×
n
∈ R
are
:=
f
ij
(y)
1
≤
i,j
≤
n
,Σ
Y
(z)
:=
ρ
ji
b
i
(y
i
)
1
≤
i,j
≤
n
,
Σ
X
(z)
diag
ρ
1
b
1
(y
1
), . . . , ρ
n
b
n
(y
n
)
.
D(z)
:=
G
Y
The smooth functions
f
ij
:
→ R
+
are assumed to be bounded from below and
above. The state space domain of the pair process
Z
n
G
Y
.
=
(X, Y )
is
G
= R
×
The infinitesimal generator
A
of the semigroup generated by the process
Z
is
given by
2
tr
Q
(z)D
2
−
μ(z)
∇
,
1
A
:= −
(13.33)
ΣΣ
.
with
Q
=
Example 13.4.4
(Volatility processes) In [68, Chap. 10.6], it is assumed that each
volatility component
Y
k
follows a mean-reverting Ornstein-Uhlenbeck process, i.e.
=
α
k
(m
k
−
=
≤
≤
c
k
(y
k
)
y
k
)
,
b
k
(y
k
)
β
k
,1
k
n
. Here,
α
k
>
0 is called the rate of
0 is the long-run mean level of
Y
k
. Under an EMM, the
drift term
c
k
becomes
c
k
(y)
=
α
k
(m
k
−
y
k
)
−
β
k
Λ
k
(y)
, for some volatility risk
premium
Λ(y)
mean reversion and
m
k
≥
(Λ
1
(y),...,Λ
n
(y))
. See [68, Chap. 2.5] for a representation of
Λ
in the one dimensional case
n
=
=
1.
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