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1
2
y∂
xx
f(x,y)
+
βρy∂
xy
f(x,y)
+
1
2
β
2
y∂
yy
f(x,y)
H
f )(z)
:=
(
A
r
2
y
∂
x
f(x,y)
1
+
−
+
α(m
−
y)∂
y
f(x,y).
(9.12)
Furthermore, we have
G
Y
= R
≥
0
. To cast the pricing equation (
9.11
) corresponding
to the Heston model in a variational formulation and to establish its well-posedness,
we change variables
y
2
).
v(t,x,
y)
:=
v(t,x,
1
/
4
(9.13)
The pricing equation for
v
becomes,
−
A
H
∂
t
v
v
+
r
v
=
0 n
J
× R × R
≥
0
,
(9.14)
g(e
x
)
v
0
=
R × R
≥
0
,
in
with
1
8
1
2
ρβ
1
2
β
2
∂
yy
f(x,
(
A
H
f )(x,
y
2
∂
xx
f(x,
:=
+
+
y)
y)
y∂
xy
f(x,
y)
y)
r
−
8
y
2
∂
x
f(x,y)
1
+
∂
y
f(x,y).
4
αm
−
β
2
1
2
+
−
αy
+
(9.15)
y
In the multi-scale SV model, assume that the market price of volatility risk
γ
i
=
0,
i
1
,...,n
v
,are
such that the coefficients
b
,
Σ
in (
9.8
)-(
9.9
) satisfy (8.2)-(8.3). Then, the generator
A
=:
A
=
1
,...,n
v
. Furthermore, assume that the functions
ξ
,
c
k
,
g
k
,
k
=
MS
of the multi-scale process
Z
is given by
n
v
1
2
ξ
2
(y)∂
xx
f(z)
1
2
MS
f )(z)
g
i
(y
i
)∂
y
i
y
i
f(z)
(
A
=
+
i
=
1
n
v
n
v
+
ξ(y)
ρ
i
g
i
(y
i
)∂
xy
i
f(z)
+
q
i,j
(y)∂
y
i
y
j
f(z)
i
=
i
=
1
1
j>i
r
2
ξ
2
(y)
∂
x
f(z)
c
i
(y
i
)
∂
y
i
f(z),
n
v
−
1
g
i
(y
i
)ρ
i
μ
r
ξ(y)
+
−
+
−
i
=
1
where
the
coefficients
q
i,j
(y)
are
given
by
q
i,j
(y)
:=
g
i
(y
i
)g
j
(y
j
)(ρ
i
ρ
j
+
i
k
=
1
L
j,k
L
i,k
)
. Hence, for the mean reverting OU model considered in Exam-
ple
9.1.1
, we find
1
2
ξ
2
(y)∂
xx
f(x,y)
1
2
β
2
∂
yy
f(x,y)
MS
f )(z)
(
A
=
+
+
ξ(y)ρβ∂
xy
f(x,y)
r
2
ξ
2
(y)
∂
x
f(x,y)
1
+
−
α(m
∂
y
f(x,y),
βρ
μ
r
ξ(y)
−
+
−
y)
−
(9.16)
where
z
=
(x, y
1
)
=
(x, y)
.
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