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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 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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