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Under these assumptions, there exists a unique solution to (
15.1
), and if we addi-
tionally assume that
2
2
E[|
Z
0
|
]
<
∞
, then
E[|
Z
t
|
]
<
∞
,
∀
t
≥
0, and there exists a
constant
C(t) >
0 such that
E[|
C(t)
1
]
;
2
2
Z
t
|
]≤
+ E[|
Z
0
|
(15.3)
see, e.g. [3].
15.1.1 Bates Models
The models of Bates [13, 14] are combinations of the jump-diffusion model of
Merton (10.6)-(10.7) and the stochastic volatility model of Heston (9.4)-(9.5). Let
n
v
∈{
. Under a risk-neutral probability measure, the log-price dynamics
X
t
=
ln
(S
t
)
of the risky underlying are
}
1
,
2
r
Y
t
d
t
Y
t
d
W
t
n
v
n
v
1
2
d
X
t
=
− κ
λ
t
−
+
+
d
J
t
,
i
=
1
i
=
1
where
J
is a compound Poisson process with state dependent intensity
λ
t
=
λ
0
+
n
v
i
1
λ
i
Y
t
,
λ
i
≥
0,
i
=
0
,...,n
v
, and jump size distribution
ν
0
as in (10.7), with
=
κ :=
R
e
μ
+
δ
2
/
2
(e
ζ
1. Each of the processes
Y
t
−
1
)ν
0
(
d
ζ)
=
−
follows a CIR
process, i.e.
β
i
Y
t
d
Y
t
Y
t
)
d
t
d
W
t
,i
=
α
i
(m
i
−
+
=
1
,...,n
v
.
are independent, and the Brownian motions
W
i
The Brownian motions
W
1
,
W
2
1
ρ
i
W
i
+
n
v
satisfy
W
t
ρ
i
W
t
, with independent Brownian motions
W
i
+
n
v
,
=
+
−
t
i
=
1
,...,n
v
. Thus, the coefficients
b
,
Σ
and
ς
l
in (
15.1
) are, for the case
n
v
=
2,
given by:
c
=
0,
⎛
1
2
+
λ
i
κ
y
i
α
1
(m
1
−
y
1
)
α
2
(m
2
−
⎞
r
−
λ
0
κ −
n
v
i
=
⎝
⎠
,
b(z)
=
(15.4)
y
2
)
⎛
⎞
√
y
1
√
y
2
0
0
β
1
1
⎝
β
1
ρ
1
√
y
1
ρ
1
√
y
2
⎠
0
−
0
Σ(z)
=
,
(15.5)
β
2
1
β
2
ρ
2
√
y
2
ρ
2
√
y
2
0
0
−
=
ζ,
0
,
0
.
ς
l
(z, ζ )
(15.6)
The Lévy density
k(ζ)
of
ν
0
(
d
ζ)
depends on
y
1
,y
2
via
λ
0
+
λ
i
y
i
1
n
v
√
2
πδ
2
e
−
(ζ
−
μ)
2
/(
2
δ
2
)
.
=
k(ζ)
(15.7)
k
=
1
Note that for
n
v
=
0 we obtain the first model of Bates [13], which
further reduces to the Heston model for
λ
0
=
1 with
λ
1
=
0.
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