Information Technology Reference
In-Depth Information
S
given
in (
15.14
). The functional setting for this pricing equation does not fit into the ab-
stract parabolic framework described in Sect. 3.2 due to absence of diffusion with
respect to the volatility coordinate
y
(compare with the operator
A
=
A
Consider now the pricing equation (
15.12
) in the BNS model with
S
). The order of
A
S
is
α<
1, since the driving process (of volatility)
is a subordinator. Thus, the first order term
∂
y
is dominant. It is therefore desirable
to remove this term, see also Sect. 10.3.
Consider in (
15.12
) the change of variables
w(t,x,y)
the jump operator appearing in
A
t,e
λt
y)
,
:=
+
κ
v(t,x
λ
and assume for simplicity
r
=
0. Let
G
R
:=
(
−
R
1
,R
1
)
×
(
0
,R
2
)
and let
Γ
0
:=
(
−
R
1
,R
1
)
×{
0
}
and
Γ
1
:=
∂G
R
\
Γ
0
. Instead of the pricing equation (
15.12
)for
v
on the unbounded domain
G
= R × R
+
, we consider the pricing equation for
w
on
the bounded domain
G
R
,i.e.
∂
t
w
+
A
J
(t)
w
+
A
(t)
=
0
in
J
×
G
R
,
w
=
0
on
J
×
Γ
1
,
(15.18)
g(e
x
)
w(
0
)
=
in
G
R
,
where the change of variables induces the operator
A
(t)
+
A
J
(t)
given by
1
2
e
λt
y(∂
xx
−
∂
x
),
A
(t)
:= −
λ
ϕ(x
ϕ(x,y,t)
k
w
(z)
d
z.
(
A
J
(t)ϕ)(x, y, t)
e
−
λt
z, t)
:= −
+
ρz,y
+
−
R
+
We will now derive a variational formulation for problem (
15.18
). To this end, de-
note by
(ϕ, φ)
the
L
2
(G
R
)
-inner product. For
ϕ,φ
C
0
∈
(G
R
)
, we associate with
A
+
A
J
(t)
the bilinear form
a(t
;
ϕ,φ)
:=
A
(t)ϕ, φ
+
A
(t)
J
(t)ϕ, φ
=
a
C
(t
;
ϕ,φ)
+
a
J
(t
;
ϕ,φ).
Define the weighted Sobolev space
(G
R
)
·
W
,
C
0
W
:=
where the norm
·
W
is given by
W
:=
√
y∂
x
v
2
2
2
v
L
2
(G
R
)
+
v
L
2
(G
R
)
.
Lemma 15.3.2
The bilinear form a
C
(t
is continuous and sat-
isfies a Gårding inequality
,
i
.
e
.
there exist constants C
i
>
0,
i
;·
,
·
)
:
W
×
W
→ R
=
1
,
2
,
3,
such that
∀
ϕ,φ
∈
W and
∀
t
∈
J there holds
a
C
(t
a
C
(t
2
2
|
;
ϕ,φ)
|≤
C
1
ϕ
W
φ
W
,
;
ϕ,ϕ)
≥
C
2
ϕ
W
−
C
3
ϕ
L
2
(G
R
)
.
Proof
By integration by parts, we have
1
2
e
λt
(y∂
x
ϕ,∂
x
φ)
1
2
e
λt
(y∂
x
ϕ,φ),
a
C
(t
;
ϕ,φ)
=
+
Search WWH ::
Custom Search