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10.6.1 Finite Difference Discretization
As in the Black-Scholes case, we replace the domain
J
×
G
by discrete grid points
(t
m
,x
i
)
and approximate the partial derivatives in (
10.24
) by difference quotients at
the grid points. Let the grid points in the log-price coordinate
x
be given by
x
i
=−
R
+
ih,
i
=
0
,
1
,...,N
+
1
,h
:=
2
R/(N
+
1
)
=
x,
(10.26)
which are equidistant with mesh width
h
, and the time levels by
t
m
=
mt, m
=
0
,
1
,...,M, t
:=
T/M.
We define the weights for
j
=
0
,
1
,
2
,...
,
(j
+
1
)h
ν
j
k
(
−
2
)
(z)
d
z
k
(
−
3
)
((j
k
(
−
3
)
(jh),
=
=
+
−
1
)h)
jh
−
jh
ν
j
k
(
−
2
)
(z)
d
z
k
(
−
3
)
(
k
(
−
3
)
(
=
=
−
jh)
−
−
(j
+
1
)h),
−
(j
+
1
)h
which satisfy
j
=
0
(ν
j
=
ν
j
)
k
(
−
2
)
(z)
d
z<
+
∞
. Then, we can discretize the
R
C
0
(G)
at the mesh point
x
i
,
i
generator (
10.24
)for
f
∈
=
1
,...,N
,by
N
−
i
R
−
x
i
z)k
(
−
2
)
(z)
d
z
jh)ν
j
∂
xx
f(x
i
+
=
∂
xx
f(x
i
+
+
O
(h)
0
j
=
0
N
−
i
(δ
xx
f)
i
+
j
ν
j
(h
2
),
=
+
O
+
O
(h)
j
=
0
i
−
1
0
∂
xx
f(x
i
+
z)k
(
−
2
)
(z)
d
z
=
(δ
xx
f)
i
−
j
ν
j
+
O
(h).
−
R
+
x
i
j
=
0
Using the
θ
-scheme for the time discretization, we obtain
Find
u
m
+
1
N
∈ R
such that for
m
=
0
,...,M
−
1
,
I
θt
G
J
u
m
+
1
=
I
θ)t
G
J
u
m
,
+
−
(
1
−
(10.27)
u
0
=
u
0
,
with
G
J
N
×
N
∈ R
defined for
i
=
1
,...,N
,
h
2
2
(ν
0
ν
1
,
1
G
i,i
=
ν
0
)
ν
1
+
−
−
h
2
2
ν
1
ν
0
,
1
G
i,i
+
1
=
ν
2
ν
0
−
−
−
h
2
2
ν
1
ν
0
,
1
G
i,i
−
1
=
ν
2
ν
0
−
−
−
h
2
2
ν
j
ν
j
−
1
,j
1
G
i,i
+
j
=
ν
j
+
1
−
−
=
2
,...,N
−
i,
h
2
2
ν
j
ν
j
−
1
,j
1
G
i,i
−
j
=
ν
j
+
1
−
−
=
2
,...,i
−
1
.
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