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1
2
Q
11
∂
x
1
x
1
+
Q
12
∂
x
1
x
2
+
1
2
Q
22
∂
x
2
x
2
−
μ
1
∂
x
1
−
μ
2
∂
x
2
.
BS
A
=
BS
by finite difference quotients, we define, for
N
1
,N
2
∈ N
To discretize
A
,agrid
G
R,R)
2
on
G
=
(
−
by
G
:= {
(x
1
,i
1
,x
2
,i
2
)
|
1
≤
i
1
≤
N
1
,
1
≤
i
2
≤
N
2
}
,
where the grid points
(x
1
,i
1
,x
2
,i
2
)
are given by
(x
1
,i
1
,x
2
,i
2
)
=
(
−
R
+
i
1
h
1
,
−
R
+
i
2
h
2
),
h
k
:=
2
R/(N
k
+
1
),
1
≤
i
k
≤
N
k
,k
=
1
,
2
.
C
4
(G)
,weset
f
i
1
,i
2
:=
For
f
∈
f(x
1
,i
1
,x
2
,i
2
)
for the function value of
f
at the grid
point
(x
1
,i
1
,x
2
,i
2
)
∈
G
and consider the difference quotients
h
−
2
1
(h
1
)
∂
x
1
x
1
f(x
1
,i
1
,x
2
,i
2
)
=
(f
i
1
−
1
,i
2
−
2
f
i
1
,i
2
+
f
i
1
+
1
,i
2
)
+
O
(δ
x
1
x
1
f)
i
1
,i
2
+
O
(h
1
),
=:
∂
x
2
x
2
f(x
1
,i
1
,x
2
,i
2
)
=
h
−
2
2
f
i
1
,i
2
+
f
i
1
,i
2
+
1
)
+
O
(h
2
)
(f
i
1
,i
2
−
1
−
2
(δ
x
2
x
2
f)
i
1
,i
2
+
O
(h
2
).
=:
Furthermore,
(
2
h
2
)
−
1
(f
i
1
,i
2
+
1
−
(h
2
)
∂
x
1
x
2
f(x
1
,i
1
,x
2
,i
2
)
=
∂
x
1
[
f
i
1
,i
2
−
1
)
+
O
]
(
4
h
1
h
2
)
−
1
f
i
1
+
1
,i
2
+
1
−
f
i
1
−
1
,i
2
−
1
=
f
i
1
−
1
,i
2
+
1
−
f
i
1
+
1
,i
2
−
1
+
+
O
(h
1
)
+
O
(h
2
)
(δ
x
1
x
2
f)
i
1
,i
2
+
O
(h
1
)
(h
2
),
=:
+
O
as well as
(
2
h
1
)
−
1
(f
i
1
+
1
,i
2
−
(h
1
)
(h
1
),
∂
x
1
f(x
1
,i
1
,x
2
,i
2
)
=
f
i
1
−
1
,i
2
)
+
O
=:
(δ
x
1
f)
i
1
,i
2
+
O
(
2
h
2
)
−
1
(f
i
1
,i
2
+
1
−
(h
2
)
(h
2
).
∂
x
2
f(x
1
,i
1
,x
2
,i
2
)
=
f
i
1
,i
2
−
1
)
+
O
=:
(δ
x
2
f)
i
1
,i
2
+
O
Denote by
v
i
1
,i
2
≈
v(t
m
,x
1
,i
1
,x
2
,i
2
)
an approximation of the option value
v
on time
level
t
m
at the grid point
(x
1
,i
1
,x
2
,i
2
)
BS
v
∈
G
. We now replace the PDE,
∂
t
v
−
A
+
rv
=
0, by the finite difference equations
m
E
i
1
,i
2
=
0
,
1
≤
i
k
≤
N
k
,k
=
1
,
2
,m
=
0
,...,M
−
1
,
(8.15)
g(e
x
i
1
,e
x
i
2
)
and homogeneous boundary condi-
with the initial condition
v
i
1
,i
2
=
tions
v
0
,i
2
=
v
i
1
,
0
=
m
i
1
,i
2
0,
i
k
=
0
,...,N
k
+
=
E
1,
m
0
,...,M
.In(
8.15
),
is the
finite difference operator given by
k
−
1
v
m
+
1
i
1
,i
2
v
i
1
,i
2
−
θ(
v)
i
1
,i
2
+
m
v)
m
+
1
i
1
,i
2
rθv
m
+
1
i
1
,i
2
E
i
1
,i
2
:=
−
F
+
(
1
−
θ)(
F
θ)v
i
1
,i
2
,
+
r(
1
−
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