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a
BS
(u
R
,v)
H
0
(G),
a.e. in
J,
+
=
∀
∈
(∂
t
u
R
,v)
0
,
v
(4.13)
=
u
0
|
G
.
u
R
(
0
)
By Proposition
4.2.1
and Theorem 3.2.2, the problem (
4.13
) is well-posed, i.e. there
exists a unique solution
u
R
∈
L
2
(
0
,T
;
H
0
(G))
∩
C
0
(
[
0
,T
];
L
2
(G))
which can be
approximated by a finite element Galerkin scheme.
4.4 Discretization
We use the finite element and the finite difference method to discretize the Black-
Scholes equation. As for the heat equation in Chap. 2, we use the variational formu-
lation of the differential equations for FEM and determine approximate solutions
that are piecewise linear. For FDM we replace the derivatives in the differential
equation by difference quotients.
4.4.1 Finite Difference Discretization
We discretize the PDE (
4.12
) directly using finite differences on a bounded domain
with homogeneous Dirichlet boundary conditions. Proceeding as in Sect. 2.3.1,we
obtain the matrix problem:
Find
u
m
+
1
N
∈ R
such that for
m
=
0
,...,M
−
1
,
I
+
θk
G
BS
u
m
+
1
=
I
−
θ)k
G
BS
u
m
,
−
(
1
(4.14)
u
0
=
u
0
,
where
G
BS
σ
2
/
2
R
(σ
2
/
2
=
+
−
r)
C
+
r
I
, is given explicitly with
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
2
−
1
01
.
.
.
.
.
.
.
.
.
.
.
.
1
h
2
1
2
h
−
1
−
1
R
=
,
C
=
.
.
.
.
.
.
.
.
.
.
.
.
.
−
1
1
−
12
−
10
4.4.2 Finite Element Discretization
S
1
We discretize (
4.13
)usingthe
θ
-scheme and the finite element space
V
N
=
T
∩
H
0
(G)
with
S
1
g(e
x
)
and proceeding exactly
:=
given as in (3.17). Setting
u
0
(x)
T
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