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σ
L
∞
(G)
+
2
:=
L
∞
(G)
Denote by
σ
the constant
σ
σ
1
/
2
σ
L
∞
(G)
. Then, for an
arbitrary
ε>
0,
1
2
σ
0
ϕ
a
LV
(ϕ, ϕ)
≥
2
2
L
2
(G)
−
σ
ϕ
L
2
(G)
ϕ
L
2
(G)
L
2
(G)
+
r
ϕ
≥
σ
0
/
2
εσ
L
2
(G)
+
r
σ/(
4
ε)
ϕ
2
2
−
−
ϕ
L
2
(G)
.
σ
0
/(
4
σ)
,wehave
Choosing
ε
=
a
LV
(ϕ, ϕ)
≥
σ
0
/
4
ϕ
2
L
2
(G)
+
(r
−
(σ/σ
0
)
2
)
ϕ
2
L
2
(G)
,
which gives, by the Poincaré inequality (3.4),
a
LV
(ϕ, ϕ)
σ
0
/
8
ϕ
2
σ
0
/(
8
C)
2
(σ/σ
0
)
2
2
L
2
(G)
≥
L
2
(G)
+
ϕ
L
2
(G)
−|
r
−
|
ϕ
σ
0
/
8min
1
,C
−
1
2
2
≥
{
}
ϕ
H
1
(G)
−
C
3
ϕ
L
2
(G)
.
By Proposition
4.5.5
, the bilinear form
a
LV
(
·
,
·
)
is continuous and satisfies a
L
2
(G)
, the weak formulation (
4.37
) admits a
Gårding inequality. Hence, for
g
∈
L
2
(J
H
0
(G))
H
1
(J
L
2
(G))
.
∈
;
∩
;
unique solution
u
4.6 Further Reading
To derive the partial differential equations, we followed the line of Lamberton and
Lapeyre [109]. Using finite differences to price options was first described in Bren-
nan and Schwartz [26]. A rigorous treatment can be found in Achdou and Piron-
neau [1]. Finite elements were first applied to finance in Wilmott et al. [161]. Error
estimates for non-smooth initial data are given in Thomée [154]. The CEV model
was introduced by Cox and Ross [45, 46], and analytic formulas can be found, for
example, in Hsu et al. [88]. See also [56]. The probabilistic argument to estimate
the localization error is due to Cont and Voltchkova in [41], even in a more gen-
eral setting of Lévy processes. The local volatility model is used to recover the
volatility smile observed in the stock market as shown in Derman and Kani [57]or
Dupire [59].
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