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In-Depth Information
d
i,j
=
={
}
Let
b(x)
(b
0
(x),...,b
d
(x))
and assume that the matrix
A(x)
a
ij
(x)
=
0
≤
≤ ··· ≤
is symmetric with real eigenvalues
λ
0
(x)
λ
d
(x)
. We can use the
eigenvalues to distinguish three types of PDEs: elliptic, parabolic and hyperbolic.
λ
1
(x)
d
+
1
, a PDE is called
Definition 2.2.2
Let
I
={
0
,...,d
}
.At
x
∈ R
(i) Elliptic
⇔
λ
i
(x)
=
0,
∀
i
∧
sign
(λ
0
(x))
=···=
sign
(λ
d
(x))
,
(ii) Parabolic
⇔∃!
j
∈
I
:
λ
j
(x)
=
0
∧
rank
(A(x), b(x))
=
d
+
1,
(iii) Hyperbolic
⇔
(λ
i
(x)
=
0
,
∀
i)
∧∃!
j
∈
I
:
sign
λ
j
(x)
=
sign
λ
k
(x)
,
k
∈
I
\{
j
}
.
The PDE is called elliptic, parabolic, hyperbolic on
G
, if it is elliptic, parabolic,
hyperbolic at all
x
∈
G
.
We give a typical example for each type:
(i) The Poisson equation
u
=
f(x)
is elliptic.
(ii) The heat equation
∂
t
u
−
u
=
f(t,x)
is parabolic (set
x
0
=
t
).
(iii) The wave equation
∂
tt
u
t
).
(iv) The Black-Scholes equation for the value of a European option price
v(t,s)
−
u
=
f(t,x)
is hyperbolic (set
x
0
=
1
2
σ
2
s
2
∂
ss
v
∂
t
v
−
−
rs∂
s
v
+
rv
=
0
,
(2.3)
with volatility
σ>
0 and interest rate
r
≥
0 is parabolic at
(t, s)
∈
(
0
,T)
×
(
0
,
∞
)
and degenerates to an ordinary differential equation as
s
→
0.
Partial differential equations arising in finance, like the Black-Scholes equa-
tion (
2.3
), are mostly of parabolic type, i.e. they are of the form
d
d
∂
t
u
−
a
ij
(x)∂
x
i
x
j
u
+
b
i
(x)∂
x
i
u
+
c(x)u
=
f(x).
i,j
=
1
i
=
1
Therefore, we introduce the basic concepts for solving parabolic equations in the
next section. For illustration purpose, we consider the heat equation. Indeed, setting
s
e
x
,
t
2
σ
−
2
τ
and
=
=
e
αx
+
βτ
u(τ, x),
rσ
−
2
,β
rσ
−
2
)
2
,
v(t,s)
=
α
=
1
/
2
−
=−
(
1
/
2
+
the Black-Scholes equation (
2.3
)for
v(t,s)
can be transformed to the heat equation
∂
τ
u
−
∂
xx
u
=
0for
u(τ, x)
.
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