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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 0 (x))
=···=
sign d (x)) ,
(ii) Parabolic
⇔∃!
j
I :
λ j (x)
=
0
rank (A(x), b(x))
=
d
+
1,
(iii) Hyperbolic
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),
2
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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