The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 260 Wednesday, February 4, 2004 12:51 PM

260 The Mathematics of Financial Modeling and Investment Management

tions of PDEs is a delicate mathematical problem. We can only give a

brief account by way of an example.

Let’s consider the diffusion equation. This equation describes the

propagation of the probability density of stock prices under the ran-

dom-walk hypothesis:

∂f 2 ∂

(^2) f
-----= a ---------
∂t ∂x^2
The Black-Scholes equation, which describes the evolution of option
prices (see Chapter 15), can be reduced to the diffusion equation.
The diffusion equation describes propagating phenomena. Call
f(t,x) the probability density that prices have value x at time t. In
finance theory, the diffusion equation describes the time-evolution of the
probability density function f(t,x) of stock prices that follow a random
walk.^7 It is therefore natural to impose initial and boundary conditions
on the distribution of prices.
In general, we distinguish two different problems related to the diffu-
sion equation: the first boundary value problem and the Cauchy initial
value problem, named after the French mathematician Augustin Cauchy
who first formulated it. The two problems refer to the same diffusion
equation but consider different domains and different initial and bound-
ary conditions. It can be demonstrated that both problems admit a
unique solution.
The first boundary value problem seeks to find in the rectangle 0 ≤x
≤l, 0 ≤t ≤T a continuous function f(t,x) that satisfies the diffusion equa-
tion in the interior Q of the rectangle plus the following initial condition,
f( 0 ,x)= φ()x, 0 ≤≤ xl
and boundary conditions,
ft( , 0 )= f 1 ()t, ft l( , )= f 2 ()t, 0 ≤≤ tT
The functions f 1 , f 2 are assumed to be continuous and f 1 (0) = φ(0), f 2 (0)
= φφφφ(l).
The Cauchy problem is related to an infinite half plane instead of a
finite rectangle. It is formulated as follows. The objective is to find for
(^7) In physics, the diffusion equation describes phenomena such as the diffusion of par-
ticles suspended in some fluid. In this case, the diffusion equation describes the den-
sity of particles at a given moment at a given point.