The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 242 Wednesday, February 4, 2004 12:51 PM

242 The Mathematics of Financial Modeling and Investment Management

imposed. Restrictions that uniquely identify a solution to a differential

equation can be of various types. For instance, one could impose that a

solution of an n-th order differential equation passes through n given

points. A common type of restriction—called an initial condition—is

obtained by imposing that the solution and some of its derivatives

assume given initial values at some initial point.

Given an ODE of order n, to ensure the uniqueness of solutions it

will generally be necessary to specify a starting point and the initial

value of n–1 derivatives. It can be demonstrated, given the differential

equation

Fx Yx[ , (),Y ()^1 ()x, ...,Y()n()x] = 0

that if the function F is continuous and all of its partial derivatives up to

order n are continuous in some region containing the values y 0 ,...,

y

(

0

n – 1 )

, then there is a unique solution y(x) of the equation in some

interval I = (M ≤x ≤ L) such that y 0 = Y(x 0 ),...,y Y(n–1)(x 0 ).^2

(

0

n – 1 )

=

Note that this theorem states that there is an interval in which the solu-

tion exists. Existence and uniqueness of solutions in a given interval is a

more delicate matter and must be examined for different classes of

equations.

The general solution of a differential equation of order n is a func-

tion of the form

y = φ(xC, 1 , ...,Cn )

that satisfies the following two conditions:

■ Condition 1. The function y = φ(x,C 1 ,...,Cn) satisfies the differential

equation for any n-tuple of values (C 1 ,...,Cn).

■ Condition 2. Given a set of initial conditions y(x 0 ) = y 0 ,...,y(n–1)(x 0 ) =

y

(

0

n–1)

that belong to the region where solutions of the equation exist,

it is possible to determine n constants in such a way that the function y

= φ(x,C 1 ,...,Cn) satisfies these conditions.

The coupling of differential equations with initial conditions embod-

ies the notion of universal determinism of classical physics. Given initial

(^2) The condition of existence and continuity of derivatives is stronger than necessary.
The Lipschitz condition, which requires that the incremental ratio be uniformly
bounded in a given interval, would suffice.