The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 240 Wednesday, February 4, 2004 12:51 PM

240 The Mathematics of Financial Modeling and Investment Management

DIFFERENTIAL EQUATIONS DEFINED

A differential equation is a condition expressed as a functional link

between one or more functions and their derivatives. It is expressed as

an equation (that is, as an equality between two terms).

A solution of a differential equation is a function that satisfies the

given condition. For example, the condition

Y′′() αx + Y′() βx + Yx()– bx()= 0

equates to zero a linear relationship between an unknown function Y(x),

its first and second derivatives Y′(x),Y′′(x), and a known function b(x).^1

The unknown function Y(x) is the solution of the equation that is to be

determined.

There are two broad types of differential equations: ordinary differ-

ential equations and partial differential equations. Ordinary differential

equations are equations or systems of equations involving only one

independent variable. Another way of saying this is that ordinary differ-

ential equations involve only total derivatives. In contrast, partial differ-

ential equations are differential equations or systems of equations

involving partial derivatives. That is, there is more than one indepen-

dent variable.

As we move from deterministic equations to stochastic equations,

we introduce stochastic differential equations. In these differential equa-

tions, a random or stochastic term is included.

ORDINARY DIFFERENTIAL EQUATIONS

In full generality, an ordinary differential equation (ODE) can be expressed

as the following relationship:

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

where Y(m)(x) denotes the m-th derivative of an unknown function Y(x). If

the equation can be solved for the n-th derivative, it can be put in the form:

Y()n()x = GxYx[ , (),Y()^1 ()x, ..., Y(n–^1 )()x]

(^1) In some equations we will denote the first and second derivatives by a single and
double prime, respectively.