9-DifferntEquations Page 239 Wednesday, February 4, 2004 12:51 PM

I

CHAPTER

9

Differential Equations and

Difference Equations

n Chapter 4, we explained how to obtain the derivative of a function.

In this chapter we will introduce differential equations. In nontechnical

terms, differential equations are equations that express a relationship

between a function and one or more derivatives (or differentials) of that

function.

It would be difficult to overemphasize the importance of differential

equations in modern science: they are used to express the vast majority

of the laws of physics and engineering principles. In economics and

finance, differential equations are used to express various laws and con-

ditions including the following:

■ The laws of deterministic quantities such as the accumulation of risk-

free bank deposits.

■ The laws that govern the evolution of price probability distributions.

■ The solution of economic variational problems, such as intertemporal

optimization.

■ Conditions of continuous hedging, such as the Black-Scholes equation

that we will describe in Chapter 15.

A large number of properties of differential equations have been

established over the last three centuries. This chapter provides only a

brief introduction to the concept of differential equations and their

properties, limiting our discussion to the principal concepts.

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