The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 249 Wednesday, February 4, 2004 12:51 PM

Differential Equations and Difference Equations 249

Solving this equation for Y(s), that is, Y(s) = g[s,y(t)(0),...,y(n–1)(0)] the

inverse Laplace transform y(t) = L–1[Y(s)] uniquely determines the solu-

tion of the equation.

Because inverse Laplace transforms are integrals, with this method,

when applicable, the solution of a differential equation is reduced to the

determination of integrals. Laplace transforms and inverse Laplace

transforms are known for large classes of functions. Because of the

important role that Laplace transforms play in solving ordinary differ-

ential equations in engineering problems, there are published reference

tables.^4 Laplace transform methods also yield closed-form solutions of

many ordinary differential equations of interest in economics and

finance.

NUMERICAL SOLUTIONS OF ORDINARY

DIFFERENTIAL EQUATIONS

Closed-form solutions are solutions that can be expressed in terms of

known functions such as polynomials or exponential functions. Before

the advent of fast digital computers, the search for closed-form solu-

tions of differential equations was an important task. Today, thanks to

the availability of high-performance computing, most problems are

solved numerically. This section looks at methods for solving ordinary

differential equations numerically.

The Finite Difference Method

Among the methods used to numerically solve ordinary differential

equations subject to initial conditions, the most common is the finite

difference method. The finite difference method is based on replacing

derivatives with difference equations; differential equations are thereby

transformed into recursive difference equations.

Key to this method of numerical solution is the fact that ODEs sub-

ject to initial conditions describe phenomena that evolve from some

starting point. In this case, the differential equation can be approxi-

mated with a system of difference equations that compute the next point

based on previous points. This would not be possible should we impose

boundary conditions instead of initial conditions. In this latter case, we

have to solve a system of linear equations.

(^4) See, for example, “Laplace Transforms,” Chapter 29 in Milton Abramowitz and
Irene A. Stegun (eds.), Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables (New York: Dover, 1972).