The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 265 Wednesday, February 4, 2004 12:51 PM

Differential Equations and Difference Equations 265

EXHIBIT 9.6 Solution of the First Boundary Problem by the Finite Difference Method

Applying the same discretization to a first boundary problem would

require the solution of a system of linear equations at every step.

Exhibit 9.6 illustrates this case.

SUMMARY

■ Derivatives can be combined to form differential equations.

■ Differential equations are conditions that must be satisfied by their

solutions.

■ Differential equations generally admit infinite solutions.

■ Initial or boundary conditions are needed to identify solutions uniquely.

■ Differential equations are the key mathematical tools for the develop-

ment of modern science; in finance they are used in arbitrage pricing, to

define stochastic processes, and to compute the time evolution of aver-

ages.

■ Ordinary differential equations include only total derivatives; partial

differential equations include partial derivatives.

■ Differential equations can be solved in closed form or with numerical

methods.