The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 256 Wednesday, February 4, 2004 12:51 PM

256 The Mathematics of Financial Modeling and Investment Management

This is a system of 1,000 equations in 1,000 unknowns. Solving the

system we compute the entire solution. In this system two equations, the

first and the last, are linked to boundary values; all other equations are

transfer equations that express the dynamics (or the law) of the system.

This is a general feature of boundary value problems. We will encounter it

again when discussing numerical solutions of partial differential equations.

In the above example, we chose a forward scheme where the derivative

is approximated with the forward difference quotient. One might use a dif-

ferent approximation scheme, computing the derivative in intervals cen-

tered around the point x. When derivatives of higher orders are involved,

the choice of the approximation scheme becomes critical. Recall that when

we approximated first and second derivatives using forward differences, we

were required to evaluate the function at two points (i,i+ 1) and three

points (i,i+ 1,i+ 2) ahead respectively. If purely forward schemes are

employed, computing higher-order derivatives requires many steps ahead.

This fact might affect the precision and stability of numerical computations.

We saw in the examples that the accuracy of a finite difference

scheme depends on the discretization interval. In general, a finite differ-

ence scheme works, that is, it is consistent and stable, if the numerical

solution converges uniformly to the exact solution when the length of

the discretization interval tends to zero. Suppose that the precision of an

approximation scheme depends on the length of the discretization inter-

val ∆x. Consider the difference δf= ˆ f()x – fx()between the approxi-

mate and the exact solutions. We say that δf→ 0 uniformly in the

interval [a,b] when ∆x→0 if, given any εarbitrarily small, it is possible

to find a ∆xsuch that δf< ε, ∀x∈ [ab, ].

NONLINEAR DYNAMICS AND CHAOS

Systems of differential equations describe dynamical systems that evolve

starting from initial conditions. A fundamental concept in the theory of

dynamical system is that of the stability of solutions. This topic has

become of paramount importance with the development of nonlinear

dynamics and with the discovery of chaotic phenomena. We can only

give a brief introductory account of this subject whose role in econom-

ics is still the subject of debate.

Intuitively, a dynamical system is considered stable if its solutions

do not change much when the system is only slightly perturbed. There

are different ways to perturb a system: changing parameters in its equa-

tions, changing the known functions of the system by a small amount,

or changing the initial conditions.