The Mathematics of Financial Modelingand Investment Management

9-DifferntEquations Page 243 Wednesday, February 4, 2004 12:51 PM

Differential Equations and Difference Equations 243

conditions, the future evolution of a system that obeys those equations is

completely determined. This notion was forcefully expressed by Pierre-

Simon Laplace in the eighteenth century: a supernatural mind who

knows the laws of physics and the initial conditions of each atom could

perfectly predict the future evolution of the universe with unlimited pre-

cision.

In the twentieth century, the notion of universal determinism was

challenged twice in the physical sciences. First in the 1920s the develop-

ment of quantum mechanics introduced the so called indeterminacy

principle which established explicit bounds to the precision of measure-

ments.^3 Later, in the 1970s, the development of nonlinear dynamics and

chaos theory showed how arbitrarily small initial differences might become

arbitrarily large: the flapping of a butterfly’s wings in the southern hemi-

sphere might cause a tornado in northern hemisphere.

SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Differential equations can be combined to form systems of differential

equations. These are sets of differential conditions that must be satisfied

simultaneously. A first-order system of differential equations is a system

of the following type:

dy 1

---------= f 1 (xy,,, 1 ...yn)

dx

dy

^2

---------= f 2 (xy 1 ...

dx

,,, yn)



.



.

.

dyn

---------= f(xy 1 ...

dx

n ,,, yn)



(^3) Actually quantum mechanics is a much deeper conceptual revolution: it challenges
the very notion of physical reality. According to the standard interpretation of quan-
tum mechanics, physical laws are mathematical recipes that link measurements in a
strictly probabilistic sense. According to quantum mechanics, physical states are
pure abstractions: they can be superposed, as the celebrated “Schrodinger’s cat”
which can be both dead and alive.