The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 134 Friday, March 12, 2004 12:39 PM

134 The Mathematics of Financial Modeling and Investment Management

INTEGRAL TRANSFORMS

Integral transforms are operations that take any function f(x) into

another function F(s) of a different variable sthrough an improper inte-

gral

∞

Fs()= ∫ Gsx ( , )fx()dx

• ∞

The function G(s,x) is referred to as the kernel of the transform. The

association is one-to-one so that fcan be uniquely recovered from its

transform F. For example, linear processes can be studied in the time

domain or in the frequency domain: The two are linked by integral

transforms. We will see how integral transforms are applied to several

applications in finance. The two most important types of integral trans-

forms are the Laplace transform and Fourier transform. We discuss both

in this section.

Laplace Transform

Given a real-valued function f, its one-sided Laplace transform is an

operator that maps f to the function L(s) = L(f(x)) defined by the

improper integral

∞

Ls()= L[fx()]= ∫ e –sxfx()dx

0

if it exists.

The Laplace transform of a real-valued function is thus a real-valued

function. The one-sided transform is the most common type of Laplace

transform used in physics and engineering. However in probability theory

Laplace transforms are applied to density functions. As these functions are

defined on the entire real axis, the two-sided Laplace transforms are used.

In probability theory, the two-sided Laplace transform is called the

moment generating function. The two-sided Laplace transform is defined

by

∞

Ls()= L[fx()]= e ()dx

• sx

∫ fx

• ∞