The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 130 Friday, March 12, 2004 12:39 PM

130 The Mathematics of Financial Modeling and Investment Management

b b

∫ ()yd = ∫

g –^1 ()

fy – 1 fgx( ())g′()x xd

a g () a

Lebesque-Stieltjes Integrals

Most applications of calculus require only the integral in the sense of

Riemann. However, a number of results in probability theory with a

bearing on economics and finance theory can be properly established

only in the framework of Lebesgue-Stieltjes integral. Let’s therefore

extend the definition of integrals by introducing the Lebesgue-Stieltjes

integral.

The integral in the sense of Riemann takes as a measure of an inter-

val its length, also called the Jordan measure. The definition of the inte-

gral can be extended in the sense of Lebesgue-Stieltjes by defining the

integral with respect to a more general Lebesgue-Stieltjes measure.

Consider a non-decreasing, left-continuous function g(x) defined on a

domain which includes the interval [xi – xi–1] and form the differences

mLi= g(xi) – g(xi–1). These quantities are a generalization of the concept

of length. They are called Lebesgue measures. Suppose that the interval

(a,b) is divided into a partition of n disjoint subintervals by the points

a = x 0 < x 1 < ... < xn = b and form the Lebesgue-Stieltjes sums

n

Sn = ∑fx()i mLi , xi ∈ (xi,xi – 1 )

i = 1

where xi * is any point in i-th subinterval of the partition.

Consider the set of all possible sums {Sn}. These sums depend on the

partition and the choice of the midpoint in each subinterval. We define

the integral of f(x) in the sense of Lebesgue-Stieltjes as the limit, if the

limit exists, of the Lebesgue-Stieltjes sums {Sn} when the maximum

length of the intervals in the partition tends to zero. We write, as in the

case of the Riemann integral:

b

I = ∫ f x()d gx()= lim Sn

a

The integral in the sense of Lebesgue-Stieltjes can be defined for a

broader class of functions than the integral in the sense of Riemann. If f

is an integrable function and g is a differentiable function, the two inte-

grals coincide. In the following chapters, all integrals are in the sense of

Riemann unless explicitly stated to be in the sense of Lebesgue-Stieltjes.