The Mathematics of Financial Modelingand Investment Management

4-PrincipCalculus Page 128 Friday, March 12, 2004 12:39 PM

128 The Mathematics of Financial Modeling and Investment Management

EXHIBIT 4.8 Riemann Sums

I = ∫ ()xd = sup SL { }

b

f x {}n = inf SU n

a

is called the proper integral of f on [a,b] in the sense of Riemann.

An alternative definition of the proper integral in the sense of Rie-

mann is often given as follows. Consider the Riemann sums:

n

()*(x

Sn = (^) ∑fxi i – xx – 1 )
i = 1
where x*
i is an arbitrary point in the interval [x^1 ,xi–1]. Call ∆xi = (xi –
xi–1) the length of the i-th interval. The proper integral I between a and
b in the sense of Riemann can then be defined as the limit (if the limit
exists) of the sums Sn when the maximum length of the subintervals
tends to zero: