The traditional theory of probability is based on measurable sets formed by operations involving infinitely many steps. In contrast, Riemann sums have only a finite number of terms. On the face of it, such sums should be relatively easy to calculate. This chapter contains a number of such calculations, using Maple 15 computer software. Random walk and Brownian path diagrams can easily be produced with Maple. The chapter investigates numerically the weak stochastic integral, for which the Riemann sum estimates does not generally converge whenever the partitions of the domain are successively refined. It demonstrates numerically and visually the meaning of the weak convergence of Riemann sum estimates of the observables in Itô’s formula. It is argued that large changes in asset prices are common. The chapter presents a series of asset prices from Thomson Reuters Datastream. The chapter focuses on the binary partitions of R<SUP>T</SUP>, and other joint-contingent observables.

Controlled Vocabulary Terms

Brownian motion; observable variable; random walk; Riemann distribution; stochastic Integrability