ABSTRACT
This chapter computes the Riemann integral of a Brownian sample path with respect to time. Since it is possible to integrate Brownian paths (and functions of Brownian motion) with respect to time, it is interesting to find out what other integrals can be calculated for Brownian motion. The Riemann—Stieltjes integral generalizes the Riemann integral. It provides an integral of one function with respect to another appropriate one. An important characteristic of a stochastic process is the quadratic variation that measures the accumulated variability of the process along its path. The quadratic variation is a path-dependent quantity. In many applications, including derivative pricing, the stochastic process is assumed to be time-homogeneous. The chapter discusses a simple example of how Girsanov’s theorem can be applied to change probability measures so as to eliminate the drift in a drifted Brownian process.
