Analysis of Financial Time Series, Third Edition

6.8 Stochastic Integral

We briefly discuss stochastic integration so that the price of an asset can be obtained under the assumption that it follows an Ito process. We deduce the integration result using Ito's formula. For a rigorous treatment on the topic, readers may consult textbooks on stochastic calculus. First, like the usual integration of a deterministic function, integration is the opposite of differentiation so that

continues to hold for a stochastic process xt. In particular, for the Wiener process wt, we have because w0 = 0. Next, consider the integration . Using the prior result and taking integration of Eq. (6.7), we have

Therefore,

This is different from the usual deterministic integration for which .

Turn to the case that xt is a geometric Brownian motion—that is, xt satisfies

where μ and σ are constant with σ > 0; see Eq. (6.8). Applying Ito's lemma to G(xt, t) = ln( ...