A Modern Theory of Random Variation: With Applications in Stochastic Calculus, Financial Mathematics, and Feynman Integration

Chapter 8

Stochastic Integration

In Chapter 1 examples of observables expressed in elementary, basic, and contingent forms are given. For instance, the contingent observable (1.4) has elementary form (1.5); and the joint-contingent observable (1.6) has elementary form (1.7). Theorem 82 of Chapter 5 examines conditions under which an elementary form Y ≃ y[R, F_Y] might be deduced from a contingent form f(X) ≃ f(x)[Ω_X, F_X], where Y and f(X) represent the same observable—so the datum satisfies y = f(x), and the sample space Ω_Y is R. This is done by deducing a distribution function F_Y from F_X.

When we have knowledge of the distribution function of an observable, then, in advance of actually carrying the experiment or measurement we can, by performing particular calculations with the distribution function, acquire knowledge of the datum, such as its expected value or variance. Conversely, if we have repeated instances of the measurement obtained in nearly identical experimental conditions, then we can use the experimental or sample data to estimate the distribution function under which the sample data were generated. This can be done, for instance, by calculating the relative frequencies of sample data.

Likewise, if we know the functional form f of a contingent observable f (X) ≃ f (x)[Ω_X, F_X], we can sometimes deduce from this the distribution function F_Y for the elementary form Y = f (X) of the contingent observable.

Much of the analysis of random variability consists of the search ...