Game-Theoretic Foundations for Probability and Finance

14A Game‐Theoretic It Calculus

In this chapter we introduce and study martingale spaces, which provide an abstract framework for studying continuous‐time game‐theoretic martingales. Martingale spaces are sufficiently powerful to support the Itô calculus, and they will be used throughout the remaining chapters of this book.

The framework of Chapter 13 was concrete. There we considered the price path of a single security, assumed to be continuous, and our sample space consisted of the security's possible price paths – i.e. all the continuous functions on . The sample space for a martingale space, in contrast, is an abstract measurable space , with a filtration as in continuous‐time measure‐theoretic probability, and the price paths of securities are adapted processes on . The martingale space consists of the measurable space, supplemented not with a probability measure but with price paths for a countable number of securities. These price paths are the space's basic martingales. For simplicity, we assume that they are continuous, but the framework can be extended to handle some discontinuous ...