Introduction

In this book we present a new approach to the theory of modern stochastic integration. The novelty is that we define a semimartingale as a stochastic process which is a “good integrator” on an elementary class of processes, rather than as a process that can be written as the sum of a local martingale and an adapted process with paths of finite variation on compacts: This approach has the advantage over the customary approach of not requiring a close analysis of the structure of martingales as a prerequisite. This is a significant advantage because such an analysis of martingales itself requires a highly technical body of knowledge known as “the general theory of processes”. Our approach has a further advantage of giving traditionally difficult and non-intuitive theorems (such as Stricker’s theorem) transparently simple proofs. We have tried to capitalize on the natural advantage of our approach by systematically choosing the simplest, least technical proofs and presentations. As an example we have used K.M. Rao’s proofs of the Doob-Meyer decomposition theorems in Chap. III, rather than the more abstract but less intuitive Doléans-Dade measure approach.