The aim of this book is to introduce stochastic analysis, keeping in mind the viewpoint of path space. The area covered by stochastic analysis is very wide, and we focus on the topics related to Brownian motions, especially the Itô calculus and the Malliavin calculus. As is widely known, a stochastic process is a mathematical model to describe a randomly developing phenomenon. Many continuous stochastic processes are driven by Brownian motions, while basic discontinuous ones are related to Poisson point processes.

The Itô calculus, named after K. Itô who introduced the calculus in 1942, is typified by stochastic integrals, Itô's formula, and stochastic differential equations. While Itô investigated those topics in terms of Brownian motions, they are now studied in the extended framework of martingales. One of the important applications of the calculus is a construction of diffusion processes through stochastic differential equations. The Malliavin calculus was introduced by P. Malliavin in the latter half of the 1970s and developed by many researchers. As he originally called it “a stochastic calculus of variation”, it is exactly a differential calculation on a path space. It opened a way to take a purely probabilistic approach to transition densities of diffusion processes, which are fundamental objects in theory and are applied to many fields in mathematics and physics.

We made the book self-contained as much as possible. Several preliminary facts in analysis and probability theory are gathered in the Appendix. Moreover, a lot of examples are presented to help the reader to easily understand the assertions. This book is organized as follows. Chapter 1 starts with fundamental facts on stochastic processes. In particular, Brownian motions and martingales are introduced and basic properties associated with them are given. In the last three sections, investigations of path space type are made; the Cameron–Martin theorem, Schilder's theorem and an analogy with path integrals are presented.

Chapter 2 introduces stochastic integrals and Itô's formula, an associated chain rule. Although Itô originally discussed them with respect to Brownian motions, we formulate them with respect to martingales in the recent manner due to J. L. Doob, H. Kunita and S. Watanabe. Moreover, several facts on continuous martingales are discussed: for example, representations of them by time changes and those via stochastic integrals with respect to Brownian motions.