Introduction to Stochastic Calculus with Applications, Second Edition
Containing many solved examples and exercises, this book gives a simple but rigorous treatment of stochastic calculus and its applications, including a range of advanced topics.
TABLE OF CONTENTS Introduction to Stochastic Calculus with Applications, Second Edition
Chapter 7: Martingales
Martingales play a central role in the modern theory of stochastic processes and stochastic calculus. Martingales constructed from a Brownian motion were considered in Section 3.3 and martingales arising in diffusions in Section 6.1. Martingales have a constant expectation, which remains the same under random stopping. Martingales converge almost surely. Stochastic integrals are martingales. These are the most important properties of martingales, which hold under some conditions.
7.1 Definitions
The main ingredient in the definition of a martingale is the concept of conditional expectation, consult Chapter 2 for its definition and properties.
Definition 7.1
A stochastic process M (t), where time t is continuous 0 ? t ? T, or discrete t = 0,1, , T, adapted to a filtration 
is a martingale if for any t, M(t) is integrable, that is, EM(t) < ? and for any t and s with 0 ? s < t ? T,
M (t) is a martingale on [0, ?) if it is integrable and the martingale property (7.1) holds for any 0 ? s < t < ?.
Definition 7.2
A stochastic process X (t), t ? 0 adapted to a filtration 
is a supermartingale (submartingale) if it is integrable, and for any t and s, 0 ? s < t ? T
If X ( t) is a supermartingale, then ? X ( t) is a submartingale. The mean of a supermartingale is...
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