Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach

CHAPTER 22

MARTINGALES

A martingale is a special stochastic process whose increments over an interval in future have zero expectation given information of the past history of the process. Martingales play a central role in the theory of stochastic processes and stochastic calculus. In this chapter, we present the definition of martingales.

22.1 Basic Concepts and Facts

Definition 22.1 (Martingale). Let I be a set. A stochastic process X = (X_t : t I) is called a martingale relative to a filtration ({_t : t I}, P) if the process satisfies the following three conditions:

(a) X is adapted to the filtration.

(b) E(|X_t|) < ∞ for all t I.

(c) E[X_t|_s] = X_s a.s. for all s ≤ t.

Definition 22.2 (Supermartingale). Let I be a set. A stochastic process X = (X_t : t I) is called a supermartingale relative to a filtration ({_t : t I}, P) if the process satisfies the following three conditions:

(a)