Martingales and Financial Mathematics in Discrete Time

4Martingales

In this chapter, we will define the concept of a “martingale”, which is a family of discrete-time stochastic processes that are especially important for applications in financial mathematics that we will see in the following chapters. It will be seen, in particular, that the simple symmetric random walk studied in Chapter 3 is a martingale. This chapter is not an exhaustive course on the theory of martingales. Notably, there is no discussion on convergence, although it is a very important aspect of the general theory of martingales. Readers who wish to have a more comprehensive introduction to the theory of martingales or their use in other fields of application may refer to [DUF 97, WIL 91].

Section 4.1 begins by defining martingales, and section 4.2 looks at the important concept of the transform of a martingale. The Doob decomposition theorem is then introduced in section 4.3. In a certain way, this theorem justifies the universality of martingales. The concept of stopping time is then introduced in section 4.4, and section 4.5 then studies the concept of martingales stopped at a stopping time. Finally, there are a series of exercises in using the concepts explained in this section 4.6.

4.1. Definition

In all that follows, we will work on a filtered probability space (Ω, , ℙ, (_n)_n∈ℕ).

DEFINITION 4.1.– A sequence of random variables (X_n)_n∈ℕ is an (_n)_n∈ℕ-martingale ...