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

CHAPTER 25

MARTINGALE CONVERGENCE THEOREMS

Martingale convergence theorems state that under certain conditions, a martingale, submartingale, or supermartingale converges to a limiting random variable. In this chapter, we shall present several martingale convergence theorems.

25.1 Basic Concepts and Facts

Definition 25.1 (Crossing of Real Numbers). Let {x_n : n ≥ 0} be a sequence of real numbers. Let a < b be real numbers. The sequence {τ_n(a, b) : n ≥ 0} of crossings with respect to the sequence {x_n : n ≥ 0} is defined by

(25.1a)

and for every n ≥ 1, we have

(25.1b)

(25.1c)

where A_2n−1 = {k ≥ τ_2n−2 : x_k ≤ a} and A_2n = {k ≥ τ_2n−1 : x_k ≥ b}.

Definition 25.2 (Crossing of Random Variables). Let{X_n : n ≥ 0} be a sequence of real random variables. Let a < b be real numbers. The sequence {τ_n(a, b) : n ≥ 0} of crossings with respect to the sequence {X_n : n ≥ 0} is defined as follows. For every ω Ω, {τ_n(a, b)(ω) : n ≥ 0} is the sequence of crossings with respect to the sequence {X_n(ω) : n ≥ 0}.

Definition 25.3 (_∞). Let {_n : n ≥ 0} be a filtration. Then _∞ is the σ-algebra generated by :

Theorem ...