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

CHAPTER 24

MARTINGALE INEQUALITIES

Martingale inequalities are important in the study of stochastic differential equations. For example, martingale inequalities are used to establish bounds of solutions to stochastic differential equations. In this chapter, we present several important martingale inequalities.

24.1 Basic Concepts and Facts

Theorem 24.1 (Doob’s Submartingale Inequality). Let {Y_t : a ≤ t ≤ b} be a right-continuous submartingale. Then for any > 0, we have

(24.1)

where Y_b⁺ = max(Y_b, 0). In particular, if the Y_t is a right-continuous martingale, then for any > 0, we have

(24.2)

24.2 Problems

24.1. Let {X_i : i = 0, 1, …, n} be a submartingale adapted to a filtration {_i : i = 0, 1, …, n}. Show that for every λ > 0, we have

where M_n = max{X_i : i = 0, 1, …, n}.

24.2 (Doob’s Maximal Inequality). Let {X_i : i = 1, 2, …, n} be a martingale or nonnegative submartingale adapted to the ...