O'Reilly logo

Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

CHAPTER 26

RANDOM WALKS

A random walk is a simple discrete stochastic process formed by successive summation of independent and identically distributed random variables. Random walks are one of the most widely studied topics in probability theory and have important applications in finance. In this chapter, we present random walks and relevant results.

26.1 Basic Concepts and Facts

Definition 26.1 (Random Walk). A random walk {Sn : n ≥ 0} starting at x is a stochastic process defined as

equation

where x R and Zi’s are independent and identically distributed random variables.

Definition 26.2 (Simple Random Walk). Let {Sn : n ≥ 0} be a random walk as defined in Definition 26.1. It is called a simple random walk if for every n ≥ 1,

equation

where p (0, 1).

26.2 Problems

26.1. Let {Sn : n ≥ 0} be a random walk as defined in Definition 26.1. Let μ = E[Z1] and . Suppose that |μ| < ∞ and σ < ∞. Show that

(a) {Sn − μn : n ≥ 0} is a martingale.
(b) {(Sn − μn)2 − σ2n : n ≥ 0} is a martingale.
(c) For any u R

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required