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.

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

where *x* **R** and *Z*_{i}’s are independent and identically distributed random variables.

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

where *p* (0, 1).

**26.1.** Let {*S*_{n} : *n* ≥ 0} be a random walk as defined in Definition 26.1. Let μ = *E*[*Z*_{1}] and . Suppose that |μ| < ∞ and σ < ∞. Show that

(a) {*S*_{n} − μ*n* : *n* ≥ 0} is a martingale.

(b) {(*S*_{n} − μ*n*)^{2} − σ^{2}*n* : *n* ≥ 0} is a martingale.

(c) For any *u* **R**

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