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TL;DR

A discrete-time process where each step is an independent random variable: \( S_n = S_0 + \sum_{i=1}^n X_i \). The simple random walk has \( X_i \in \{-1, +1\} \) with equal probability.

By Valenke Exam Prep Team·Last updated 2026-06-03

Random Walk

A discrete-time process where each step is an independent random variable: \( S_n = S_0 + \sum_{i=1}^n X_i \). The simple random walk has \( X_i \in \{-1, +1\} \) with equal probability.

Why it matters for interviews

The discrete analog of Brownian motion. Random walk problems (gambler's ruin, first passage times, ballot problems) are staples of quant interviews. The efficient market hypothesis posits that prices follow a random walk.

Definition and Mathematical Foundation

A discrete-time process where each step is an independent random variable: \( S_n = S_0 + \sum_{i=1}^n X_i \). The simple random walk has \( X_i \in \{-1, +1\} \) with equal probability.

Application in Quantitative Finance

The discrete analog of Brownian motion. Random walk problems (gambler's ruin, first passage times, ballot problems) are staples of quant interviews. The efficient market hypothesis posits that prices follow a random walk.

Related Terms

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Frequently Asked Questions

What is the gambler's ruin problem?▼
A gambler with initial wealth \( a \) plays fair coin flips against an opponent with wealth \( b \). The probability the gambler goes broke is \( b/(a+b) \). This is solved using martingale methods or difference equations.
What is the reflection principle?▼
A combinatorial technique: the number of paths from (0,0) to (n,k) that touch or cross zero equals the total paths from (0,0) to (n,-k). It is used to derive first passage time distributions.
Is a random walk recurrent or transient?▼
In 1D and 2D, the simple random walk is recurrent (returns to origin with probability 1). In 3D and higher, it is transient. This is Polya's recurrence theorem.