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TL;DR

A continuous-time stochastic process \( W_t \) with independent, normally distributed increments: \( W_t - W_s \sim N(0, t-s) \) for \( t > s \), starting at \( W_0 = 0 \).

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

Brownian Motion

A continuous-time stochastic process \( W_t \) with independent, normally distributed increments: \( W_t - W_s \sim N(0, t-s) \) for \( t > s \), starting at \( W_0 = 0 \).

Why it matters for interviews

The foundation of continuous-time finance. Black-Scholes, stochastic volatility models, and virtually all derivatives pricing frameworks are built on Brownian motion. A must-know for quant interviews.

Definition and Mathematical Foundation

A continuous-time stochastic process \( W_t \) with independent, normally distributed increments: \( W_t - W_s \sim N(0, t-s) \) for \( t > s \), starting at \( W_0 = 0 \).

Application in Quantitative Finance

The foundation of continuous-time finance. Black-Scholes, stochastic volatility models, and virtually all derivatives pricing frameworks are built on Brownian motion. A must-know for quant interviews.

Related Concepts

Brownian Motion (Wiener Process)

Related Terms

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Frequently Asked Questions

What are the key properties of Brownian motion?▼
Continuous paths, independent increments, Gaussian increments \( W_t - W_s \sim N(0, t-s) \), stationary increments, and almost surely nowhere differentiable paths.
What is geometric Brownian motion?▼
GBM models asset prices as \( dS = \mu S dt + \sigma S dW \), solved as \( S_t = S_0 e^{(\mu - \sigma^2/2)t + \sigma W_t} \). It ensures prices stay positive and is the basis of Black-Scholes.
Is Brownian motion a martingale?▼
Yes. Standard Brownian motion is a martingale: \( E[W_t | \mathcal{F}_s] = W_s \) for \( s < t \). This martingale property is central to risk-neutral pricing.