What is the difference between a Brownian motion, a Wiener process, a random walk, white noise, and a Lévy process? Why does it matter?

Today, we rarely stop to consider random walks explicitly, especially in the context of statistical and financial models where they are often assumed as a foundational component. The concept, however, wasn’t always known. It wasn’t until the observations of Robert Brown in 1827 that we began to formalize the idea of particles moving in seemingly random patterns, a phenomenon that came to be known as Brownian motion. Yet, even after Brown’s observations, it took decades for the mathematical formalism of random walks to be developed.

But why has Brownian motion, and random walks more broadly, become so fundamental to modern finance? The answer lies in their ability to model uncertainty and unpredictability, particularly in the fluctuations of asset prices. As we know from empirical studies, asset prices often follow a path that resembles a random walk, where future movements are independent of past behavior and are driven by a series of random shocks. This randomness is the core assumption behind models such as the Black-Scholes equation, which revolutionized the pricing of financial derivatives. The random walk concept allows for the quantification of risk and the prediction of probable outcomes in the inherently uncertain financial markets.

Brownian motion, originally observed by Robert Brown in 1827, refers to the random movement of particles suspended in a fluid, a phenomenon later formalized in mathematical terms to describe a continuous-time stochastic process. The Wiener process, introduced by Norbert Wiener, is the mathematical model of Brownian motion and serves as a fundamental building block in probability theory and finance. It is characterized by independent and normally distributed increments, with continuous paths and no memory of past movements.

The Wiener process (and thus the Brownian motion) is characterized by:

- It starts at \(0\), \(W(0)=0\).
- It has independent increments.
- The increments \(W(t+s) - W(t)\) are normally distributed with mean \(0\) and variance, \(s\), i.e., \(W(t+s) - W(t) \sim N(0,s)\).

A random walk is a discrete-time stochastic process where an object, such as a price or position, takes a series of successive random steps, each of which is determined by chance. In its simplest form, each step is equally likely to move up or down by a fixed amount, and the outcome of each step is independent of previous steps. Random walks are used to model a wide range of phenomena, from stock price movements to diffusion in physics. A basic example of a random walk is the random walk on the integer number line, Z, which begins at 0 and moves either +1 or -1 at each step with equal probability.

The general definition of a random walk is as follows. Let \(X_0, X_1, X_2, \dots\)
represent a sequence of random variables, where \(X_0\) is the starting position and
each subsequent \(\( X_n \)\) \((for n \geq 1)\) is the position after \(n\) steps. The
process is called a *random walk* if:

where \(\xi_1, \xi_2, \dots\) are independent and identically distributed (i.i.d.) random variables that represent the increments or “steps” taken at each time point.

The Brownian motion or Wiener process can be approximated by a simple random walk along the integer number line (+1 or -1) if the step-size is sufficiently small. When the step size goes to zero, the random walk converges to the Wiener process (Donsker’s theorem by use of the CLT).

Note also that the random walk hypothesis is a financial theory that suggests stock market prices follow a random walk, meaning price changes are random and therefore cannot be predicted.

A Gaussian random walk is used when each step in the random walk is normally (Gaussian) distributed, rather than taking discrete values like \(+1\) or \(−1\) in a simple symmetric random walk. This type of random walk has continuous, normally distributed increments. Gaussian random walks are particularly useful in contexts where continuous changes and normal distributions are more realistic assumptions, such as in financial modeling or certain physical processes. If the mean is non-zero the random walk will vary around a trend.

White noise is a type of stochastic process characterized by a sequence of random variables that are independent and identically distributed (i.i.d.), with a mean of zero and constant variance. It is often used to model purely random, uncorrelated disturbances in time series analysis, where each value in the sequence is completely independent of the others. In the context of finance, white noise is important because it represents the unpredictable “noise” component in models of asset prices, interest rates, or economic variables.

In the random walk model described above, if \(\xi_i\) has mean \(0\) it is white noise, since it has an i.i.d. distribution with mean zero and constant variance.

A Lévy process is a generalization of the above processes and can be viewed as the continuous-time counterpart of a random walk. The most prominent examples of Lévy processes are the Wiener process (or Brownian motion) and the Poisson process. Other significant examples include the Gamma process, the Pascal process, and the Meixner process. While Brownian motion with drift has continuous paths, most other Lévy processes exhibit jumps or discontinuities. Importantly, all Lévy processes belong to the class of additive processes, meaning they can be decomposed into independent increments over time.

The Black-Scholes model, introduced in 1973 by Fischer Black, Myron Scholes, and later expanded by Robert Merton, is a foundational framework in modern finance for pricing European-style options. The model is based on the idea that the price of a financial asset follows a stochastic process, specifically a geometric Brownian motion, which assumes continuous price changes and a constant volatility. By solving the resulting partial differential equation, the model provides a closed-form solution for the fair price of an option, accounting for variables such as the current asset price, the option’s strike price, time to expiration, risk-free interest rate, and the asset’s volatility. This model has had a profound impact on financial markets, providing a theoretical foundation for the massive growth in derivatives trading.

Boyle (1977) showed that we can find the fair value of an option, by simulating various random paths and looking at the expectation. This is one of the great advantages with using a random walk in our modelling, it is easily used in simulations.

- Black and Sholes, 1973, The Pricing of Options and Corporate Liabilities, JPE.
- Boyle, 1977, Options: A Monte Carlo Approach, JFE.
- Wiener, 1923, Differential Space, Journal of Mathematics and Physics.