Economic Modelling · Stochastic models for security prices

Random Walks and Brownian Motion Explained

Updated 11 October 2026 · Fact-checked

A random walk adds independent random steps over discrete time. Standard Brownian motion (a Wiener process) is its continuous-time limit: B(0) = 0, increments over disjoint intervals are independent, B(t) − B(s) ~ N(0, t − s), and paths are continuous but nowhere differentiable. You solve questions by using these normal increment rules.

Understand Random Walks and Brownian Motion

A random walk is a process built by adding up independent random steps. Let X₁, X₂, ... be independent and identically distributed with mean μ and variance σ². Then S(n) = X₁ + ... + X_n is a random walk. It moves at discrete times 0, 1, 2, and so on.

A simple symmetric random walk has steps of +1 or −1, each with probability ½. Its mean is 0 and its variance after n steps is n. The variance grows in proportion to time, so the standard deviation grows like √n, not like n. This is the key idea.

Now shrink the time step and the step size together. Take steps of size √h every h units of time. By the central limit theorem, the position at time t becomes normally distributed with mean 0 and variance t. The limit is standard Brownian motion, also called a Wiener process, written B(t) or W(t).

The Wiener process is defined by these properties:

B(0) = 0.
Increments over non-overlapping intervals are independent.
For s < t, B(t) − B(s) is normal with mean 0 and variance t − s.
Sample paths are continuous.

A path is continuous but has no derivative at any point. It wiggles at every scale. Because of this, ordinary calculus fails. This is why stochastic calculus and Ito's lemma are needed for the lognormal and Black-Scholes models.

A random walk is discrete in time and (usually) in space. Brownian motion is continuous in both. Brownian motion with drift is X(t) = μt + σB(t). It has mean μt and variance σ²t. It is the building block for share price models.

Key rules to remember

Random walk: S(n) = X₁ + X₂ + ... + X_n; The X_i are independent and identically distributed. E[S(n)] = nμ and Var[S(n)] = nσ².
Simple symmetric random walk: P(X = +1) = P(X = −1) = ½; E[S(n)] = 0; Var[S(n)] = n; Variance grows linearly with the number of steps.
Standard Brownian motion increments: B(t) − B(s) ~ N(0, t − s) for 0 ≤ s < t; The variance is the time gap, not the standard deviation. Increments over disjoint intervals are independent.
Distribution of B(t): B(t) ~ N(0, t), since B(0) = 0; Standard deviation is √t.
Covariance: Cov(B(s), B(t)) = min(s, t); Follows from independent increments. Useful for sums and averages of values at different times.
Brownian motion with drift: X(t) = μt + σB(t), so X(t) ~ N(μt, σ²t); Increments X(t) − X(s) ~ N(μ(t − s), σ²(t − s)).
Quadratic variation: (dB)² = dt; The reason Ito's lemma has an extra term. Paths have non-zero quadratic variation but are not differentiable.
Scaling limit: Steps ±√h every h time units → B(t) as h → 0; Var at time t is (t ÷ h) × h = t.

How to solve Random Walks and Brownian Motion questions

Use this method for any question on random walks or Brownian motion, whether it asks for a probability, a variance, a covariance or a property.

1Write down the process clearly: random walk, Brownian motion, or Brownian motion with drift. Note B(0) and any drift μ and volatility σ.
2Express what you need as a sum of independent increments. Split B(t) into B(s) plus [B(t) − B(s)] if the times differ.
3Find the mean and variance of each piece. For a Wiener process the variance of an increment equals the time gap.
4Add means. Add variances only for independent increments. Include covariance terms if the pieces overlap.
5State the distribution as N(mean, variance). Use the conditional form given B(s) = b when the question gives the past value.
6Standardise: Z = (value − mean) ÷ √variance. Read the probability from the normal table.
7For property questions, name the property used (independent increments, normal increments, continuity, non-differentiability) and say why it applies.
8Check that you used variance, not standard deviation, and that the time units match.

Quickest way: Variance equals time gap

When to use it: Use this for MCQs and short calculations that ask for a distribution, probability or variance of a Brownian motion quantity.

Identify the time gap and use it directly as the variance: N(0, gap) for standard Brownian motion.
For drift and volatility: mean = μ × gap, variance = σ² × gap.
If the question conditions on B(s), the future change B(t) − B(s) is independent of the past. The mean of B(t) is B(s).
For covariance, take the smaller time: Cov(B(s), B(t)) = min(s, t).
Standardise with the square root of the variance and look up the normal table.

Common mistakes in Random Walks and Brownian Motion

Using the time gap as the standard deviation instead of the variance.
Students treat B(t) − B(s) ~ N(0, t − s) as if the second parameter were σ.
Fix: The second parameter is variance. The standard deviation is √(t − s). Write the square root before standardising.
Saying Brownian motion paths are differentiable because they are continuous.
Students link continuity with smoothness from ordinary calculus.
Fix: State that paths are continuous everywhere but differentiable nowhere. This is why stochastic calculus is needed.
Adding variances of overlapping increments, such as B(2) and B(5), as if independent.
Students forget B(5) includes B(2).
Fix: Write B(5) = B(2) + [B(5) − B(2)] and use independence of the two parts. Or use Cov = min(s, t).
Believing the random walk variance grows like n² or that the spread grows linearly with time.
Students mix up adding means (which grow like n) with adding variances.
Fix: Variance of a sum of n independent steps is nσ². The standard deviation grows like √n.
Ignoring drift when the question gives a process X(t) = μt + σB(t).
Students apply standard Brownian motion results automatically.
Fix: Use mean μt and variance σ²t. Only the B(t) part has mean zero.
Saying a random walk and Brownian motion are the same thing.
Both are described as random movements.
Fix: A random walk is discrete time. Brownian motion is the continuous-time limit with normal increments.

Worked examples

Example 1

B(t) is a standard Brownian motion. Find P(B(4) > 3).

Show the solution

B(0) = 0, so B(4) ~ N(0, 4).
The standard deviation is √4 = 2.
Standardise: Z = (3 − 0) ÷ 2 = 1.5.
P(B(4) > 3) = 1 − Φ(1.5) = 1 − 0.9332 = 0.0668.

Answer: P(B(4) > 3) ≈ 0.0668

Example 2

X(t) = 0.05t + 0.2B(t), where B is standard Brownian motion. Find the distribution of X(9) − X(4) and the probability that it is negative.

Show the solution

The increment is over a time gap of 9 − 4 = 5.
X(9) − X(4) = 0.05 × 5 + 0.2 × [B(9) − B(4)].
The mean is 0.05 × 5 = 0.25.
The variance is 0.2² × 5 = 0.04 × 5 = 0.2, so the standard deviation is √0.2 ≈ 0.4472.
So X(9) − X(4) ~ N(0.25, 0.2).
P(negative) = Φ((0 − 0.25) ÷ 0.4472) = Φ(−0.559).
Φ(0.559) ≈ 0.712, so Φ(−0.559) ≈ 0.288.

Answer: X(9) − X(4) ~ N(0.25, 0.2); probability of a negative increment ≈ 0.288

Exam tips

Write the distribution in the form N(mean, variance) every time. Examiners give marks for stating it clearly.
For written questions on properties, list all four defining properties and explain the link to random walks through the central limit theorem.
Be ready to explain why non-differentiable paths mean ordinary calculus fails and Ito's lemma is needed.
In computer-based paper work, simulate a path by summing independent N(0, h) increments and show the time step h and the formula used.
Show the standardising step in full, and state any normal table value you use.

Practice questions from Stochastic models for security prices

Random Walks and Brownian Motion: frequently asked questions

What is the difference between a random walk and Brownian motion?

A random walk moves in discrete steps at discrete times. Brownian motion moves continuously in time with normally distributed increments. Brownian motion is the limit of a suitably scaled random walk as the step size and time step shrink.

What is a Wiener process in actuarial science?

It is another name for standard Brownian motion. It starts at zero, has independent normal increments with variance equal to the time gap, and has continuous paths. It is the basis for models of share prices and other stochastic asset models.

Why is Brownian motion not differentiable?

Over a small time h the change is of order √h, not h. So the ratio change ÷ h grows without limit as h shrinks. The path has no tangent at any point.

Is Brownian motion a martingale?

Standard Brownian motion is a martingale, because the expected future value given the past equals the current value. Brownian motion with non-zero drift is not.