Brownian Motion

How can we model a quantity that changes randomly at every moment? Imagine tracking the position of a tiny particle suspended in water. Collisions with surrounding molecules push it back and forth. We will model one coordinate of its displacement using Brownian motion: a random curve whose uncertainty grows with elapsed time.

Before you begin: You should know means, variances, independence, the normal distribution, and the central limit theorem. No stochastic calculus is needed. Allow about 45–60 minutes, including the experiments and exercises.

By the end, you should be able to explain the square-root-of-time scale, calculate probabilities for positions and increments, and distinguish independent increments from independent positions.

1. From a random variable to a random path

A random variable describes one uncertain number. A stochastic process is a collection of random variables indexed by time, written .

• Fix a time, say : is a random variable, with a distribution across repetitions.
• Run the experiment once: the entire curve is one sample path.

An average over many paths can be zero even though an individual path wanders far from zero. Keep these two views in mind throughout the tutorial.

2. Start with a coin-toss walk

Divide one unit of time into equal intervals. At each step, independently move up or down with equal probability:

Why divide by ? Since each toss has mean zero and variance one, independence gives

At time one, the variance is one for every . Dividing each step by instead would give variance , making the randomness vanish as we refine the time grid. Leaving the steps unscaled would give variance , which grows without bound.

Predict, then explore: If the number of steps quadruples, what happens to the size of a single step? What happens to the variance at time one?

Step size: . The plot joins the simulated positions with straight lines. Changing the step count generates a different approximation; it does not reveal hidden detail in the same path. The vertical axis is fixed for comparison; unusually large excursions can extend beyond the plot.

The central limit theorem tells us that approaches a standard normal distribution. There is a stronger theorem saying these interpolated random walks converge in distribution, as random curves, to Brownian motion. We will use this as motivation, without proving that theorem.

3. The Brownian motion model

A standard Brownian motion satisfies four properties:

1. Starts at zero: .
2. Independent increments: Changes over non-overlapping time intervals are independent.
3. Normal increments: For , .
4. Continuous paths: With probability one, the curve has no jumps.

Throughout this page, the second argument of is the variance, so its standard deviation is .

An increment is a change in position, . Its distribution depends on the elapsed time , rather than the starting time. This is called stationary increments. It does not mean that the positions have the same distribution at all times.

Taking immediately gives

Four times as much elapsed time produces twice the standard deviation. For each fixed positive time , about 95% of positions fall between and . That is a statement about a fixed time, not about an entire path staying inside those bounds.

4. One path, many repetitions

The next experiment simulates 600 independent paths over four time units. Each grid increment is normal with variance . The positions have the exact Brownian joint distribution at the grid times; the straight lines between them are only a drawing approximation.

Try this: Move the observation time from one to four. Predict the change in spread before looking at the histogram. Move the time back: the simulated paths stay the same until you generate a new sample.

Blue bars show the simulated positions; the orange curve is the theoretical density. The histogram uses fixed axes so you can compare its spread across observation times. Positions outside its horizontal range are not shown.

The simulated summaries fluctuate. A mean near zero describes an average over repetitions; it does not force any individual curve toward zero.

5. Calculate with increments

Let for . Standardize by dividing by the standard deviation, not the variance.

Example: a position at a fixed time

Since ,

Example: a change in position

The change from time two to time five has distribution . Therefore,

Example: knowing the current position

If we observe , the next increment remains independent of the past and has distribution . Thus the conditional distribution of is , and

This uses a conditional distribution for a continuous observation; we are not dividing by , which is zero.

Try your own threshold

This calculates a position probability at a fixed time. The probability of ever crossing a threshold before that time is a different question.

6. Independent changes, dependent positions

Although and are independent, and are not. They share the first unit of motion:

For , the same decomposition gives

So , and for positive times, . For instance, . Overlapping increments can also be dependent because they share some motion.

7. Continuous does not mean smooth

A continuous curve need not have a derivative. Over an interval of length , a Brownian increment has standard deviation . Its average slope therefore has standard deviation

As the interval shrinks, these slopes become more variable. This suggests why ordinary calculus is not enough to describe Brownian paths. A deeper theorem states that, with probability one, a Brownian path is nowhere differentiable. The slope calculation motivates this theorem; it does not prove it. Straight segments in our plots are a simulation artifact.

For a model with a starting position, a trend, and a different noise scale, we can use

Then . Here is the drift per unit time and sets the size of random fluctuations. Standard Brownian motion has and . This is an idealized model: independent normal changes and unrestricted positions should be checked against the setting being modeled.

8. Check your understanding

Try each problem before expanding its solution.

1. Find the distribution, variance, and standard deviation of .
2. Calculate .
3. Are and independent? What about and ?
4. Given , find the conditional mean and variance of , and the probability it exceeds 3.
5. Let . Find the distribution of and calculate .
6. Explain why is smaller than the probability that a path crosses level 1 at some time before time one. Does a histogram of endpoints answer the crossing question?

Further reading

For another treatment of the definition and covariance calculation, see Hossein Pishro-Nik's Introduction to Probability, Statistics, and Random Processes, §11.4.2. For a more advanced follow-up, see Steven Lalley's Brownian Motion lecture notes.