LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
Brownian motion is the random process underneath the usual stochastic price models. Three properties define it in this lesson. It starts at zero with probability one. It is almost surely continuous. Its increments are independent and stationary, and an increment is normal with variance equal to the length of the step. A five-day notebook makes those properties visible, and it shows why an integral against Brownian motion collapses to the ending value.
Why it matters
Simulated paths in earlier sessions draw a normal shock with standard deviation equal to the square root of the step. That draw is not a coding habit. It is the increment property. The integral that appears in the closed form for geometric Brownian motion, the integral of a constant times the Brownian increment, collapses for the same reason. The lesson wants both facts on one page, with a picture. Formal statements are credited to a book it has recommended before, on mathematical modeling and computation in finance.
The idea in plain words
Take a clock from today out to a horizon. The lesson's concrete clock is five days, with a point at 0, 1, 2, 3, 4, and 5. The vertical axis is the process. Connecting the points is one path.
The first property puts the day-zero point at zero. The careful statement is that the probability of that event is one. The lesson distinguishes probability one from a casual claim of certainty, and compares the distinction to the way a limit can settle at a value. For the notebook, the path starts at zero.
The second property is almost-sure continuity. The lesson's formal sketch is a probability limit, as a mesh gets finer, equal to one. It suggests looking up the measure-theoretic wording if you need it, and it asks you to carry the phrase "almost surely continuous" into the pictures.
The third property is the one the code uses. The change from one time to a later time does not depend on the path so far, and the distribution of the change depends on the times only through the gap. That change is normal with variance equal to the gap. A one-day gap has variance one, so a standard normal draw is the right shock. A shorter gap needs the square root of the gap as the standard deviation, which is how the earlier path simulators were built.
How to use it
Import NumPy and Matplotlib. Build the time grid 0 through 5. Start a list at 0. For each later day, append a standard normal draw. Plot time against the list. That is one path. Repeat a thousand times, storing each path. A scatter of every point shows a single dot at time zero and, at each later day, a cloud centered on zero. The lesson describes the cloud as running out to about three on either side, dense in the middle, with a thin tail, which is the shape it wants you to recognize as standard normal. If the cloud is shifted or far wider than that, the draw is not the unit normal the one-day gap requires.
The integral is the second picture. Integrating the Brownian increment from the start to day five equals the process at day five minus the process at day zero. The start is zero, so the integral equals the last point. The lesson reads one path's last point as about 0.73791 and treats that number as the value of the integral for that path. Intermediate days do not enter. The same would be true of a gain-and-loss total: if the book was flat five days ago, today's level is the whole change. Sigma in front of the integral is a constant scale. Set it to one and the ending value is the whole answer.
What usually goes wrong
Each day's cloud in the lesson is a fresh standard normal, the increment. A cumulative path adds those increments, so the position's spread grows with time, and the integral of the increments is the last position. Do not center the next draw on the previous point.
Probability one is not a promise that every finite sample starts at exactly zero by accident. The code sets the first point to zero.
A path that uses the same random draw every day has dependent increments and breaks the third property. Each step needs its own draw.
Questions people ask
What is the variance of an increment?
The length of the time gap. A one-day gap in the sketch has variance one, so the draw is a standard normal.
What is the integral from day zero to day five?
The process on day five, because the value on day zero is zero. One path in the lesson ends near 0.73791, and that ending is the integral for that path.
How is this different from a stock-price path?
Price models such as geometric Brownian motion transform the increments, so prices are dependent. The Brownian increments underneath stay independent and stationary.