LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A Brownian bridge is a random path that starts at zero and is forced back to zero at a chosen end time. The Karhunen–Loève expansion writes that path as an infinite sum of sine waves, each multiplied by an independent normal draw and by an eigenvalue of the bridge covariance. Truncate the sum and you have a practical simulator.
Why it matters
A model price is an average over scenarios. The wrong dependence across time makes that average a different price from the one the model claims, and the quote can contain an arbitrage relative to the model. The bridge is the simplest process tied to two anchors, such as a close and the next open. It is also the continuous version of diagonalizing a covariance matrix, which is the habit later Volterra and rough-volatility simulators still need.
The idea in plain words
Work with a square-integrable process: the expected square of its value is finite, so the covariance between any two times on the interval is finite too. The lecture sets the mean to zero. The covariance kernel is then the expected product of the process at those two times.
The theorem rebuilds the process from that kernel. Solve an integral equation: integrate the kernel against a candidate function and require the result to equal an eigenvalue times the same function. The functions are the modes. Draw each coefficient as a normal with mean zero and variance equal to its eigenvalue, or draw a standard normal and multiply by the square root of the eigenvalue. The two writings match, because scaling a unit-variance normal by a square root produces a variable whose variance is the original number.
On a grid the kernel is a matrix. A draw is the eigenvectors times the square root of the eigenvalues times a standard normal vector, and the covariance of that product is the matrix you started with.
Brownian motion starts at zero, is continuous with probability one, and has independent normal increments whose variance equals the step length. The bridge subtracts, at each date, elapsed time over the horizon times the value at the horizon. Its covariance is the minimum of the two times, minus their product divided by the horizon.
How to use it
Confirm the bridge is square-integrable, write the kernel, pose the integral equation, solve it, and optionally rescale each eigenfunction so the integral of its square equals one. Then write the series.
On the unit interval, split the minimum in the kernel into two integrals, subtract the product of the times, and differentiate twice. What remains is a second-order equation. Its solutions are sines and cosines in time over the square root of the eigenvalue. A zero at the left endpoint removes the cosine. A zero at the right endpoint forces each eigenvalue to be one over k pi squared. The squared-sine identity then puts a factor of the square root of two in front, because the cosine piece of that identity integrates to zero. The bridge is that factor times a sum of standard normals times the sine of k pi t, divided by k pi. That division is the square root of the eigenvalue, so each mode has the variance the kernel asks for.
A computer cannot sum forever. Stop after finitely many modes. How many you need follows how fast the eigenvalues shrink. The lecture's picture uses 50 modes and five paths, and mentions a cutoff on the order of 500 as a computational choice. The time step and the path count add error when a sample covariance is compared with the formula. More modes and more paths pull that sample toward the theoretical kernel by the law of large numbers, if the draws really come from this series.
If the integral equation has no closed solution, build the covariance on your grid and use the eigenvector product. That discrete draw targets those time steps. It is not a chopped-off copy of the sine series.
What usually goes wrong
One shortcut treats a price, a volatility series, or a strategy's gain as one frozen random variable. The link between times is the object to get right. Another error uses the eigenvalue both as a variance and as an extra square-root scale. A third treats a short truncation, or a handful of paths, as the kernel. Many processes never yield a differential equation you can finish by hand, and the matrix construction is what remains.
Questions people ask
What does the series give you?
Sample paths whose covariance matches the bridge, once the truncation and the path count are large enough.
Why do the eigenvalues fall like one over k squared?
Those values make a sine of k pi t vanish at both ends of the unit interval and still solve the equation that came from the kernel.
Does a correct simulator remove bad quotes?
It keeps the paths inside the model you stated. The model can still be a poor picture of the market. The expansion is about matching the covariance you claimed.