LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
The cookbook is an open-source set of simulation recipes for a handful of Gaussian and Gaussian-related processes. Each recipe is either a continuous-time path you discretize numerically, or a discrete-time construction, with code you can lift into your own project. The tour names Brownian motion, the Brownian bridge, fractional Brownian motion, a Volterra process, and a Markovian lift of the rough case.
Why it matters
Pricing code needs a path you can step forward, plus a check that it has the covariance you meant. The lecture's standard is to put the empirical covariance next to the theoretical one and see whether they agree as you add paths or change a roughness dial.
The resource is a tour, not a new theorem. It points at lectures and snippets that already exist.
The idea in plain words
Brownian motion is the engine the other recipes sit on. The linked lecture runs about forty minutes. One route is a Karhunen-Loève expansion, a series you truncate and simulate. Another is the definition itself: stack normal increments. Python snippets are attached so the same step can live in a project rather than only on a slide.
The Brownian bridge is the path that is pinned at both ends. There is a continuous-time recipe and a discrete-time counterpart, and again a simulation. The page shows empirical covariance against the theoretical structure. That comparison is the point of the format. You are supposed to see the algorithm, then see whether the cloud of paths reproduces the second-moment shape the formula claims.
Fractional Brownian motion is the older article. It uses a Davies-Harte scheme, which is efficient, and it is not causal. The method is a spectral decomposition, so building the path uses knowledge of the whole trajectory. You can still simulate, and the lecture says the scheme works. Without a causal representation, later steps that want to move forward from "now" get stuck.
That is why the Volterra recipe is next. Its simulation needs every increment, the global path, so it is relatively slow. Markovian lifting is the alternative the cookbook implements in code. The lecture notes that the earlier lifting article did not ship code. The cookbook page is the discretization, written out, plus a simulation tab. You can draw smoother paths and rougher ones, set how many Ornstein-Uhlenbeck factors to keep, and watch the empirical covariance move toward the theoretical matrix as the factor count and the path count rise. Changing the roughness dial is part of the same check. The lift is much cheaper than storing the whole path, and it remains an approximation.
How to use it
Pick the process by the constraint you actually have. If the path is free at the right edge, start with Brownian motion and the normal-increment construction. If a price or a rate is pinned at a future date, use the bridge and read the covariance panel before you trust the draws. If you need long memory and you can build the whole path at once, Davies-Harte is the efficient spectral route the page points to. If the next step has to be causal, the Volterra version states the cost, and the lift is the approximation that replaces that cost with a chosen number of mean-reverting factors.
Raise the path count when you compare covariances. A single pretty path will not show whether the empirical matrix has settled on the theoretical one. Raise the factor count on the lift when the rough covariance is still off, and stop when adding factors stops moving the picture. The lecture is explicit that you do not get the exact kernel back.
What usually goes wrong
A spectral fractional path gets dropped into a trading loop that is only allowed to see the present. Davies-Harte was not built as that kind of causal step. The lecture separates it from the Volterra and lifted versions for that reason.
The lift gets treated as exact once the picture looks close. More factors improve the match. They do not remove the word approximation.
The covariance check gets skipped because the paths look plausible. Every recipe in the tour is organized around that check, empirical against theoretical, not around a single sample path.
Questions people ask
Which recipe is the starting point?
Brownian motion. The lecture calls it the engine for the rest, and it shows both a Karhunen-Loève truncation and a construction from normal increments, with Python attached.
Why is Davies-Harte not enough for a live step?
It is a spectral method, so the path is built with global knowledge. It is efficient for simulation. It is not a causal representation, which is why the Volterra page and the lift are there.
Does adding Ornstein-Uhlenbeck factors recover the rough path exactly?
No. The lecture says the empirical covariance moves toward the theoretical one as you add factors and paths, and that the result is still an approximation, bought at a lower cost.