Skorokhod Embedding

Every mean-zero law hides inside a Brownian path — you only have to know when to stop watching.

The idea

Theorem (Skorokhod's embedding theorem).

Let $\mu$ be a probability distribution on the real line with mean $0$ and finite variance, and let $W$ be a standard Brownian motion started at $0$. Then there is a stopping time $T$ such that $W(T)$ has exactly the distribution $\mu$.

A stopping time is a random time whose decision "have we stopped by now?" uses only the path seen so far, never the future. In other words, the theorem says we can produce any such distribution from Brownian motion by choosing when to stop watching it. Nothing is added to the process and nothing is transformed; the stopping rule alone shapes the law of $W(T)$.

The mean-zero hypothesis is forced: the Optional Stopping Theorem freezes the mean of a stopped Brownian motion at its start, $\mathbb{E}[W(T)] = W(0) = 0$, so a law with nonzero mean can never be the law of $W(T)$. The content of the theorem is that no further obstruction exists.

The embedding lets arguments run from Brownian motion to $\mu$: any fact about $W$ at a stopping time becomes a fact about $\mu$. Applied repeatedly, it plants an entire random walk — a sum of independent draws from $\mu$ — inside one Brownian path, which is the standard route to comparing the two processes.

Ways to work on it

• Walkthrough. The embedding question, the two-point exit construction, and why the expected stopping time equals the target's variance.
• Proof. Verify the optional stopping hypotheses once, then derive the exit law and the expected duration.
• Practice. Exit probabilities and expected embedding times, from an interval and from a target law.
• Hardest. Embed a target law with three atoms and compute the expected embedding time.

Not sure where to start? Take the ten-question placement test.