Introduction
The Skorokhod Embedding theorem says the following:
With this theorem, one can embed any random walk whose increments are given by the distribution of into a Brownian motion, allowing us to study random walks through studying Brownian motion.
Skorokhod, however, did not prove this statement exactly. He relied on “external randomisation” to achieve this result. Dubins however realised the distribution of within the natural filtration of Brownian motion itselfessentially proving a manner of probability integral transform for Brownian motion.
Despite the clear significance of the theorem, I have not managed to find an exact rigorous treatment of the proof. Proofs can be found in Mörters and Peres (2010), Dubins (1968), and Meyer (1971) among other resources, but they all have faults in rigour somewhere or the other, and the steps to make it rigorous is not completely straightforward either. Some of the unjustified statements actually require the optionally stopping theorem which is nowhere mentioned in these proofs. I present a clear formalisation.
Preliminary results
First, we shall state Wald’s lemma (for a proof, see Mörters and Peres (2010, theorems 2.44 and 2.48)), which allows us to prove the embedding theorem for simple random walks.
Now, we shall state some standard results on martingales. Their proofs can be found in Durret (2019, theorems 4.4.6, 4.6.3) and Mörters and Peres (2010, proposition 2.42).
Dubins’ embedding
Dubins (1968) showed how a random variable having finite variance can be approximated by employing a sort of binary search algorithm. Consider a random variable . We let to be the expectation of i.e., our “best guess” when no other information is present. Now, add the information whether or not. Our next “best guess” is the expectation on conditioned on this information. We repeat this for successive . The resultant sequence is a uniformly integrable martingale that converges to both almost surely and in .
We will show that our sequence of guesses form a binary-splitting martingale.