Skorokhod's embedding theorem

In mathematics and probability theory, Skorokhod's embedding theorem is either or both of two theorems that allow one to regard any suitable collection of random variables as a Wiener process (Brownian motion) evaluated at a collection of stopping times. Both results are named for the Ukrainian mathematician A.V. Skorokhod.

Skorokhod's first embedding theorem

Let X be a real-valued random variable with expected value 0 and finite variance; let W denote a canonical real-valued Wiener process. Then there is a stopping time (with respect to the natural filtration of W), τ, such that Wτ has the same distribution as X,

\mathbb{E}[\tau] = \mathbb{E}[X^{2}]

and

\mathbb{E}[\tau^{2}] \leq 4 \mathbb{E}[X^{4}].

Skorokhod's second embedding theorem

Let X1, X2, ... be a sequence of independent and identically distributed random variables, each with expected value 0 and finite variance, and let

S_{n} = X_{1} + \cdots + X_{n}.

Then there is a sequence of stopping times τ1 ≤ τ2 ≤ ... such that the W_{\tau_{n}} have the same joint distributions as the partial sums Sn and τ1, τ2 − τ1, τ3 − τ2, ... are independent and identically distributed random variables satisfying

\mathbb{E}[\tau_{n} - \tau_{n - 1}] = \mathbb{E}[X_{1}^{2}]

and

\mathbb{E}[(\tau_{n} - \tau_{n - 1})^{2}] \leq 4 \mathbb{E}[X_{1}^{4}].

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