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.

Contents

Skorokhod's first embedding theorem

Skorokhod's second embedding theorem

References

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 ''W''_''τ'' has the same distribution as ''X'',
:$mathbb\{E\}[\; au]\; =\; mathbb\{E\}[X^\{2\}]$
and
:$mathbb\{E\}[\; au^\{2\}]\; leq\; 4\; mathbb\{E\}[X^\{4\}].$
(Naturally, the above inequality is trivial unless ''X'' has finite fourth moment.)

Skorokhod's second embedding theorem

Let ''X''₁, ''X''₂, ... 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 non-decreasing (a.k.a. weakly increasing) sequence ''τ''₁, ''τ''₂, ... of stopping times such that the $W\_\{\; au\_\{n\}\}$ have the same joint distributions as the partial sums ''S''_''n'' and ''τ''₁, ''τ''₂ − ''τ''₁, ''τ''₃ − ''τ''₂, ... are independent and identically distributed random variables satisfying
:$mathbb\{E\}[\; au\_\{n\}\; -\; au\_\{n\; -\; 1\}]\; =\; mathbb\{E\}[X\_\{1\}^\{2\}]$
and
:$mathbb\{E\}[(\; au\_\{n\}\; -\; au\_\{n\; -\; 1\})^\{2\}]\; leq\; 4\; mathbb\{E\}[X\_\{1\}^\{4\}].$

References

★ Probability and Measure, , Patrick, Billingsley, John Wiley & Sons, Inc., 1995, ISBN 0-471-00710-2 (Theorems 37.6, 37.7)