Skorokhod's embedding theorem explained

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 W_τ has the same distribution as X,

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

and

\operatorname{E}[\tau ^{2]} \leq 4 \operatorname{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} + … + 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 S_n and τ₁, τ₂ - τ₁, τ₃ - τ₂, ... are independent and identically distributed random variables satisfying\operatorname{E}[\tau _{n} - \tau _{n} ] = \operatorname{E}[X ^{2 ]} _{1}

and

\operatorname{E}[(\tau _{n} - \tau _{n} ) ^{2]} \le 4 \operatorname{E}[X ^{4 ].} _{1}

References

• Book: Billingsley, Patrick . Probability and Measure . John Wiley & Sons, Inc. . New York . 1995 . 0-471-00710-2. (Theorems 37.6, 37.7)