Skorokhod's embedding theorem: Difference between revisions

Revision as of 12:04, 15 March 2018

Revision as of 17:34, 5 April 2020

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

:<math>\mathbboperatorname{E}[\tau] = \mathbboperatorname{E}[X^{2}]</math>

and

:<math>\mathbboperatorname{E}[\tau^{2}] \leq 4 \mathbboperatorname{E}[X^{4}].</math>

==Skorokhod's second embedding theorem==

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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

:<math>S_{n}S_n = X_{1}X_1 + \cdots + X_{n}X_n.</math>

Then there is a sequence of stopping times ''&tau;''₁ &le; ''&tau;''₂ &le; ... such that the <math>W_{\tau_{n}}</math> have the same joint distributions as the partial sums ''S''_''n'' and ''&tau;''₁, ''&tau;''₂ &minus; ''&tau;''₁, ''&tau;''₃ &minus; ''&tau;''₂, ... are independent and identically distributed random variables satisfying

:<math>\mathbboperatorname{E}[\tau_{n}tau_n - \tau_{n - 1}] = \mathbboperatorname{E}[X_{1}X_1^{2}]</math>

and

:<math>\mathbboperatorname{E}[(\tau_{n} - \tau_{n - 1})^{2}] \leq 4 \mathbboperatorname{E}[X_{1}X_1^{4}].</math>

==References==