Skorokhod's embedding theorem (original) (raw)

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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

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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,

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

and

E ⁡ [ τ 2 ] ≤ 4 E ⁡ [ X 4 ] . {\displaystyle \operatorname {E} [\tau ^{2}]\leq 4\operatorname {E} [X^{4}].} {\displaystyle \operatorname {E} [\tau ^{2}]\leq 4\operatorname {E} [X^{4}].}

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

S n = X 1 + ⋯ + X n . {\displaystyle S_{n}=X_{1}+\cdots +X_{n}.} {\displaystyle S_{n}=X_{1}+\cdots +X_{n}.}

Then there is a sequence of stopping times _τ_1 ≤ _τ_2 ≤ ... such that the W τ n {\displaystyle W_{\tau _{n}}} {\displaystyle W_{\tau _{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

E ⁡ [ τ n − τ n − 1 ] = E ⁡ [ X 1 2 ] {\displaystyle \operatorname {E} [\tau _{n}-\tau _{n-1}]=\operatorname {E} [X_{1}^{2}]} {\displaystyle \operatorname {E} [\tau _{n}-\tau _{n-1}]=\operatorname {E} [X_{1}^{2}]}

and

E ⁡ [ ( τ n − τ n − 1 ) 2 ] ≤ 4 E ⁡ [ X 1 4 ] . {\displaystyle \operatorname {E} [(\tau _{n}-\tau _{n-1})^{2}]\leq 4\operatorname {E} [X_{1}^{4}].} {\displaystyle \operatorname {E} [(\tau _{n}-\tau _{n-1})^{2}]\leq 4\operatorname {E} [X_{1}^{4}].}