Skorokhod's embedding theorem

<h2 id="skorokhods-first-embedding-theorem">Skorokhod's first embedding theorem</h2>
Let X be a <a href="/facts/Real_number/R02gw5Pb">real</a>-valued random variable with <a href="/facts/Expected_value/1XV0JKL8">expected value</a> 0 and finite <a href="/facts/Variance/ULBJKXD1">variance</a>; let W denote a canonical real-valued Wiener process. Then there is a stopping time (with respect to the natural <a href="/facts/Filtration_(abstract_algebra)/jL7AG71G">filtration</a> of W), τ, such that Wτ has the same distribution as X,

E
        ⁡
        [
        τ
        ]
        =
        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}].}

<h2 id="skorokhods-second-embedding-theorem">Skorokhod's second embedding theorem</h2>
Let X1, X2, ... be a sequence of <a href="/facts/Independent_and_identically_distributed_random_variables/othIRaWt">independent and identically distributed random variables</a>, each with expected value 0 and finite variance, and let

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

Then there is a sequence of stopping times τ1 ≤ τ2 ≤ ... such that the 
 
 
 
 
 W
 
 
 τ
 
 n
 
 
 
 
 
 
 {\displaystyle 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

E
        ⁡
        [
        
          τ
          
            n
          
        
        −
        
          τ
          
            n
            −
            1
          
        
        ]
        =
        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}].}

<ul><li>Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-00710-2. (Theorems 37.6, 37.7)</li></ul>

Skorokhod's embedding theorem open-in-new

Skorokhod's embedding theorem