Kolmogorov continuity theorem

<h2 id="statement">Statement</h2>
<p>Let 
  
    
      
        (
        S
        ,
        d
        )
      
    
    {\displaystyle (S,d)}
  
 be some complete separable metric space, and let 
  
    
      
        X
        :
        [
        0
        ,
        +
        ∞
        )
        ×
        Ω
        →
        S
      
    
    {\displaystyle X\colon [0,+\infty )\times \Omega \to S}
  
 be a stochastic process. Suppose that for all times 
  
    
      
        T
        >
        0
      
    
    {\displaystyle T>0}
  
, there exist positive constants 
  
    
      
        α
        ,
        β
        ,
        K
      
    
    {\displaystyle \alpha ,\beta ,K}
  
 such that
</p>

E
        
        [
        d
        (
        
          X
          
            t
          
        
        ,
        
          X
          
            s
          
        
        
          )
          
            α
          
        
        ]
        ≤
        K
        
          |
        
        t
        −
        s
        
          
            |
          
          
            1
            +
            β
          
        
      
    
    {\displaystyle \mathbb {E} [d(X_{t},X_{s})^{\alpha }]\leq K|t-s|^{1+\beta }}

<p>for all 
  
    
      
        0
        ≤
        s
        ,
        t
        ≤
        T
      
    
    {\displaystyle 0\leq s,t\leq T}
  
. Then there exists a modification 
  
    
      
        
          
            
              X
              ~
            
          
        
      
    
    {\displaystyle {\tilde {X}}}
  
 of 
  
    
      
        X
      
    
    {\displaystyle X}
  
 that is a continuous process, i.e. a process 
  
    
      
        
          
            
              X
              ~
            
          
        
        :
        [
        0
        ,
        +
        ∞
        )
        ×
        Ω
        →
        S
      
    
    {\displaystyle {\tilde {X}}\colon [0,+\infty )\times \Omega \to S}
  
 such that
</p>
<ul><li>
  
    
      
        
          
            
              X
              ~
            
          
        
      
    
    {\displaystyle {\tilde {X}}}
  
 is <a href="/facts/Sample-continuous_process/sWFG0MEJ">sample-continuous</a>;</li>
<li>for every time 
  
    
      
        t
        ≥
        0
      
    
    {\displaystyle t\geq 0}
  
, 
  
    
      
        
          P
        
        (
        
          X
          
            t
          
        
        =
        
          
            
              
                X
                ~
              
            
          
          
            t
          
        
        )
        =
        1.
      
    
    {\displaystyle \mathbb {P} (X_{t}={\tilde {X}}_{t})=1.}
  
</li></ul>
<p>Furthermore, the paths of 
  
    
      
        
          
            
              X
              ~
            
          
        
      
    
    {\displaystyle {\tilde {X}}}
  
 are locally <a href="/facts/H%C3%B6lder_continuity/CI0yp6kD">
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  
-Hölder-continuous</a> for every 
  
    
      
        0
        <
        γ
        <
        
          
            
              β
              α
            
          
        
      
    
    {\displaystyle 0<\gamma <{\tfrac {\beta }{\alpha }}}
  
.
</p>
<h2 id="example">Example</h2>
<p>In the case of <a href="/facts/Brownian_motion/Th0g0wkr">Brownian motion</a> on 
  
    
      
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {R} ^{n}}
  
, the choice of constants 
  
    
      
        α
        =
        4
      
    
    {\displaystyle \alpha =4}
  
, 
  
    
      
        β
        =
        1
      
    
    {\displaystyle \beta =1}
  
, 
  
    
      
        K
        =
        n
        (
        n
        +
        2
        )
      
    
    {\displaystyle K=n(n+2)}
  
 will work in the Kolmogorov continuity theorem. Moreover, for any positive integer 
  
    
      
        m
      
    
    {\displaystyle m}
  
, the constants 
  
    
      
        α
        =
        2
        m
      
    
    {\displaystyle \alpha =2m}
  
, 
  
    
      
        β
        =
        m
        −
        1
      
    
    {\displaystyle \beta =m-1}
  
 will work, for some positive value of 
  
    
      
        K
      
    
    {\displaystyle K}
  
 that depends on 
  
    
      
        n
      
    
    {\displaystyle n}
  
 and 
  
    
      
        m
      
    
    {\displaystyle m}
  
.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Kolmogorov_extension_theorem/2vgha79k">Kolmogorov extension theorem</a></li></ul>

<ul><li>Daniel W. Stroock, S. R. Srinivasa Varadhan (1997). <i>Multidimensional Diffusion Processes</i>. Springer, Berlin. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-662-22201-0. p. 51</li></ul>

Kolmogorov continuity theorem open-in-new

Kolmogorov continuity theorem