title: Sample-continuous process
text: # Sample-continuous process
  
  In mathematics, a sample-continuous process is a stochastic process whose sample paths are almost surely continuous functions.
  
  ## Definition
  
  Let (Ω, Σ, P) be a probability space. Let X : I × Ω → S be a stochastic process, where the index set I and state space S are both topological spaces. Then the process X is called sample-continuous (or almost surely continuous, or simply continuous) if the map X(ω) : I → S is continuous as a function of topological spaces for P-almost all ω in Ω.
  
  In many examples, the index set I is an interval of time, [0, T] or [0, +∞), and the state space S is the real line or n-dimensional Euclidean space $R^n.$
  
  ## Examples
  
  * Brownian motion (the Wiener process) on Euclidean space is sample-continuous.
  * For "nice" parameters of the equations, solutions to stochastic differential equations are sample-continuous. See the existence and uniqueness theorem in the stochastic differential equations article for some sufficient conditions to ensure sample continuity.
  * The process X : [0, +∞) × Ω → R that makes equiprobable jumps up or down every unit time according to
  
         ${\begin{cases}X_{t}\sim \mathrm {Unif} (\{X_{t-1}-1,X_{t-1}+1\}),&t{\mbox{ an integer;}}\\X_{t}=X_{\lfloor t\rfloor },&t{\mbox{ not an integer;}}\end{cases}}$
      is not sample-continuous. In fact, it is surely discontinuous.
  
  ## Properties
  
  * For sample-continuous processes, the finite-dimensional distributions determine the law, and vice versa.
  