In
probability theory and
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, a continuous-time stochastic process, or a continuous-space-time stochastic process is a
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
for which the index variable takes a continuous set of values, as contrasted with a
discrete-time process for which the index variable takes only distinct values. An alternative terminology uses continuous parameter as being more inclusive.
A more restricted class of processes are the
continuous stochastic processes; here the term often (but not always
[Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', OUP. (Entry for "continuous process")]) implies both that the index variable is continuous and that sample paths of the process are continuous. Given the possible confusion, caution is needed.
[
Continuous-time stochastic processes that are constructed from discrete-time processes via a waiting time distribution are called continuous-time random walks.]
Examples
An example of a continuous-time stochastic process for which sample paths are not continuous is a Poisson process. An example with continuous paths is the Ornstein–Uhlenbeck process.
See also
* Continuous signal
References
{{Stochastic processes, state=collapsed
Stochastic processes