probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

and

statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, 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 in a probability space, where the index of the family often has the interpretation of time. Sto ...

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 alwaysDodge, 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.

References

{{Stochastic processes, state=collapsed Stochastic processes

Examples

See also

References