Continuous-time stochastic process explained

In probability theory and statistics, a continuous-time stochastic process, or a continuous-space-time stochastic process is a stochastic process 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.^[1]

A more restricted class of processes are the continuous stochastic processes; here the term often (but not always^[2]) implies both that the index variable is continuous and that sample paths of the process are continuous. Given the possible confusion, caution is needed.^[2]

Continuous-time stochastic processes that are constructed from discrete-time processes via a waiting time distribution are called continuous-time random walks.^[3]

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

Notes and References

1. Parzen, E. (1962) Stochastic Processes, Holden-Day. (Chapter 6)
2. Dodge, Y. (2006) The Oxford Dictionary of Statistical Terms, OUP. (Entry for "continuous process")
3. Book: Paul. Wolfgang. Baschnagel. Jörg. Stochastic Processes: From Physics to Finance. 20 June 2022. 2013-07-11. Springer Science & Business Media. 9783319003276. 72–74.