continuous-time stochastic process

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.Parzen, E. (1962) Stochastic Processes, Holden-Day. {{ISBN|0-8162-6664-6}} (Chapter 6)

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. {{ISBN|0-19-920613-9}} (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.{{cite book|last1=Paul|first1=Wolfgang|last2=Baschnagel|first2=Jörg|title=Stochastic Processes: From Physics to Finance|url=https://books.google.com/books?id=OWANAAAAQBAJ&pg=PA72|accessdate=20 June 2022|date=2013-07-11|publisher=Springer Science & Business Media|isbn=9783319003276|pages=72–74}}