Continuous-time stochastic process
Encyclopedia
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.
A more restricted class of processes are the continuous stochastic process
es: here the term often (but not always) implies both that the index variable is continuous and that sample paths of the process are continuous. Given the possible confusion, caution is needed.
. An example with continuous paths is the Ornstein–Uhlenbeck process.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, a continuous-time stochastic process, or a continuous-space-time stochastic process is a stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic 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.
A more restricted class of processes are the continuous stochastic process
Continuous stochastic process
In the probability theory, a continuous stochastic process is a type of stochastic process that may be said to be "continuous" as a function of its "time" or index parameter. Continuity is a nice property for a process to have, since it implies that they are well-behaved in some sense, and,...
es: here the term often (but not always) implies both that the index variable is continuous and that sample paths of the process are continuous. Given the possible confusion, caution is needed.
Examples
An example of a continuous-time stochastic process for which sample paths are not continuous is a Poisson processPoisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...
. An example with continuous paths is the Ornstein–Uhlenbeck process.