In
probability theory
Probability theory 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 expressing it through a set ...
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
In 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 (the sample paths of) a process to have ...
es; 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 walk
In mathematics, a continuous-time random walk (CTRW) is a generalization of a random walk where the wandering particle waits for a random time between jumps. It is a stochastic jump process with arbitrary distributions of jump lengths and waiting ...
s.
Examples
An example of a continuous-time stochastic process for which sample paths are not continuous is a Poisson process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
. An example with continuous paths is the Ornstein–Uhlenbeck process
In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
See also
* Continuous signal
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
Discrete time views values of variables as occurring at distinct, separate "po ...
References
{{Stochastic processes, state=collapsed
Stochastic processes