Let be a probability space. A stochastic process is a collection
of random variables defined on , where is a set, called the index set of the process . is usually (but not always) a subset of . is sometimes known as a random function.
Given any , the possible values of are called the states of the process at . The set of all states (for all ) of a stochastic process is called its state space.
If is discrete, then the stochastic process is a discrete-time process. If is an interval of , then is a continuous-time process. If can be linearly ordered, then is also known as the time.
A stochastic process with state space can be thought of in either of following three ways.
As a collection of random variables, , for each in the index set .
As a function in two variables and ,
The process is said to be measurable, or, jointly measurable if it is -measurable. Here, and are the Borel -algebras on and respectively.
In terms of the sample paths. Each maps to a function
Many common examples of stochastic processes have sample paths which are either continuous or cadlag.
Examples. The following list is some of the most common and important stochastic processes:
a random walk, as well as its limiting case, a Brownian motion, or a Wiener process
Markov process; a Markov chain is a Markov process whose state space is discrete
Sometimes, a stochastic process is also called a random process, although a stochastic process is generally linked to any “time” dependent process. In a random process, the index set may not be linearly ordered, as in the case of a random field, where the index set may be, for example, the unit sphere .
In statistics, a stochastic process is often known as a time series, where the index set is a finite (or at most countable) ordered sequence of real numbers.
|Date of creation||2013-03-22 14:39:10|
|Last modified on||2013-03-22 14:39:10|
|Last modified by||gel (22282)|