In probability theory, a stochastic (/stoʊˈkæstɪk/) process, or often random process, is a collection of random variables, representing the evolution of some system of random values over time. This is the probabilistic counterpart to a deterministic process (or deterministic system). Instead of describing a process which can only evolve in one way (as in the case, for example, of solutions of an ordinary differential equation), in a stochastic or random process there is some indeterminacy: even if the initial condition (or starting point) is known, there are several (often infinitely many) directions in which the process may evolve.
In the simple case of discrete time, as opposed to continuous time, a stochastic process is a sequence of random variables (for example, see Markov chain, also known as discrete-time Markov chain). The random variables corresponding to various times may be completely different, the only requirement being that these different random quantities all take values in the same space (the codomain of the function). One approach may be to model these random variables as random functions of one or several deterministic arguments (in most cases, the time parameter). Although the random values of a stochastic process at different times may be independent random variables, in most commonly considered situations they exhibit complicated statistical dependence.
In probability theory and statistics, a discrete-time stochastic process is a stochastic process for which the index variable takes a discrete set of values, as contrasted with a continuous-time process for which the index variable takes values in a continuous range. An alternative terminology uses discrete parameter as being more inclusive.
A more restricted class of processes are those with discrete time and discrete state space. The apparently simpler terms "discrete process" or "discontinuous process" may cause confusion with processes having continuous time and discrete state space. Given the possible confusion, caution is needed.
Examples of discrete-time stochastic processes are random walks and branching processes, for which the state space may be either continuous or discrete. Important examples of discrete time and continuous state space processes are models conventionally used in time series analysis: for example, the autoregressive, vector autoregressive, moving average, ARMA, ARIMA and ARCH models.