# stochastic process

Stochastic processes are families (also: sets) of random variables X(t), which are characterized by a parameter t. Here, t traverses a suitable parameter space R and is interpreted as time in most applications. T is then a sequence of time points or a time interval. In contrast to "classical" stochastics, which focuses on the temporally static case, the theory of stochastic processes investigates the temporally dynamic aspects in the behavior of random variables.

Since stochastic processes are ultimately generalized random experiments, they are also referred to as random functions or random processes.

To simplify the presentation and in view of the oversized number of applications, in the following the parameter t is interpreted as time in the presentation of the theoretical basics. Thus, X(t) is the random variable of interest at time t, while T covers the entire period under consideration. Z denotes the set of all states (also called realizations) that X(t) can assume.

## Definition of the stochastic process

A stochastic process with parameter space T and state space Z is the set of random variables.

If the parameter set is finite or countably infinite, it is called a stochastic process with discrete time. Such processes can be written down in sequence of random variables: {X(1), X(2) ... }. Conversely, any sequence of random variables can be interpreted as a stochastic process with discrete time. If T is an interval, then it is called a stochastic process with continuous time.

The stochastic process (see figure) is called discrete if its state space Z is a finite or a countably infinite set. In contrast, a continuous stochastic process exists if Z is an interval.

## The time relation of stochastic processes

There are discrete stochastic processes with discrete time, discrete stochastic processes with continuous time, continuous stochastic processes with discrete time and continuous stochastic processes with continuous time.

To capture possible dependencies of random quantities on deterministic parameters, one passes from the concept of random quantity to that of stochastic process (also: random process). For example, the movement of the molecules of a gas, the water level in a reservoir, wind excitations, earthquake excitations, the number of customers in a queue, the number of telephone calls, precipitation quantity and many others, considered as a function of time, are stochastic processes. Practically significant types of stochastic processes are Markov processes, Gaussian processes, stationary processes, Poisson processes, Wieners processes, diffusion processes, processes with independent increments, martingales, branching processes and point processes.

The classification of stochastic processes can be done according to various features, such as the distribution functions, the covariance functions, the parameter set and the realizations.