Stochastic process

A stochastic process is a mathematical description of time-ordered, random processes. The theory of stochastic processes provides a substantial extension of probability theory, and forms the basis for the stochastic analysis. Although simple stochastic processes were studied long ago, todays valid formal theory was not developed until the early 20th century, especially by Paul Lévy and Andrei Kolmogorov.

Definition

Let be a probability space, a σ - algebra provided with a space (usually the real numbers with the Borel σ - algebra) and an index set, mostly. A stochastic process is then a family of random variables, ie a mapping

Such that for all a - is measurable map. An alternative formulation provides that a single random variable, one ( provided with a suitable σ - algebra ) set of functions. With suitable choice of these two definitions coincide.

The question of the existence of stochastic processes with certain properties is released from the sets of Daniell - Kolmogorov and Ionescu Tulcea largely.

Classification

The most basic classification of stochastic processes in different classes is via the index set and the set of values ​​:

• Is countable ( approximately ), so the process is called discrete-time, otherwise time ever.
• Is finite or countable, one speaks of value-discrete processes or point processes.

In addition, stochastic processes are sub-divided according to stochastic properties in various process classes. The most important class here is that of Markov processes, which are characterized by a kind of " memoryless ". The most studied processes belong to this class. Within the Markov processes ( the discrete-time case, one speaks of Markov chains ) are again the Lévy processes of importance, representing a stochastic equivalent of the linear maps. More process classes are martingales, Gaussian processes and Ito processes.

Paths

For each you get a picture. These images are called the paths of the process. Frequently it is called instead of the paths by the trajectories or the realizations of the stochastic process.

Is specifically and (or a more general topological space ), so one can speak of continuity properties of the paths. This is called a continuous-time stochastic process continuous, right-continuous, left -continuous or càdlàg if all paths of the process have the corresponding property. The Wiener process has continuous paths, down two of which are seen in the picture to the examples. The Poisson process is an example of a continuous-time, discrete-value càdlàg process; So he's right -sided continuous paths, where the left-hand limit exists at every point.

Stochastic processes versus time series

In addition to the theory of stochastic processes, there is also the mathematical discipline of time series analysis, which largely operates independently. By definition, stochastic processes and time series one and the same, yet shall designate the areas differences: While the time series analysis is understood as a branch of statistics and tries to specific models (such as ARMA models ) to time-ordered data to adapt, is in the stochastic processes, the Stochastics and the special structure of random functions (such as continuity, differentiability, measurability or variation in relation to certain filtrations ) in the foreground.

Examples

• A simple example of a discrete-time point process is the symmetric random walk, here illustrated by a game: A player begins at the time with an initial investment of 10 euros a game in which he successively repeatedly tossing a coin. In the "head", he won a Euro in " number" losing an. The account balance after t games is now a stochastic process ( with deterministic initial distribution ). More specifically, it is X is a Lévy process and a martingale.
• One of the most important stochastic processes (also called " Brownian motion ") of the Wiener process. The individual states are normally distributed with variance linearly anwachsender. The Wiener process is used in the stochastic integration, financial mathematics and physics.

• Other examples: Bernoulli process, Brownian bridge, Brownian motion Cracked, Markov chain, Ornstein - Uhlenbeck process, Poisson process, white noise