Stochastic process

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The Brownian Bridge , a stochastic process

A stochastic process (also random process ) is the mathematical description of temporally ordered, random processes. The theory of stochastic processes represents an essential extension of probability theory and forms the basis for stochastic analysis . Although simple stochastic processes were studied a long time ago, the formal theory that is valid today was not developed until the beginning of the 20th century, primarily by Paul Lévy and Andrei Kolmogorow .


Let be a probability space , a space provided with a σ-algebra (mostly the real numbers with Borel's σ-algebra ) and an index set , mostly which in applications often represents the set of the observed points in time. A stochastic process is then a family of random variables , i.e. a mapping


so that there is a - - measurable image for everyone . The set is also called the state space of the process; it contains all values ​​that the process can assume.

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

The question of the existence of stochastic processes with certain properties is largely solved with the theorem of Daniell-Kolmogorow and the theorem of Ionescu-Tulcea (named after Cassius Ionescu-Tulcea ).


Below are some criteria according to which stochastic processes are classified. A more detailed description can be found in the list of stochastic processes .

The most basic division of stochastic processes into different classes takes place via the index set and the value set :

Discrete and continuous index set

  • If countable (for example ), the process is called a time-discrete stochastic process or a somewhat imprecise discrete stochastic process
  • Otherwise the process is called a time-continuous stochastic process .

Discrete and constant set of values

  • If finite or countable, one speaks of value-discrete processes.
  • If , one speaks of a real-valued process .

Multi-dimensional index set

  • Then the stochastic process is often called random field, random field or English. random field. Often is or , especially for geostatistical models .


In addition, stochastic processes are classified analogously to random variables according to whether the expected value and the variance exist or assume special values.

  • A real-valued stochastic process is called integrable if applies to all .
  • A real-valued stochastic process is called integrable square if applies to all .
  • A real-valued stochastic process is called centered if applies to all .

Stochastic dependencies

Furthermore, stochastic processes are classified by means of the structure of their stochastic dependencies; these are usually defined via the conditional expected value . These classes include:

Markov trials
Their likelihood of entering a state depends on the state they were in before it, but not on the entire past of the process. Markov processes thus have a "short memory".
Martingales as well as sub- and supermartingales
Martingales model a fair game. If you have already won a certain amount at a point in time, the expected value for future profits is precisely this amount already won.

Other properties: paths and increments

Furthermore, processes can be classified as follows:

  • One can examine the properties of the paths and subdivide the processes accordingly: processes with continuous paths, processes with limited paths, etc. An example of a stochastic process with almost certainly continuous paths is the Wiener process .
  • One considers the so-called increments of the process, i.e. terms of the kind for indices . Depending on the required property of the increments, one then obtains processes with stationary increments , processes with independent increments or also processes with normally distributed increments. For example, the Lévy processes are precisely the stochastic processes with independent, stationary increments.


For each one you get a picture . These maps are called the paths of the process. Often, instead of the paths, one also speaks of the trajectories or the realizations of the stochastic process.

If special and (or a more general topological space ), one can speak of continuity properties of the paths. A time- continuous stochastic process is called continuous , right-side continuous , left-side continuous or càdlàg , if all paths of the process have the corresponding property. The Wiener process has continuous paths, two of which can be seen in the picture for the examples below. The Poisson process is an example of a time-continuous, value-discrete càdlàg process ; so it has continuous paths on the right-hand side where the left-hand limes exist 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 operates largely independently of it. By definition, stochastic processes and time series are one and the same, but the areas show differences: While time series analysis is a sub-area of statistics and tries to adapt special models (such as ARMA models ) to time-ordered data, the stochastic processes stand for Stochastics and the special structure of the random functions (such as continuity , differentiability , variation or measurability with regard to certain filterings ) are in the foreground.


A Standard Wiener process on the time interval [0.3]; the expected value and the standard deviation are also shown

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