Stochastic process / Time series analysis

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A stochastic process is the mathematical description of temporally arranged, coincidental procedures. The theory of the stochastic processes represents a substantial extension of the probability theory and forms the basis for the stochastic analysis. Although simple stochastic processes were already long ago studied, the today valid formal theory became only at the beginning 20. Century develops, particularly by Paul and Andrei Nikolajewitsch Kolmogorow.

Definition

Are (\ omega, \ mathcal {F}, P) a probability area, (Z, \ mathcal {Z}) an area provided with a sigma algebra (mostly the real numbers with Borel sigma algebra) and T an index quantity, mostly T \ in \ {\ N_0, \ R_ {+} \}. A stochastic process X is then a family of variates X_ t:\; \ Omega \ tons of Z, \; t \ in T, thus an illustration

of X:\Omega \times T \ tons of Z, \; (\ omega, t) \ mapsto X_t (\ omega),

so that for all t \ in T the reduced illustration X_ t:\; \ omega \ mapsto X_t (\ omega) \; \ mathcal {F} - \ mathcal {Z} - is measurable. An alternative formulation plans that X is only one variate \ omega \ ton (H, \ mathcal {H}), whereby H \ subseteq Z^T (with a suitable sigma algebra provided) a quantity of functions f: T \ tons of Z is. With suitable choice these two definitions collapse.

Organization

The most fundamental organization of stochastic processes into different classes is made by the index quantity of T and the worth tightness Z:

If T (about T= \ N_0) is countable, then the process is called time-discretely, otherwise time constant.
If Z is finite or countable, one speaks of worth-constant processes or point processes.

Beyond that stochastic processes are divided still after stochastic characteristics into different process classes. The most important class is here those of the Markov processes, which are characterised by a kind Most examined processes belong to this class. Within the Markov processes (in the time-constant case one speaks also of Markov chains) are again the processes of importance, which represent a stochastic equivalent to the linear illustrations. Further process classes are Martingale, Gauss processes and Ito processes

Stochastic processes versus time series

Apart from the theory of the stochastic processes there is also the mathematical discipline of the time series analysis, which operates to a large extent independently of it. Stochastic processes and time series are by definition in and the same, yet the areas exhibit differences: While the time series analysis understands itself and tries as subsection of the statistics to adapt special models (as for instance ARMA models) to temporally arranged data the stochastics and the special structure of the Zufallsfunktionen (about steadiness, differentiability variation or measurability concerning certain filtrations) in the foreground stand with the stochastic processes.

Examples

A simple example of a time-discrete point process is the symmetrical random mills, here illustrated with a gambling: a player begins a play at the time t=0 with a starting capital of 10", with which he throws successively a coin again and again. With "head" it wins a euro, with "number" loses it one. The account balance X_t, \; t \ in \ N_0nach t plays is now a stochastic process (with deterministic starting distribution of X_0=10). More exactly regarded it concerns with X around a process, and a Martingal.

in the theory most important stochastic process is surely the Viennese process (also "Brownian movement" mentioned). Here the individual conditions are normaldistributed with linear increasing variance. The Viennese process applies in the stochastic integration, mathematics of finance and physics.