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Stochastic Process is a mapping from a one-dimensional set T into the set of all possible random variables. Set T is called "time" since, typically, a stochastic process is used to describe a random phenomenon which evolves over time. If T has continuum of values, the process is said to have continuous time. If T is a sequence of numbers, the process is said to have discrete time. In the latter case the stochastic process is also known as a time series.

On the one hand, stochastic process is more general than random variable or random vector. On the other hand, it is a particular case of random field, which is a mapping from any n-dimensional set D into the set of all random variables... Four major types of problems are studied with regard to stochastic processes:

1] calculation of various characteristics of a stochastic process, the characteristics being distribution functions, moments, spectrum of the whole trajectory or its fragments, etc;

2] estimation of the parameters of a stochastic process;

3] filtering a stochastic process;

4] forecasting a stochastic process.

Depending on whether the studied process is defined in discrete time or continuous time, the mathematical techniques are quite different. For that reason time series analysis is regarded as a separate field of statistics. Researchers and practitioners distinguish between several overlapping classes of stochastic processes, which can be studied in depth because of certain simplifying assumptions about their inner structure: stationary processes, Markov processes (chains), martingales, semi-martingales (diffusions, jump-diffusions, etc), hidden Markov models, counting processes, renewal processes, Gaussian processes, Gaussian mixtures (spherically invariant processes) and processes with orthogonal increments.



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