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In mathematics, the theory of stochastic processes is an important contribution to probability theory, [29] and continues to be an active topic of research for both theory and applications. [30] [31] [32] The word stochastic is used to describe other terms and objects in mathematics.
An increment of a stochastic process is the difference between two random variables of the same stochastic process. For a stochastic process with an index set that can be interpreted as time, an increment is how much the stochastic process changes over a certain time period.
A stochastic simulation is a simulation of a system that has variables that can change stochastically (randomly) with individual probabilities. [ 1 ] Realizations of these random variables are generated and inserted into a model of the system.
In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty. A stochastic program is an optimization problem in which some or all problem parameters are uncertain, but follow known probability distributions .
If a stochastic process is strict-sense stationary and has finite second moments, it is wide-sense stationary. [2]: p. 299 If two stochastic processes are jointly (M + N)-th-order stationary, this does not guarantee that the individual processes are M-th- respectively N-th-order stationary. [1]: p. 159
In probability theory and statistics, a stochastic order quantifies the concept of one random variable being "bigger" than another. These are usually partial orders , so that one random variable A {\displaystyle A} may be neither stochastically greater than, less than, nor equal to another random variable B {\displaystyle B} .
Especially non-stochastic and non-Bayesian signal processing, without any hidden variables. 2. Estimation: The smoothing problem (or Smoothing in the sense of estimation) uses Bayesian and state-space models to estimate the hidden state variables.
Stochastic optimization (SO) are optimization methods that generate and use random variables. For stochastic optimization problems, the objective functions or constraints are random. Stochastic optimization also include methods with random iterates .