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1. Stochastic process: random process, probability process, random function of time A process (that is, a variation with time of the state of a certain system) whose course depends on chance and for which probabilities for some courses are given. A typical example ... (Mathematics) [100%] 2024-01-12 [Stochastic processes]
2. Stochastic process: A stochastic process, or sometimes random process, is the counterpart of a deterministic process (or deterministic system) considered in probability theory. Instead of dealing only with one possible 'reality' of how the process might evolve under time (as it is ... [100%] 2023-08-24
3. Stochastic process: In probability theory and related fields, a stochastic (/stəˈkæstɪk/) or random process is a mathematical object usually defined as a sequence of random variables, where the index of the sequence has the interpretation of time. Stochastic processes are widely used ... (Collection of random variables) [100%] 2023-10-02 [Stochastic processes] [Stochastic models]...
4. Stochastic process: In probability theory and related fields, a stochastic (/stəˈkæstɪk/) or random process is a mathematical object usually defined as a sequence of random variables, where the index of the sequence has the interpretation of time. Stochastic processes are widely used ... (Collection of random variables) [100%] 2023-11-08 [Stochastic processes] [Stochastic models]...
5. Stochastic process: A stochastic process is a probabilistic model of a system that evolves non-deterministically. Statisticians determine the system by approximating a probability distribution, which assigns a level of certainty to particular evolutions. [100%] 2023-02-19 [Statistics]
6. Quantum stochastic processes: Quantum theory emerged as a new mechanics, but it was soon realized that it was also a new probability theory. The difference between classical and quantum probability is usually taken to be the fact that in the former probabilities of ... (Mathematics) [93%] 2023-10-06
7. Stochastic portfolio theory: Stochastic portfolio theory (SPT) is a mathematical theory for analyzing stock market structure and portfolio behavior introduced by E. Robert Fernholz in 2002. (Mathematical theory for analyzing stock market structure and portfolio behavior) [88%] 2023-10-04 [Mathematical theorems] [Portfolio theories]...
8. Stochastic processes and boundary value problems: In mathematics, some boundary value problems can be solved using the methods of stochastic analysis. Perhaps the most celebrated example is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. [84%] 2023-09-29 [Partial differential equations] [Stochastic differential equations]...
9. Stochastic process, renewable: innovation stochastic process A stochastic process with a fairly "simple" structure, constructed from an input process and containing all necessary information about this process. Innovation stochastic processes have been used in the problem of linear prediction of stationary time series ... (Mathematics) [81%] 2023-10-19
10. Additive stochastic process: A real-valued stochastic process $ X = \{ {X ( t ) } : {t \in \mathbf R _ {+} } \} $ such that for each integer $ n \geq 1 $ and $ 0 \leq t _ {0} < \dots < t _ {n} $ the random variables $ X ( t _ {0} ) ,X ( t ... (Mathematics) [81%] 2023-10-18
11. Stochastic process, differentiable: A stochastic process $ X ( t) $ such that the limit $$ \lim\limits _ {\Delta t \rightarrow 0 } \ \frac{X ( t + \Delta t ) - X ( t) }{\Delta t } = \ X ^ \prime ( t) $$ exists; it is called the derivative of the stochastic process $ X ( t ... (Mathematics) [81%] 2023-10-02
12. Stochastic process, compatible: adapted stochastic process A family of random variables $X=(X_t(\omega))_{t\geq0}$ defined on a measurable space $(\Omega,\mathcal F)$, with an increasing family $\mathbf F=(\mathcal F_t)_{t\geq0}$ of sub-$\sigma$-fields $\mathcal F_t\subseteq ... (Mathematics) [81%] 2023-10-28
13. Stochastic point process: point process A stochastic process corresponding to a sequence of random variables $ \{ t _ {i} \} $, $ \dots < t _ {-} 1 < t _ {0} \leq 0 < t _ {1} < t _ {2} < \dots $, on the real line $ \mathbf R ^ {1} $. Each value ... (Mathematics) [81%] 2024-01-12
14. Stochastic process, generalized: A stochastic process $ X $ depending on a continuous (time) argument $ t $ and such that its values at fixed moments of time do not, in general, exist, but the process has only "smoothed values" $ X ( \phi ) $ describing the results of measuring ... (Mathematics) [81%] 2023-10-22
15. Stationary stochastic process: stochastic process, homogeneous in time A stochastic process $ X( t) $ whose statistical characteristics do not change in the course of time $ t $, i.e. are invariant relative to translations in time: $ t \rightarrow t + a $, $ X( t) \rightarrow X( t ... (Mathematics) [81%] 2023-11-14 [Stochastic processes]
16. Controlled stochastic process: A stochastic process whose probabilistic characteristics may be changed (controlled) in the course of its evolution in pursuance of some objective, normally the minimization (maximization) of a functional (the control objective) representing the quality of the control. Various types of ... (Mathematics) [81%] 2023-12-02
17. Stochastic processes, filtering of: filtration of stochastic processes The problem of estimating the value of a stochastic process $ Z ( t) $ at the current moment $ t $ given the past of another stochastic process related to it. For example, estimate a stationary process $ Z ( t) $ given ... (Mathematics) [81%] 2023-11-15
18. Stochastic processes, prediction of: extrapolation of stochastic processes The problem of estimating the values of a stochastic process $ X ( t) $ in the future $ t > s $ from its observed values up to the current moment of time $ s $. Usually one has in mind the extrapolation ... (Mathematics) [81%] 2023-10-21
19. Stochastic processes, interpolation of: The problem of estimating the values of a stochastic process $ X ( t) $ on some interval $ a < t < b $ using its observed values outside this interval. Usually one has in mind the interpolation estimator $ \widehat{X} ( t) $ for which the mean ... (Mathematics) [81%] 2023-09-11
20. Statistical problems in the theory of stochastic processes: A branch of mathematical statistics devoted to statistical inferences on the basis of observations represented as a random process. In the most common formulation, the values of a random function $ x( t) $ for $ t \in T $ are observed, and on ... (Mathematics) [80%] 2023-10-13