Continuous-time stochastic process

A continuous-time stochastic process is a type of stochastic process where the time index t is a continuous variable. This means that the process can take on values at any point in time, rather than being restricted to discrete time intervals. The process is stochastic, meaning that it involves random variables that change over time. Examples of continuous-time stochastic processes include Brownian motion and Poisson processes.From: Batch Distillation [2019]

Stochastic differential equations

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Published in Alfio Borzì, Modelling with Ordinary Differential Equations, 2020

Alfio Borzì

Notice that the random values in Y share the same sample space Ω, and map to the space R equipped with its Borel σ-algebra. The set X can be discrete, e.g., X=Z, or continuous, e.g., X = [0, ∞). Usually, the variable x has the meaning of time (we use x, t, s, interchangeably). In this case, if X is discrete, we have a discrete-time stochastic process and, if X is continuous, we have a continuous-time stochastic process. Our random walk is a discrete-time discrete-space stochastic process.

Robust L ₂-L _∞ Controller Design of a Class of Uncertain Stochastic Systems with Wiener Process Disturbances

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Published in IETE Journal of Research, 2022

Feng Xue, Hong-hai Wang, De-xiang Zhang

In Equations (1), (4), we model the stochastic systems which contain unknown disturbance and Wiener process input. In fact, the Wiener process is a continuous-time stochastic process which can also be called standard Brownian motion process or Brownian motion. It can be used to represent the integral of a white noise Gaussian process. Therefore, it is useful as the modeling noise in electronics engineering (see Brownian noise), instrument errors in filtering theory and unknown forces in control theory.

Continuous-time stochastic process

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Robust L 2-L ∞ Controller Design of a Class of Uncertain Stochastic Systems with Wiener Process Disturbances

Robust L ₂-L _∞ Controller Design of a Class of Uncertain Stochastic Systems with Wiener Process Disturbances