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A Markov process is a stochastic process that satisfies the Markov property ... second-order Markov effects may also play a role in the growth of some polymer chains.
Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. [1] [2] A stationary Gauss–Markov process is unique [citation needed] up to rescaling; such a process is also known as an Ornstein–Uhlenbeck process.
Markov processes are stochastic processes, traditionally in discrete or continuous time, that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is ...
If a stochastic process is N-th-order stationary, then it is also M-th-order stationary for all . If a stochastic process is second order stationary (=) and has finite second moments, then it is also wide-sense stationary. [1]: p. 159
A Markov decision process is a Markov chain in which state transitions depend on the current state and an action vector that is applied to the system. Typically, a Markov decision process is used to compute a policy of actions that will maximize some utility with respect to expected rewards.
The "Markov" in "Markov decision process" refers to the underlying structure of state transitions that still follow the Markov property. The process is called a "decision process" because it involves making decisions that influence these state transitions, extending the concept of a Markov chain into the realm of decision-making under uncertainty.
[1] [2]: 10 It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix. The stochastic matrix was first developed by Andrey Markov at the beginning of the 20th century, and has found use throughout a wide variety of scientific fields, including probability theory , statistics, mathematical finance and ...
A process is a semimartingale on , if for every () the random variable () is an -semimartingale, i.e. the composition of any smooth function with the process is a real-valued semimartingale. It can be shown that any M {\displaystyle M} -semimartingale is a solution of a stochastic differential equation on M {\displaystyle M} .