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  2. Continuous-time stochastic process - Wikipedia

    en.wikipedia.org/wiki/Continuous-time_stochastic...

    An alternative terminology uses continuous parameter as being more inclusive. [1] A more restricted class of processes are the continuous stochastic processes; here the term often (but not always [2]) implies both that the index variable is continuous and that sample paths of the process are continuous. Given the possible confusion, caution is ...

  3. Stochastic control - Wikipedia

    en.wikipedia.org/wiki/Stochastic_control

    If the model is in continuous time, the controller knows the state of the system at each instant of time. The objective is to maximize either an integral of, for example, a concave function of a state variable over a horizon from time zero (the present) to a terminal time T, or a concave function of a state variable at some future date T. As ...

  4. Jump process - Wikipedia

    en.wikipedia.org/wiki/Jump_process

    Poisson process, an example of a jump process; Continuous-time Markov chain (CTMC), an example of a jump process and a generalization of the Poisson process; Counting process, an example of a jump process and a generalization of the Poisson process in a different direction than that of CTMCs; Interacting particle system, an example of a jump ...

  5. Stochastic process - Wikipedia

    en.wikipedia.org/wiki/Stochastic_process

    If a continuous-time real-valued stochastic process meets certain moment conditions on its increments, then the Kolmogorov continuity theorem says that there exists a modification of this process that has continuous sample paths with probability one, so the stochastic process has a continuous modification or version.

  6. Markov chain - Wikipedia

    en.wikipedia.org/wiki/Markov_chain

    Another discrete-time process that may be derived from a continuous-time Markov chain is a δ-skeleton—the (discrete-time) Markov chain formed by observing X(t) at intervals of δ units of time. The random variables X (0), X (δ), X (2δ), ... give the sequence of states visited by the δ-skeleton.

  7. Discrete time and continuous time - Wikipedia

    en.wikipedia.org/wiki/Discrete_time_and...

    Discrete time is often employed when empirical measurements are involved, because normally it is only possible to measure variables sequentially. For example, while economic activity actually occurs continuously, there being no moment when the economy is totally in a pause, it is only possible to measure economic activity discretely.

  8. Continuous-time Markov chain - Wikipedia

    en.wikipedia.org/wiki/Continuous-time_Markov_chain

    A continuous-time Markov chain (CTMC) is a continuous stochastic process in which, for each state, the process will change state according to an exponential random variable and then move to a different state as specified by the probabilities of a stochastic matrix. An equivalent formulation describes the process as changing state according to ...

  9. Optimal control - Wikipedia

    en.wikipedia.org/wiki/Optimal_control

    The examples thus far have shown continuous time systems and control solutions. In fact, as optimal control solutions are now often implemented digitally , contemporary control theory is now primarily concerned with discrete time systems and solutions.