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Subject Item
dbr:Continuous-time_Markov_chain
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Markovprocess Processus de Markov à temps continu Continuous-time Markov chain Ланцюги Маркова з неперервним часом
rdfs:comment
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 the least value of a set of exponential random variables, one for each possible state it can move to, with the parameters determined by the current state. En théorie des probabilités, un processus de Markov à temps continu, ou chaîne de Markov à temps continu est une variante à temps continu du processus de Markov. Plus précisément, c'est un modèle mathématique à valeur dans un ensemble dénombrable, les états, dans lequel le temps passé dans chacun des états est une variable aléatoire réelle positive, suivant une loi exponentielle. Cet objet est utilisé pour modéliser l'évolution de certains systèmes, comme les files d'attente. En Markovprocess, uppkallad efter den ryske matematikern Markov, är inom matematiken en tidskontinuerlig stokastisk process med Markovegenskapen, det vill säga att processens förlopp kan bestämmas utifrån dess befintliga tillstånd utan kännedom om det förflutna. Det tidsdiskreta fallet kallas en Markovkedja. I stället för Markovkedjans övergångssannoliketer har Markovprocessen övergångsintensiteter och tiderna mellan övergångarna mellan tillstånden är exponentialfördelade. Teorin för Markovprocesser används bl. a. inom fysik, ekonomi och reglerteknik. У теорії ймовірностей ланцюгом Маркова з неперервним часом називається випадковий процес { X(t) : t ≥ 0 } визначений у неперервному часовому проміжку, що приймає значення у деякій скінченній чи зліченній множині і задовольняє . Відмінність цього виду ланцюгів Маркова від дискретних ланцюгів Маркова полягає в тому, що переходи між станами можуть відбуватися в будь-які моменти часу і час наступного переходу теж є випадковою величиною.
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Theorem: Existence of solution to Kolmogorov backward equations.
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En théorie des probabilités, un processus de Markov à temps continu, ou chaîne de Markov à temps continu est une variante à temps continu du processus de Markov. Plus précisément, c'est un modèle mathématique à valeur dans un ensemble dénombrable, les états, dans lequel le temps passé dans chacun des états est une variable aléatoire réelle positive, suivant une loi exponentielle. Cet objet est utilisé pour modéliser l'évolution de certains systèmes, comme les files d'attente. У теорії ймовірностей ланцюгом Маркова з неперервним часом називається випадковий процес { X(t) : t ≥ 0 } визначений у неперервному часовому проміжку, що приймає значення у деякій скінченній чи зліченній множині і задовольняє . Відмінність цього виду ланцюгів Маркова від дискретних ланцюгів Маркова полягає в тому, що переходи між станами можуть відбуватися в будь-які моменти часу і час наступного переходу теж є випадковою величиною. 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 the least value of a set of exponential random variables, one for each possible state it can move to, with the parameters determined by the current state. An example of a CTMC with three states is as follows: the process makes a transition after the amount of time specified by the holding time—an exponential random variable , where i is its current state. Each random variable is independent and such that , and . When a transition is to be made, the process moves according to the jump chain, a discrete-time Markov chain with stochastic matrix: Equivalently, by the property of competing exponentials, this CTMC changes state from state i according to the minimum of two random variables, which are independent and such that for where the parameters are given by the Q-matrix Each non-diagonal entry can be computed as the probability that the jump chain moves from state i to state j, divided by the expected holding time of state i. The diagonal entries are chosen so that each row sums to 0. A CTMC satisfies the Markov property, that its behavior depends only on its current state and not on its past behavior, due to the memorylessness of the exponential distribution and of discrete-time Markov chains. En Markovprocess, uppkallad efter den ryske matematikern Markov, är inom matematiken en tidskontinuerlig stokastisk process med Markovegenskapen, det vill säga att processens förlopp kan bestämmas utifrån dess befintliga tillstånd utan kännedom om det förflutna. Det tidsdiskreta fallet kallas en Markovkedja. I stället för Markovkedjans övergångssannoliketer har Markovprocessen övergångsintensiteter och tiderna mellan övergångarna mellan tillstånden är exponentialfördelade. Teorin för Markovprocesser används bl. a. inom fysik, ekonomi och reglerteknik.
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There exists such that for all the entry is differentiable and satisfies the Kolmogorov backward equations:
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