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Markov chains and continuous-time Markov processes are useful in chemistry when physical systems closely approximate the Markov property. For example, imagine a large number n of molecules in solution in state A, each of which can undergo a chemical reaction to state B with a certain average rate. Perhaps the molecule is an enzyme, and the ...
A Markov chain with two states, A and E. In probability, a discrete-time Markov chain (DTMC) is a sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past.
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.
In probability theory, Kemeny’s constant is the expected number of time steps required for a Markov chain to transition from a starting state i to a random destination state sampled from the Markov chain's stationary distribution.
Like the discrete-time Markov decision processes, in continuous-time Markov decision processes the agent aims at finding the optimal policy which could maximize the expected cumulated reward. The only difference with the standard case stays in the fact that, due to the continuous nature of the time variable, the sum is replaced by an integral:
The first exit time (from A) is defined to be the first hit time for S \ A, the complement of A in S. Confusingly, this is also often denoted by τ A. [1] The first return time is defined to be the first hit time for the singleton set {X 0 (ω)}, which is usually a given deterministic element of the state space, such as the origin of the ...
A semi-Markov process (defined in the above bullet point) in which all the holding times are exponentially distributed is called a continuous-time Markov chain. In other words, if the inter-arrival times are exponentially distributed and if the waiting time in a state and the next state reached are independent, we have a continuous-time Markov ...
Consider this figure depicting a section of a Markov chain with states i, j, k and l and the corresponding transition probabilities. Here Kolmogorov's criterion implies that the product of probabilities when traversing through any closed loop must be equal, so the product around the loop i to j to l to k returning to i must be equal to the loop the other way round,