In probability theory, a continuous-time Markov chain is a stochastic process { X(t) : t ≥ 0 } that enjoys the Markov property and takes values from amongst the elements of a set called the state space. The Markov property states that at any times s > t > 0, the conditional probability distribution of the process at time s given the whole history of the process up to and including time t, depends only on the state of the process at time t. In effect, the state of the process at time s is conditionally independent of the history of the process before time t, given the state of the process at time t.

Mathematical definitions

Intuitively, one can define a Markov chain as follows. Let X(t) be the random variable describing the state of the process at time t. Now prescribe that in some small increment of time from t to t+h, the probability that the process makes a transition to some state j, given that it started in some state i at time t, is given by

where o(h) represents a quantity that goes to zero as h goes to zero (see the article on order notation). Hence, over a sufficiently small interval of time, the probability of a particular transition is roughly proportional to the duration of that interval.

Continuous-time Markov chains are most easily defined by specifying the transition rates q_ij, and these are typically given as the ij-th elements of the transition rate matrix, Q (sometimes called a Q-matrix by convention). Q is a finite matrix according to whether or not the state space of the process is finite (it may be countably infinite, for example in a Poisson process where the state space is the non-negative integers). The most intuitive continuous-time Markov chains have Q-matrices that are:

• conservative—the i-th diagonal element q_ii of Q is given by

• stable—for any given state i, all elements q_ij (and q_ii) are finite.

(Note, however, that a Q-matrix may be non-conservative, unstable or both.) When the Q-matrix is both stable and conservative, the probability that no transition happens in some time r is

Therefore, the probability distribution of the waiting time until the first transition is an exponential distribution with rate parameter q_i (= −q_ii), and continuous-time Markov chains are thus memoryless processes.

Related processes

Given that a process that started in state i has experienced a transition out of state i, the conditional probability that the transition is into state j is

Using these probabilities, the sequence of states visited by the process (the so-called jump chain) can be described by a discrete-time Markov chain. The transition matrix P of the jump chain has elements p_ij = q_ij/q_i, i ≠ j, and p_ii = 0.

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.