BUILT-IN MATHEMATICA SYMBOL

# ContinuousMarkovProcess

ContinuousMarkovProcess[i0, q]
represents a continuous-time finite-state Markov process with transition rate matrix q and initial state .

ContinuousMarkovProcess[p0, q]
represents a Markov process with initial state probability vector .

ContinuousMarkovProcess[..., m, ]
represents a Markov process with transition matrix m and transition rates .

ContinuousMarkovProcess[..., g]
represents a Markov process transition rate matrix from the graph g.

## DetailsDetails

• ContinuousMarkovProcess is also known as a continuous-time Markov chain.
• ContinuousMarkovProcess is a continuous-time and discrete-state random process.
• The states of ContinuousMarkovProcess are integers between 1 and , where is the length of transition rate matrix q.
• For infinitesimal time dt, gives the probability that the process transitions from state i to state j over the next dt units of time qi, jdt=Probability[x[t+dt]=jx[t]=i].
• The time the process stays in state i before transitioning follows ExponentialDistribution[-qii].
• The transition matrix m specifies conditional transition probabilities mi, j=Probability[x[tk+1]=jx[tk]=i], where is the state at time , and the transition rate specifies that the time between events in state follows .
• The transition matrix in the case of a graph g is constructed to give equal probability of transitioning to each incident vertex with unit transition rates.
• ContinuousMarkovProcess allows q to be an × matrix where and for with rows that sum to 0, can be an integer between 1 and , is a vector of length of non-negative elements that sum to 1, m is an × matrix with non-negative elements and rows that sum to 1, and is a vector of length with positive elements.
• ContinuousMarkovProcess can be used with such functions as MarkovProcessProperties, PDF, Probability, and RandomFunction.

## ExamplesExamplesopen allclose all

### Basic Examples (2)Basic Examples (2)

Define a continuous-time Markov process:

Simulate it:

 Out[2]=
 Out[3]=

Find the PDF for the state at time t:

 Out[2]=

Find the long-run proportion of time the process is in state 2:

 Out[3]=