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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.
Details
- 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 q
i, j
dt=Probability[x[t+dt]=j
x[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 ExponentialDistribution[i]. - 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.
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