SOLUTIONS

BUILTIN MATHEMATICA SYMBOL
ContinuousMarkovProcess
ContinuousMarkovProcess[i_{0}, q]
represents a continuoustime finitestate Markov process with transition rate matrix q and initial state .
ContinuousMarkovProcess[p_{0}, 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 continuoustime Markov chain.
 ContinuousMarkovProcess is a continuoustime and discretestate 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[q_{ii}].
 The transition matrix m specifies conditional transition probabilities mi, j=Probability[x[t_{k+1}]=jx[t_{k}]=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 nonnegative elements that sum to 1, m is an × matrix with nonnegative 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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