CS2-04: Time-homogeneous Markov jump processes
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AGOGO CDO
Page 27
The Poisson process revisited
We have already mentioned (at the end of Section 2) that the Poisson process can be formulated
as a Markov jump process. We now revisit this idea.
Consider the Markov jump process with state space S = {0, 1, 2, ...} and transition rates:
if j = i
if j = i 1
otherwise
ij
0
The diagram representation is:
0
1
2
i
3
i +1
Recall that, in a Poisson process, events occur one at a time, and it is impossible to move to a
lower-numbered state. So, when it leaves state i , the process must enter state i 1 .
The generator matrix A in Kolmogorov’s equations is:
0
A
0
This leads to the forward equations:
pi 0 (t )
pij (t )
pi 0 (t )
pij
1(t )
pij (t ), j
0
essentially identical to (4.3) and (4.4).
It is interesting also to consider the backward equations:
pij (t )
pij (t )
pi
1, j (t )
which of course have the same solution as the forward equations despite looking dissimilar.
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