3 limiting behavior recopying the rst term on the

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Unformatted text preview: j ) (4.9) 4.2. COMPUTING THE TRANSITION PROBABILITY 125 Introducing matrix notation again, we can write p0 = pt Q t (4.10) Comparing (4.10) with (4.7) we see that pt Q = Qpt and that the two forms of Kolmogorov’s di↵erential equations correspond to writing the rate matrix on the left or the right. While we are on the subject of the choices, we should remember that in general for matrices AB 6= BA, so it is somewhat remarkable that pt Q = Qpt . The key to the fact that these matrices commute is that pt = eQt is made up of powers of Q: Q · eQt = 1 X n=0 Q· 1 X (Qt)n (Qt)n = · Q = eQt · Q n! n! n=0 To illustrate the use of Kolmogorov’s equations we will now consider some examples. The simplest possible is Example 4.7. Poisson process. Let X (t) be the number of arrivals up to time t in a Poisson process with rate . In order to go from i arrivals at time s to j arrivals at time t + s we must have j i and have exactly j i arrivals in t units of time, so ( t)j i pt (i, j ) = e t (4.11) (j i)! To check the di↵erential equation,...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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