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Unformatted text preview: we have to ﬁrst ﬁgure out what it is. Using
the more explicit form of the backwards equation, (4.6), and plugging in our
rates, we have
p0 (i, j ) = pt (i + 1, j )
pt (i, j )
t
To check this we have to di↵erentiate the formula in (4.11).
When j > i we have that the derivative of (4.11) is
e t( t)j (j i i)! +e When j = i, pt (i, i) = e
e t t = t ( t)j
(j i i1 1)! = pt (i, j ) + pt (i + 1, j ) , so the derivative is
pt (i, i) = pt (i, i) + pt (i + 1, i) since pt (i + 1, i) = 0.
The second simplest example is:
Example 4.8. Twostate chains. For concreteness, we can suppose that the
state space is {1, 2}. In this case, there are only two ﬂip rates q (1, 2) = and
q (2, 1) = µ, so when we ﬁll in the diagonal with minus the sum of the ﬂip rates
on that row we get
✓
◆
Q= µ µ Writing out the backward equation in matrix form, (4.7), now we have
✓0
◆✓
◆✓
◆
pt (1, 1) p0 (1, 2)
pt (1, 1) pt (1, 2)
t
=
p0 (2, 1) p0 (2, 2)
µ
µ
pt (2, 1) pt (2, 2)
t
t 126 CHAPTER 4. CONTI...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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