This preview shows page 1. Sign up to view the full content.
Unformatted text preview: of course, has a transition
probability
pt (i, j ) = P (Xt = j X0 = i) Our next question is: How do you compute the transition probability pt from
the jump rates q ?
Our road to the answer starts by using the Chapman–Kolmogorov equations,
Theorem 4.1, and then taking the k = i term out of the sum.
!
X
pt+h (i, j ) pt (i, j ) =
ph (i, k )pt (k, j )
pt (i, j )
0 =@ k X
k6=i 1 ph (i, k )pt (k, j )A + [ph (i, i) 1] pt (i, j ) (4.3) Our goal is to divide each side by h and let h ! 0 to compute
p0 (i, j ) = lim
t h!0 pt+h (i, j ) pt (i, j )
h By the deﬁnition of the jump rates
q (i, j ) = lim h!0 ph (i, j )
h for i 6= j Ignoring the detail of interchanging the limit and the sum, which we will do
throughout this chapter, we have
X
1X
lim
ph (i, k )pt (k, j ) =
q (i, k )pt (k, j )
(4.4)
h!0 h
k6=i k6=i 124 CHAPTER 4. CONTINUOUS TIME MARKOV CHAINS For the other term we note that 1
ph (i, i)
h!0
h
lim 1 and we have
lim h!0 = lim h!0 ph (i, i)
h ph (i, i) =
X ph (i, k )
k6=i 1 P h pt (i, j ) = k6=i ph (i, k ), so X = q (i, k ) = i k6=i i pt (i, j ) (4.5) Combining (4.4) and (4.5) with (4.3) and the deﬁnition of...
View Full
Document
 Spring '10
 DURRETT
 The Land

Click to edit the document details