For concreteness we can suppose that the state space

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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...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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