For concreteness we can suppose that the state space

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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

Ask a homework question - tutors are online