Lecture 12

# Thus transition probability matrix 6 4 chapman

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

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: probability matrix: 6 4. Chapman-Kolmogorov equation So far we have mentioned one-step transition probabilities, i.e., probability of An given equation provides a relation for multiple steps as follows: p n + m = ij where n m=0 12 n C-K np m: kj 1 sp t: 2 ik  k :::. For continuous time, this can be written as: p s + t = ij where s Xp A ,1 . Xp ik  kj k t  0: In matrix notation, P s + t = P sP t: Set the initial transition probability matrix as P 0 = I i.e., p ij 0 Deﬁne, q ij =1 (identity matrix) if i = j else 0 if i 6= j: lim p t , p 0 = lim p tt for j 6= i: !0 !0 t  q := lim p t , p 0 = lim p tt , 1 : !0 !0 t := ij ij ij t t ii ii ii ii t t Note that q ’s are instantaneous rate. In matrix notation, Q = lim P tt , I : !0 t where, Q = q : Now, recall the relation ij Xp t = 1: ij  j If we move 1 from RHS to LHS of this equation and then divide by t and let t ! 0, we get lim fp 1t=t + ::::: + p t , 1=t + :::: + p t=t + :::g = 0 !0 t i which implies ii q...
View Full Document

## This document was uploaded on 03/19/2014 for the course CS 6030 at Western Michigan.

Ask a homework question - tutors are online