Dear team Please inform if i am wrong pij(h) = {h µij + o (h) if i not equal to j and is 1+ h µij + o (h) if i = j) The equation above signifies that I am in state "I " moving to state j in small time period h . The probability of it is h multiplied by the transition from moving from I to j state. Now for second equation when I =j it means that I am state" I " during time period h and probability is 1+ h into transition to put me to state I during time period h Please confirm Regards Suresh sharma
The formula you have written is correct. This is given on page 16 of Chapter 5. pij(h) is the probability that we are in state j at the end of an interval of length h, given that we are in state i now. If i is not equal to j, this means that we need to change state over the interval. The probability that we make only one direct transition from state i to state j in the interval is h*mu_ij. The probability that we make multiple transitions in the interval, but still end up in state j at the end is o(h). o(h) is vanishingly small, as a very short period of time isn't really long enough to make multiple transitions. This gives the pij(h) formula when i is not equal to j. If i is equal to j, then pii(h) is the probability that we start out in state i, and are in state i at time h. To think about this, note that at the end of any interval, we are either in state i, or are somewhere else. So, the probability of being in state i at the end of the interval is: 1 - probability of going to any other state = 1 - (sum over i not equal to j) pij(h) Then using the result for i not equal to j from above, this is: 1 - (sum over i not equal to j) h*mu_ij +o(h) and as shown on page 17 of chapter 5: mu_ii = - (sum over i not equal to j) mu_ij giving the result for i = j: pii(h) = 1 + h*mu_ii +o(h) So, i equal to j is telling us about the probability of remaining in state i, and the formula is obtained by viewing this as 1 - P(leaving state i).
