Hi Guys, Sorry, another question from me Ive always assumed that the boundary condition is P_ij(0)=1, however, im wondering if this isnt a given? Is there actually some logic behind the boundary condition? where do we find it from? i cant find anything on it in the notes Thanks, Molly
Hi Molly The boundary condition is one of the following, depending on whether the destination state is the same as the starting state: p_ii(0) = 1 p_ij(0) = 0 for j not equal to i The logic behind this is that in 0 time we have no time to move between states. So if we start in State i, then after 0 time we must still be in State i. Hope this helps! Andy
Thank you so much andy that was exactly what i was looking for. when we are using the integral equations and the transition probabilities have both s,t in them, are the rules different? For example would p_ij(0,t)=0 and p_ii(0,t)=1?
Hi Molly p_ij(0,t) is not necessarily 0. This is the probability of going from State i to State j from time 0 to time t, which may be possible. p_ii(0,t) is not necessarily 1. This is the probability of starting in State i at time 0 and ending in State i at time t. Depending on the state space, this doesn't have to be guaranteed. The results here are that: p_ij(s,s) = 0 if i is not j p_ij(s,s) = 1 if i is j or, equivalently: p_ij(t,t) = 0 if i is not j p_ij(t,t) = 1 if i is j Hope this helps! Andy
