Boundary Condition

[Molly]

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

 

[Andrew Martin]

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

 

[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?

 

[Andrew Martin]

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

 

[Molly]

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

Thats great thank you so much!