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Chapman-Kolmogorov equations, continuous time Markov chains, why do they have this solution.
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$P^'(t)=QP(t)=P(t)Q$
These are the Chapman-Kolmogorov backward and forward equations for a countable sate space continuous time Markov Chain. They have the solution,
$
P(t)=e^Qt
$
I have two questions
1) in the finite state space case, why is this a solution?
2) In the case where we have an infinite state space and the above are the infinite matrices how do we get this as a solution? I know this is a much harder question so maybe just a brief outline of an answer will help. Does the problem come from bringing the derivative inside the infinite sum?
probability stochastic-processes markov-chains
asked Aug 2 at 18:37
Monty
15212
add a comment |Â
up vote
0
down vote
favorite
$P^'(t)=QP(t)=P(t)Q$
These are the Chapman-Kolmogorov backward and forward equations for a countable sate space continuous time Markov Chain. They have the solution,
$
P(t)=e^Qt
$
I have two questions
1) in the finite state space case, why is this a solution?
2) In the case where we have an infinite state space and the above are the infinite matrices how do we get this as a solution? I know this is a much harder question so maybe just a brief outline of an answer will help. Does the problem come from bringing the derivative inside the infinite sum?
probability stochastic-processes markov-chains
asked Aug 2 at 18:37
Monty
15212
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
$P^'(t)=QP(t)=P(t)Q$
These are the Chapman-Kolmogorov backward and forward equations for a countable sate space continuous time Markov Chain. They have the solution,
$
P(t)=e^Qt
$
I have two questions
1) in the finite state space case, why is this a solution?
2) In the case where we have an infinite state space and the above are the infinite matrices how do we get this as a solution? I know this is a much harder question so maybe just a brief outline of an answer will help. Does the problem come from bringing the derivative inside the infinite sum?
probability stochastic-processes markov-chains
asked Aug 2 at 18:37
Monty
15212
$P^'(t)=QP(t)=P(t)Q$
These are the Chapman-Kolmogorov backward and forward equations for a countable sate space continuous time Markov Chain. They have the solution,
$
P(t)=e^Qt
$
I have two questions
1) in the finite state space case, why is this a solution?
2) In the case where we have an infinite state space and the above are the infinite matrices how do we get this as a solution? I know this is a much harder question so maybe just a brief outline of an answer will help. Does the problem come from bringing the derivative inside the infinite sum?
probability stochastic-processes markov-chains
asked Aug 2 at 18:37
Monty
15212
edited Aug 2 at 22:19
asked Aug 2 at 18:37
Monty
15212
asked Aug 2 at 18:37
Monty
15212
asked Aug 2 at 18:37
Monty
15212
15212
add a comment |Â
add a comment |Â
1 Answer
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1) This just the solution of this ODE. You can compute it using for instance the definition of the matrix exponential:
$$e^A = sum_j=0^infty frac1j!A^j.$$
We have $$P(t) = e^tQ = sum_j=0^infty frac1j!t^jQ^j.$$
When differentiating $P(t)$ w.r.t. to $t$, you obtain $$P'(t) = sum_j=1^infty frac1j!(jt^j-1)Q^j = left(sum_j=1^infty frac1(j-1)!t^j-1Q^j-1right)Q = P(t)Q.$$
Equivalently you can bracket the single $Q$ out to the left and obtain $P'(t) = QP(t)$.
2) I think, when using the definition of the matrix exponential above, you can prove the same statement also for the infinite state space case.
answered Aug 2 at 18:57
Jonas
259210
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
1) This just the solution of this ODE. You can compute it using for instance the definition of the matrix exponential:
$$e^A = sum_j=0^infty frac1j!A^j.$$
We have $$P(t) = e^tQ = sum_j=0^infty frac1j!t^jQ^j.$$
When differentiating $P(t)$ w.r.t. to $t$, you obtain $$P'(t) = sum_j=1^infty frac1j!(jt^j-1)Q^j = left(sum_j=1^infty frac1(j-1)!t^j-1Q^j-1right)Q = P(t)Q.$$
Equivalently you can bracket the single $Q$ out to the left and obtain $P'(t) = QP(t)$.
2) I think, when using the definition of the matrix exponential above, you can prove the same statement also for the infinite state space case.
answered Aug 2 at 18:57
Jonas
259210
add a comment |Â
up vote
0
down vote
1) This just the solution of this ODE. You can compute it using for instance the definition of the matrix exponential:
$$e^A = sum_j=0^infty frac1j!A^j.$$
We have $$P(t) = e^tQ = sum_j=0^infty frac1j!t^jQ^j.$$
When differentiating $P(t)$ w.r.t. to $t$, you obtain $$P'(t) = sum_j=1^infty frac1j!(jt^j-1)Q^j = left(sum_j=1^infty frac1(j-1)!t^j-1Q^j-1right)Q = P(t)Q.$$
Equivalently you can bracket the single $Q$ out to the left and obtain $P'(t) = QP(t)$.
2) I think, when using the definition of the matrix exponential above, you can prove the same statement also for the infinite state space case.
answered Aug 2 at 18:57
Jonas
259210
add a comment |Â
up vote
0
down vote
up vote
0
down vote
1) This just the solution of this ODE. You can compute it using for instance the definition of the matrix exponential:
$$e^A = sum_j=0^infty frac1j!A^j.$$
We have $$P(t) = e^tQ = sum_j=0^infty frac1j!t^jQ^j.$$
When differentiating $P(t)$ w.r.t. to $t$, you obtain $$P'(t) = sum_j=1^infty frac1j!(jt^j-1)Q^j = left(sum_j=1^infty frac1(j-1)!t^j-1Q^j-1right)Q = P(t)Q.$$
Equivalently you can bracket the single $Q$ out to the left and obtain $P'(t) = QP(t)$.
2) I think, when using the definition of the matrix exponential above, you can prove the same statement also for the infinite state space case.
answered Aug 2 at 18:57
Jonas
259210
1) This just the solution of this ODE. You can compute it using for instance the definition of the matrix exponential:
$$e^A = sum_j=0^infty frac1j!A^j.$$
We have $$P(t) = e^tQ = sum_j=0^infty frac1j!t^jQ^j.$$
When differentiating $P(t)$ w.r.t. to $t$, you obtain $$P'(t) = sum_j=1^infty frac1j!(jt^j-1)Q^j = left(sum_j=1^infty frac1(j-1)!t^j-1Q^j-1right)Q = P(t)Q.$$
Equivalently you can bracket the single $Q$ out to the left and obtain $P'(t) = QP(t)$.
2) I think, when using the definition of the matrix exponential above, you can prove the same statement also for the infinite state space case.
answered Aug 2 at 18:57
Jonas
259210
answered Aug 2 at 18:57
Jonas
259210
answered Aug 2 at 18:57
Jonas
259210
answered Aug 2 at 18:57
Jonas
259210
259210
add a comment |Â
add a comment |Â
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