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해당되는 게시물 1건
2008/02/15 10:24
derivative of Pii(0) of a continuous time Markov chain
2008/02/15 10:24 2008/02/15 10:24
Let LaTeX equation be the transition probability from state i to state j in time t of a discrete state Markov process, that is:
  1. LaTeX equation (transition probability is always non-negative)

  2. LaTeX equation (the total out going probability from a state is 1)

  3. LaTeX equation (the probability through state k is consistent)

  4. LaTeX equation (for infinitesimal time the probability of staying at the current state is 1)
Theorem
For every i

LaTeX equation

exists but may be infinite.

Proof (adapted from Karlin and Taylor 1981)
First we define LaTeX equation,

LaTeX equation

LaTeX equation

It is easy to see LaTeX equation therefore LaTeX equation for all t. Now we put

LaTeX equation

LaTeX equation and if it is not infinite, there for every LaTeX equation, there exists a LaTeX equation such that LaTeX equation.

For each t, we decompose as LaTeX equation via division with natural number n and reminder LaTeX equation. Then,

LaTeX equation

LaTeX equation

The equality is because as LaTeX equation, LaTeX equation. Since LaTeX equation was arbitrary, we have

LaTeX equation

for when LaTeX equation, the limit is also infinite.
Finally,

LaTeX equation.

Reference
  • Samuel Karlin, Howard M. Taylor. A second course in stochastic processes, Academic press 1981 (ISBN 0-12-398650-8)
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