| course |
course_year |
question_number |
tags |
title |
year |
Markov Chains |
II |
38 |
|
A3.1 B3.1 |
2001 |
(i) Explain what is meant by the transition semigroup $\left{P_{t}\right}$ of a Markov chain $X$ in continuous time. If the state space is finite, show under assumptions to be stated clearly, that $P_{t}^{\prime}=G P_{t}$ for some matrix $G$. Show that a distribution $\pi$ satisfies $\pi G=0$ if and only if $\pi P_{t}=\pi$ for all $t \geqslant 0$, and explain the importance of such $\pi$.
(ii) Let $X$ be a continuous-time Markov chain on the state space $S={1,2}$ with generator
$$G=\left(\begin{array}{cc}
-\beta & \beta \\
\gamma & -\gamma
\end{array}\right), \quad \text { where } \beta, \gamma>0 .$$
Show that the transition semigroup $P_{t}=\exp (t G)$ is given by
$$(\beta+\gamma) P_{t}=\left(\begin{array}{cc}
\gamma+\beta h(t) & \beta(1-h(t)) \\
\gamma(1-h(t)) & \beta+\gamma h(t)
\end{array}\right),$$
where $h(t)=e^{-t(\beta+\gamma)}$.
For $0<\alpha<1$, let
$$H(\alpha)=\left(\begin{array}{cc}
\alpha & 1-\alpha \\
1-\alpha & \alpha
\end{array}\right)$$
For a continuous-time chain $X$, let $M$ be a matrix with $(i, j)$ entry
$P(X(1)=j \mid X(0)=i)$, for $i, j \in S$. Show that there is a chain $X$ with $M=H(\alpha)$ if and only if $\alpha>\frac{1}{2}$.