First, here is how we defined induced Markov chains:
Suppose that $(X,E,P)$ is an irreducible Markov chain, where $X=(X_i)_{i\in\mathbb{N}_0}$, $E$ is the state space and $P=(p_{i,j})_{i,j\in E}$ is the transition matrix. Suppose the Markov chain to be substochastic, which means $\sum_{j\in E}p_{i,j}\leqslant 1$ for all $i\in E$.
Let $F$ be an arbitrary non-empty subset of $E$. We define $$ p_{x,y}^F=\mathbb{P}_x(t(F)<\infty, X_{t(F)}=y), $$ where $\mathbb{P}_x(\cdot)$ means the probability of $\cdot$, starting in $x$ and where $t(F)$ is the first returning time of $F$.
If $y\notin F$, then $p_{x,y}^F=0$. If $y\in F$, then $$ p_{x,y}^F=\sum_{n=1}^{\infty}\sum_{x_1,x_2,\ldots,x_{n-1}\in E\setminus F}p_{x,x_1}p_{x_1,x_2}\ldots p_{x_{n-1},y}. $$ The Markov chain $(X_{|F}=(X_i{_{|F}})_{i\in\mathbb{N}_0},F,P^F)$ is called the induced Markov chain of $(X,E,P)$ on $F$.
Now the following task is given:
Show that $(X_{|F}=(X_i{_{|F}})_{i\in\mathbb{N}_0},F,P^F)$ is in fact a Markov chain. Is it true that if $P$ is stochastic (that is: $\sum_{j\in E}p_{i,j}=1$ for all $i\in E$) then $P^F$ is stochastic?
(1.) I think to show that the induced Markov chain is in fact a Markov chain means to show that it fullfills the Markov property. But I do not know how I can do that.
(2.) It is $$ \sum_{y\in F}p_{x,y}^F=\mathbb{P}_x(t(F)<\infty). $$
I think any transient irreducible Markov chain is is an example for
$$ \sum_{y\in F}p_{x,y}^F=\mathbb{P}_x(t(F)<\infty)<1, $$ isn't it?
In this case $P^F$ is not stochastic, no matter if $P$ is.
So the statement that from $P$ stochastic it follows that $P^F$ is stochastic should be false in general.
Am I right resp. is that justification ok?
It would be really great to get some feedback from you.
Thank you