I'll give the introduction but it's not the main problem.
We have E a countable space of states. Q is an irreducible transition matrix. Let ( denotes the initial probability) be the canonical Markov chain with the transition matrix Q.
Let such that
Prove that is a martingale.
So I have no problem showing the equality of the conditional expectation, and showing that it's measurable wrt the filtration.
But I don't know how to show that is integrable...
So my questions :
Just to be sure : does it mean to show that is finite ?
A friend said that there were martingales with an infinite expectation... How so ?
Is something positive always integrable ?
Would the hypothesis that is a Markov chain be useful ? Or the hypothesis that Q is irreducible ?
Erm... That's all for now.