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Browsing the web/books extensively didn't bring much light to my questions, so i would be grateful for any help ^^

Let $X_t$ be a Homogeneous Markov Process in state space $G$, and $B \subset G$. Let the stopping time $\tau_B$ be defined by $\tau_B := inf( t \geq 0 : X_t \in B)$.

It is assumed that there is an $n \geq 1$ such that \begin{equation} \inf_{x\notin B} \mathbb{P}_x(X_n \in B) = \epsilon > 0 \end{equation}

I want to show

  1. That there exists an $\alpha > 0$ such that \begin{equation} \sup_{x} \mathbb{E}_x [exp^{\alpha\tau_B}] < \infty \end{equation}

and then

  1. To find a lower bound on $\alpha$ that depends on $n$ and $\epsilon$.

This not being my area, i am struggling to get my head around it. I have seen works on Birth and Death processes, for which the jump up/down probabilities are given. They then express the expectation directly given those probabilities. But here the only information that is given is our assumption telling us (as far as i understand) that no matter where on the state space we are (at least outside of $B$), there is always a probability to enter our region of interest $B$ in $n$ steps.

From that, could I conclude that $\sup_x \mathbb{E}_x[\tau_B] < \infty $ (and would that hold true for moments of higher orders?)? Then using Taylor expansions, we would have \begin{equation} \sup_x\mathbb{E}_x[exp^{\alpha\tau_B}] = 1 + \alpha\sup_x\mathbb{E}_x[\tau_B] + ... \end{equation} Then if all the moments are finite, then this expension would be finite as well. That does not sound very rigorous at all however, if even true - and it doesn't answer the question about $\alpha$.

Thank you for any input!

1 Answers 1

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First, for $x\notin B$, $$ \Bbb P_x[\tau_B>1]\le\Bbb P_x[X_1\notin B]\le(1-\epsilon). $$ Next, by the Markov property at time $1$, $$ \Bbb P_x[\tau_B>2]\le\Bbb P_x[X_1\notin B, X_2\notin B]=\int_{B^c}\Bbb P_x[X_1\in dy]\cdot\Bbb P_y[X_1\notin B]\le(1-\epsilon)^2. $$ In general, for $n=1,2,\ldots,$ $$ \Bbb P_x[\tau_B>n]\le(1-\epsilon)^n. $$ For general $t\ge0$, there is a unique non-negative integer $n$ such that $t\in[n.n+1)$, in which case $$ \Bbb P_x[\tau_B>t]\le\Bbb P_x[\tau_B>n]. $$ From here it's not hard to get $$ \Bbb P_x[\tau_B>t]\le Ce^{-\beta t},\qquad t>0, $$ where $\beta:=-\log(1-\epsilon)$, and $C>0$ is a suitable constant.