I have some problems with the following exercise and I need some hints (no complete solutions please) to solve it.
Let $(N_t)_{t\geq 0}$ be a Poisson($\lambda$) process and $c>\lambda$. Now let $X_t = ct - N_t$. There exists a unique $R>0$ such that $e^{-RX_t}$ is a martingale (I've already noticed that $\lambda (e^R - 1)-Rc = 0$). Let $u\geq 0$ and $S=\inf \{t:X_t < -u\}$. Show that the process $e^{-RX_{S\land t}}$ converges and compute the limit.
My ideas:
The process $e^{-RX_{S\land t}}$ is a strictly positive martingale which means that it converges (by the martingale convergence theorem). The problem is that I don't know how to compute the limit. It seems clear to me that $X_t \rightarrow \infty$ because $E[X_t] = ct - \lambda t=(c-\lambda)t$ which tends to infinity because $c-\lambda > 0$. I don't know how to formally prove that though. But this directly implies that $e^{-RX_t} \rightarrow 0$.
But how do I handle the stopping time? I have no idea how I could get rid of it. I mean there are two possible limits, depending on whether the process stops or not. If it stops, the limit will be $e^{Ru}$ and if it doesn't it will be $0$. That's really confusing and I really don't know how to tackle this problem.
It would be great if someone could help me with this and give me some hints, maybe about some theorems that come in handy. And please don't post a complete solution.
Thanks in advance!
EDIT: Okay I think I just forgot that random variables are functions and that it suffices if there is a pointwise limit. I think it has to be $e^{Ru} 1_{S<\infty}$ but I still have to formally prove $X_t \rightarrow \infty$ and I have no idea how to do that.