Can someone please point me to a reference/answer me this question?
From the law of iterated logarithm, we see that Brownian motion with drift converge to $\infty$ or $-\infty$.
For a Poisson processes $N_t$ with rate $\lambda$, is there a similiar thing? As an exercise, I am consider what happens if the Brownian motion exponential martingale (plus an additional drift r) is replaced by a Poisson one. More explicitly, that is
$\exp(aN_t-(e^a-1)\lambda t + rt)$
how would this behave as $t$ tend to $\infty$?