The question looks like a gambling problem. But I am not sure whether it is the same or similar to gambler's ruin problem.
Assume I have wealth $W_t$. At each step $t$, I encounter a random shock $x_t\in[0,\infty)$ drawn from the same distribution with continuous differentiable cdf $G(x)$. If $x_t\ge W_t$, I go bankrupt, otherwise I survive. Conditional on surviving, my wealth at the next step is $$ W_{t+1}=W_t-x_t+w $$ $w>0$ is my risk-less income. The question is whether I will go bankrupt almost surely if $t$ goes to forever. If not, in such model, what is the necessary and sufficient condition for it.
Assuming that "$I_t$" means "$x_t$", I think that if your "shock" is drawn from a positive distribution with mean $m$ and variance $v$ such that $m > w$ or ( $m = w$ and $v > 0$ ) or $v = \infty$, then almost surely you will go bankrupt, and in all other cases the probability of going bankrupt will be strictly less than 1. I think you should try proving this case by case.