This question is a part of an larger question in actuarial mathematics. It is a model of an insurance firm with periodic stochastic outflows of $X_i$, an initial wealth of u and some income paid periodically c. Then we told a specific form of the probability of not being ruined and told to find it explicitly.
The math i hope is:
Let $(X_i)_{i\geq0}$ be iid with $P(X_1=0)=1-p$ and $P(X_1=2)=p$ (the stochastic claims), $S_n=\sum _{i =0}^n X_i$ (sum of outflows), $c$ be some constant (the periodic income) and $u$ some large constant (initial wealth). I would like to show that $P(\forall n: S_n \leq n\cdot c+u)$ (probability of not being ruined) is on the form $1-ab^u$ and determine a and b.
I've realized I'm probably supposed to look at the complement and somehow use $X_i$ is almost binomial, but I can't seem to get anywhere.