I would like to calculate the gambler´s ruin probability in a generalization of the classic gambler´s ruin problem. In the classical model, the gambler´s wealth could be modeled by an additive random walk with an absorving barrier at $x=0$. In this generalization, the gambler´s wealth follows a multiplicative random walk instead.
His initial wealth is $W_{0}$ and his opponent is infinitely wealthy, so he never goes bankrupt. With probability $p$ he wins the game and his current wealth is multiplied by the factor $(1+r_{1})$, with $r_{1}$ fixed and $0<r_{1}<\infty$. With probability $1-p$, he loses the game and his current wealth is multiplied by $(1+r_{2})$ for $-1\leq r_{2}<0$.
If $W_{n}\leq0$, for any $n$, the gambler is ruined and the game is over.
After N trials, his wealth will be given by $$W_{N}=W_{0}(1+r_{1})^{S}(1+r_{2})^{N-S}$$ where $S$ is the numbers of success (wins), and $N-S$ is the numbers of loses.
As well pointed out below by @Minus One-Twelfth, ruin in the technical sense, $W_{n}\leq0$ is not possible to happen, for strictly positive $r_{2}$. So, lets define ruin for an arbitrary positive small $\epsilon$ as being $W_{n}\leq\epsilon$.
What is the probability that he is ruined after playing this game consecutively N times? In other words, what is $P(W_{n}\leq\epsilon)$?
Thank you in advance.