Asymptotic properties of the degeneration probability for semi-Markov multiplication processes

In the paper, processes $Y(t), t\geq 0$, are considered defined as follows:
1) $Y(0)-x$;
2) sample paths of $Y(t)$ are right continuous and $Y'(t)=-1$ everywhere except at points $t_i=\sum_{k=1}^i \tau_k$, where
$$ Y(t_n+0)=\gamma_n Y(t_n-0), $$
$\{\tau_i\}_1^{\infty}$ and $\{\gamma\}_1^{\infty}$ being independent sequences of independent positive random variables.
Let $\zeta=\inf\{t: Y(t)\leq 0\}$. The probability
$$ f(x)=\mathbf{P}(\zeta<\infty|Y(0)=x) $$
is called the degeneration probability. Under wide conditions upon $\{\tau_i\}$ and $\{\gamma_i\}$, asymptotic behavior of $f(x)$ as $x\to 0$ or $x\to\infty$ is studied.