Asymptotic properties of the probability of the degeneration after time $t$ for semi-Markov processes of multiplication

Let $Y(t)$ be the process defined by
1) $Y(0)=x$,
2) $Y(t)=x\prod\limits_{i=1}^{\nu(t)}\gamma_i-\sum\limits_{i=1}^{\nu(t)}\tau_i\gamma_i\dots\gamma_{\nu(t)}-\gamma(t)$ where $\{\tau_i\}_1^\infty$ and $\{\gamma_i\}_1^\infty$ are independent sequences of independent identically distributed positive random variables and
\begin{gather*} \nu(t)=\sup\biggl\{n\colon\sum_{i=1}^n\tau_i\le t\biggr\}, \\ \gamma(t)=t-\sum_{i=1}^{\nu(t)}\tau_i. \end{gather*}
Let
\begin{gather*} \zeta_x=\inf\{t\colon Y(t)\le0\mid Y(0)=x\}, \\ f(x,t)=\mathbf P(\zeta_x\ge t). \end{gather*}

In the paper, asymptotic properties of $f(x,t)$ for $x>0$ as $t\to\infty$ are studied.