Asymptotic properties of the extinction probability for a Markov multiplication process

For sequences $\{\tau_i\}$, $\{\gamma_i\}$ of independent positive random variables the following process is constructed: $Y(0)=x$, $dY/dt=-1$ everywhere except points $t_n=\tau_1+\dots+\tau_n$ where $Y(t_n)=\gamma_n Y(t_n-0)=Y(t_n+0)$. Limit theorems are proved concerning the behaviour of the extinction probability
$$ f(x)=\mathbf P(\inf\{Y(t),t\ge 0\}<0),\qquad x\to\infty. $$