Semi-Markov Processes of Multiplication with Drift

For sequences $\tau_1,\tau_2,\dots$; $\gamma_1,\gamma_2,\dots$ of independent positive random variables the following process is constructed: $Y(0)=x$, $\dfrac{dY}{dt}=-1$ everywhere except at the points $t_k=\sum\limits_{i=1}^k\tau_i$ for which $Y(t_i)=Y(t_i+0)=\gamma_iY(t_i-0)$. Limit theorems are proved concerning the behaviour of $Y(t)$ and $Y(t_n)$ when $t,n\to\infty$.