On perpetuities with light tails

In the paper we consider the asymptotics of logarithmic tails of a perpetuity $$R\sum _{j=1}^{\mathrm{\infty }}{Q}_j\prod _{k=1}^{j-1}{M}_k,(M_n,Q_n{)}_{n=1}^{\mathrm{\infty }}arei\mathrm{.}i\mathrm{.}d\mathrm{.}copiesof(M,Q),$$ in the case when $\mathbb{P}(M\in [0,1))=1$ and $Q$ has all exponential moments. If $M$ and $Q$ are independent, under regular variation assumptions, we find the precise asymptotics of $-\log \mathbb{P}(R>x)$ as $x\to \mathrm{\infty }$. Moreover, we deal with the case of dependent $M$ and $Q$ and give asymptotic bounds for $-\log \mathbb{P}(R>x)$. It turns out that dependence structure between $M$ and $Q$ has a significant impact on the asymptotic rate of logarithmic tails of $R$. Such phenomenon is not observed in the case of heavy-tailed perpetuities.