Pointwise estimates for first passage times of perpetuity sequences

We consider first passage times ${\tau }_u=\inf \{n:Y_n>u\}$ for the perpetuity sequence $$Y_n=B_1+A_1{B}_2+\cdots +(A_1\dots A_{n-1})B_n,$$ where $(A_n,B_n)$ are i.i.d. random variables with values in ${\mathbb R} ^+\times {\mathbb R}$\mathrm{.}Recently,anumberoflimittheoremsrelatedto$\tau_u$ were proved including the law of large numbers, the central limit theorem and large deviations theorems. We obtain a precise asymptotics of the sequence $\mathbb{P}[{\tau }_u=\log u\mathrm{/}\rho ]$, $\rho >0$, $u\to \mathrm{\infty }$ which considerably improves the previous results. There, probabilities ${\mathbb P}[\tau_u \in I_u]$wereidentified,forsomelargeintervals$I_u$around$k_u$,withlengthsgrowingatleastas$\log\log u$. Remarkable analogies and differences to random walks are discussed.