Exact asymptotics for the distribution of the time of attaining the maximum for a trajectory of a compound Poisson process with linear drift

We consider the random process $at-\nu_+(pt)+\nu_-(-qt)$, $ t\in(-\infty,\infty)$, where $\nu_-$ and $\nu_+$ are independent standard Poisson processes if $t\geq 0$ and $\nu_-(t)=\nu_+(t)=0$ if $t<0$. Under certain conditions on the parameters $a$, $p$, and $q$, we study the distribution function $G=G(x)$ of the time of attaining the maximum for a trajectory of this process. In the present article, we find an exact asymptotics for the tails of $G$. We also find a connection between this problem and the statistical problem of estimation of an unknown discontinuity point of a density function.