Asymptotics for the distribution of the time of attaining the maximum for a trajectory of a Poisson process with linear drift and intensity switch

We find the exact asymptotics of the distribution of the time when the trajectory of the process $Y(t)=at-\nu_+(pt)+\nu_-(-qt)$, $t\in(-\infty,\infty)$ attains its maximum, where $\nu_{\pm}(t)$ are independent standard Poisson processes extended by zero on the negative semiaxis. The parameters $a$, $p$, $q$ are assumed just to satisfy the condition $\mathbf{E}Y(t)<0$, $t\neq 0$.