Geodesic
Exact asymptotics for the distribution of the time of attaining the maximum for a trajectory of a compound Poisson process with linear drift
Matematičeskie trudy, Tome 22 (2019) no. 2, pp. 134-156 
V. E. Mosyagin. Exact asymptotics for the distribution of the time of attaining the maximum for a trajectory of a compound Poisson process with linear drift. Matematičeskie trudy, Tome 22 (2019) no. 2, pp. 134-156. http://geodesic.mathdoc.fr/item/MT_2019_22_2_a7/
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     author = {V. E. Mosyagin},
     title = {Exact asymptotics for the~distribution of the~time of attaining the~maximum for a~trajectory of a~compound {Poisson} process with linear drift},
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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 $t0$. 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.

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