On subexponential tails for the maxima of negatively driven compound renewal and L\'evy processes

We study subexponential tail asymptotics for the distribution of the maximum $M_t:={\sup }_{u\in [0,t]}{X}_u$ of a process $X_t$ with negative drift for the entire range of $t>0$. We consider compound renewal processes with linear drift and L'evy processes. For both we also formulate and prove the principle of a single big jump for their maxima. The class of compound renewal processes particularly includes Cram'er-Lundberg risk process.