On the Moment of the First Intersection of a Level and the Behavior at Infinity for one Class of Processes with Independent Increments

Let $\mathfrak{S}$ be the class of homogeneous processes $\xi(t)$ with independent increments and without positive jumps, and let $\mathfrak{S}^+$, $\mathfrak{S}^-$ be its subclasses such that
\begin{align*} \mathbf P\Bigl\{\sup_t\xi(t)=\infty\Bigr\}&=1, \quad \xi(t)\in\mathfrak{S}^+, \\ \mathbf P\Bigl\{\inf_t\xi(t)=-\infty\Bigr\}&=1, \quad \xi(t)\in\mathfrak{S}^-. \end{align*}
It is proved that $\xi\in\mathfrak{S}^+$ (resp. $\xi\in\mathfrak{S}^-$) if and only if $\gamma={\mathbf E}\xi(1)\geqq 0$ (resp. $\gamma\leqq 0$). The exact expression for the distribution of the first intersection moment $\tau_x$ of a level $x>0$ by a process $\xi\in\mathfrak{S}^+$ is found. We have also found the exact expression for the probability $P(x)={\mathbf P}\{\sup_t\xi(t)\geqq x\}$ as well as the implicit dependence of $Q(x)={\mathbf P}\{\inf_t\xi(t)\leqq-x\}$ on the distribution of $\xi(t)$. Asymptotic and other properties of the probabilities $P$, $Q$ are investigated.