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 \[ \begin{gathered} {\bf P}\left\{ {\mathop {\sup }\limits_t \xi (t) = \infty } \right\} = 1,\qquad \xi (t) \in \mathfrak{S}^ + , \hfill \\ {\bf P}\left\{ {\mathop {\inf }\limits_t \xi (t) = - \infty } \right\} = 1,\qquad \xi (t) \in \mathfrak{S}^ - . \hfill \\ \end{gathered} \] It is proved that $\xi \in \mathfrak{S}^ + $ (resp. $\xi \in \mathfrak{S}^ - $) if and only if $\gamma = {\bf E}\xi (1) \geqq 0$ (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 $P(x) = {\bf P}\{ {\sup _t \xi (t) \geqq x} \}$ as well as the implicit dependence of $Q(x) = {\bf P}\{ {\inf _t \xi (t) \leqq - x} \}$ on the distribution of $\xi (t)$. Asymptotic and other properties of the probabilities P, Q are investigated.
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