The First Passage Time of a Level and the Behavior at Infinity for a Class of Processes with Independent Increments

Let S be the class of homogeneous processes ξ (t) with independent increments and without positive jumps, and let S^ +, S^ - be its subclasses such that \[ {gathered} { P}\{ { { }_t ξ (t) = ∞ } \} = 1, ξ (t) ∈ S^ + , \\ { P}\{ { {inf }_t ξ (t) = - ∞ } \} = 1, ξ (t) ∈ S^ - . \\ {gathered} \] It is proved that ξ ∈ S^ + (resp. ξ ∈ S^ -) if and only if γ = Eξ (1) 0 (resp. γ 0). The exact expression for the distribution of the first intersection moment τ ₓ of a level $x > 0$ by a process ξ ∈ S^ + is found. We have also found the exact expression for the probability P(x) = P\ ₜ ξ (t) x \ as well as the implicit dependence of Q(x) = P\ inf ₜ ξ (t) - x \ on the distribution of ξ (t). Asymptotic and other properties of the probabilities P, Q are investigated.