Abstract:
In the paper we generalize an observation of Keilson [1]. Let $X(t)$ be a left continuous homogeneous stochastic process with independent increments and let us suppose that its trajectories are continuous from above ($X(t+0)-X(t)\le0$) with probability 1. For such processes the indentity
$$
h(x,t)=\frac xtf(x,t)
$$
is obtained where $f(x,t)$ and $h(x,t)$ are generalized densities for $X(t)$ and for the first passage time of the level $x>0$ respectively.