This article is cited in
18 papers
On the first passage time of a given level for processes with independent increments
D. V. Gusak,
V. S. Korolyuk Kiev
Abstract:
The distribution of the first passage time of a non-negative level for a homogeneous process with independent increments
$\xi(t)$ is studied,
$\xi(t)$ having a bounded variation, and its characteristic function being of the form
$\mathbf Me^{i\alpha\xi(t)}=e^{i\psi(\alpha)}$, where
$$
\psi(\alpha)=i\alpha a+\int_{-\infty}^0(e^{i\alpha x}-1)\,dM(x)+\int_0^\infty(e^{i\alpha x}-1)\,dN(x).
$$
The double transformation of the distribution considered is shown to be
$$
\theta(s,\alpha)=
\begin{cases}
-\frac{\varkappa^+(s,0)}{\pi^+(s,\alpha)}&(a\le0),
\\
-\frac1{1-i\alpha a}\cdot\frac{\varkappa^+(s,0)}{\varkappa^+(s,\alpha)}&(a>0),
\end{cases}
$$
where
$\varkappa^+(s,\alpha)$ is determined by the factorization identity
$$
\frac{s-\psi(\alpha)}{1-i\alpha a}=\varkappa^+(s,\alpha)\varkappa^-(s,\alpha)\quad(s>0,\ -\infty<\alpha<\infty).
$$
Received: 01.08.1966
Fulltext:
PDF file (413 kB)