Prelimit and limit generalizations of the Pollaczek–Khinchin formula

The moment generating function of the nondegenerate distribution of the maximum $\xi ^+=\sup _{0\leq t<\infty }\xi (t)$ of a compound Poisson process \[ \xi (t)=at+S(t), \quad a<0, \qquad S(t)=\sum _{k\leq \nu (t)}\xi _k, \quad \xi _k>0, \] where $\nu (t)$ is a simple Poisson process with intensity $\lambda >0$, is determined via the well-known Pollaczek–Khinchin formula if $m=\mathsf {E}\xi (1)<0$.

We obtain a prelimit generalization of this formula that determines the Laplace–Carson transform of the moment generating function of the maximum $\xi ^+(t)=\sup _{0\leq t’\leq t}\xi (t’)$, $0<t<\infty$, and the moment generating function of $\xi ^+=\xi ^+(\infty )$ under the assumption that $m<0$ for homogeneous processes $\xi (t)$ with independent increments and of bounded variation. Relationships of a different type between characteristic functions of $\xi ^+(\theta _s)$ $(\mathsf {P}\{\theta _s>t\}=e^{-st},\ s,t>0)$ and of $\xi ^+$ are also obtained by using earlier results presented by the author.