On some generalizations of the Pollachek–Khinchine formula

D. V. Gusak (Husak)

Teoriâ slučajnyh processov, Tome 16 (2010) no. 1, pp. 49-56

Résumé

For the skip-free Poisson process $\xi(t) \ (t\geq0, \xi(0)=0),$ $$ \xi(t)=at+S(t), \ a0, \ S(t)=\sum_{k\leq\nu(t)}\xi_k, \ \xi_k>0, \xi(0)=0, $$ where $\nu(t)$ is a simple Poisson process with intensity $\lambda>0,$ the moment generating function (m.g.f.) of $\xi^+=\sup_{0\leq t\infty}\xi(t)$ is defined by the well-known Pollachek–Khinchine formula under the condition $m=E\xi (1)0$ (see [1-3]). For a homogeneous process $\xi(t)$ with bounded variation, we establish prelimit and limit generalizations of this formula, which define the m.g.f. of $$ \xi^+(\theta_s)=\sup_{0\leq t\leq\theta_s}\xi (t), \ \xi^+=\lim_{s\to0}\xi^+(\theta_s) \left(P\{\theta_s>t\}=e^{-st}, \ s>0\right). $$ These generalizations are essentially based on the condition $P\{\tau ^+(0)= \gamma ^+(0)=0\}=0,$ where $(\tau ^+(0),\gamma ^+(0))$ is the initial ladder point of $\xi (t)\ (t\geq0, \xi(0)=0)$. Some another relations for the m.g.f. of $\xi^+(\theta_s)$ and $\xi^+$ are established for the general lower semicontinuous process $\xi(t)$ on the base of results in [3-5].

Détail

Keywords: Semicontinuous processes, Pollachek–Khinchine formula.

Bibliographie

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[2] G. P. Klimov, Stochastic Systems of Queueing Theory, Nauka, Moscow, 1966 (in Russian)

[3] S. Asmussen, Ruin Probability, World Scientific, Singapore, 2000

[4] D. V. Gusak(Husak), “Boundary problems for processes with independent increments in the risk theory”, Proceedings of Institute of Mathematics, 67 (2007) (in Ukrainian)

[5] D. V. Gusak (Husak), “Distribution of overjump functionals of semicontinuous homogeneous processes with independent increments”, Ukr. Math. Journ., 54:3 (2002), 371–379

[6] A. V. Skorokhod, Random Processes with Independent Increments, Nauka, Moscow, 1964 (in Russian)