Sojourn time of almost semicontinuous integral-valued processes in a fixed state (Q1759966)

The author deals with the almost lower semicontinuous integer-valued process \(\xi(t)\) which crosses a negative level \(x<0\) only with the help of negative geometrically distributed jumps \(\xi_k<0\) with moment generating function of the negative parts of jumps \[ p(z) =\operatorname{E}[z^{\xi_k}|\xi_k<0] =(1-b)/(z-b),\quad 0\leq b < 1. \] The cumulant of the process \(\xi(t)\) is of the form \[ k(z) =\ln \operatorname{E} z^{\xi(t)} =\lambda(p p_{(1)}(z)+q(1-b)/(z-b)-1),\quad |z|=1,\;p+q=1, \] where \(p_{(1)}(z)=\operatorname{E}[z^{\xi_1}|\xi_1>0]\), \(0<b<1\), is the parameter of the geometric distribution of \(\xi_k<0\), and \(\lambda > 0\) is the intensity of jumps \(\xi_k\). The author derives relations in terms of the roots \(z_s < 1 < \hat{z}_s\) of the Lundberg equation \(k(z)=s\) for the moment generating function of the sojourn time of \(\xi(t)\) in a fixed state. For more results on distribution of sojourn times of almost continuous integer-valued processes \(\xi(t)\), see the author book [Processes with independent increments in risk theory (Ukrainian). Kyïv: Instytut Matematyky NAN Ukraïny (2011; Zbl 1249.60001)].