Exit from interval and entry into interval, jumping and crossing over interval by Poisson process with exponential component. (Ukrainian, English) Zbl 1164.60395

Kadankova, T.

Teor. Jmovirn. Mat. Stat. 75, 21-35 (2006); translation in Theory Probab. Math. Stat. 75, 23-39 (2007).

Let \(\eta\in(0,\infty)\) be a positive-valued random variable, and let \(\gamma\) be a random variable with exponential distribution \(P(\gamma>x)=e^{-\lambda x}, x\geq0\). Let us define the Poisson process \(\xi(t)\in\mathbb R\) with cumulant \(k(p)={1\over t}\ln E[e^{-p\xi(t)}]=c\int_{-\infty}^{\infty}(e^{-xp}-1)\,dF(x)\), where \(F(x)=ae^{\lambda x}I_{\{x\leq0\}}+(a+(1-a)P(\eta<x) )I_{\{x>0\}}\), \(a\in(0,1)\), \(\lambda>0, c>0, \text{Re}\,p=0\). For fixed \(B>0\), let \(y\in[0,B]\), \(\chi=\inf\{t:\, \xi(t)\notin [-y,x]\}\), \(A^{x}=\{\xi(\chi)>x\}\), \(A_{y}=\{\xi(\chi)<-y\}\), \(X=(\xi(\chi)-x)I_{A^{x}}+(-\xi(\chi)-y)I_{A_{y}}\). The author obtains the integral transform of the joint distribution of \((\chi,X)\). For exponentially distributed intervals, the distribution of \((\sup_{u\leq t}\xi(u), \xi(t),\inf_{u\leq t}\xi(u))\) and the distribution of the number of crossing over an interval from above and from below are derived.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

60J05	Discrete-time Markov processes on general state spaces
60J45	Probabilistic potential theory

Keywords:

exit and entry; jumping and crossing over interval; Poisson process with exponential component; integral transform