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

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.

MSC:

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