## Stability of transient quasi-homogeneous Markov semigroups and estimate of ruin probability.(Ukrainian, English)Zbl 1164.60405

Teor. Jmovirn. Mat. Stat. 75, 36-44 (2006); translation in Theory Probab. Math. Stat. 75, 41-50 (2007).
A bounded semigroup $$(P_{st}, 0\leq s\leq t<\infty)\subset L({\mathcal F})$$ is called quasi-homogeneous if there exists non-trivial homogeneous bonded semigroup $$(Q_{t-s}, 0\leq s\leq t<\infty)$$ with infinitesimal operator $$A$$: $$\lim_{h\to0}h^{-1}(Q_{h}f-f)\equiv Af, \forall f\in{\mathcal F}_0\subset {\mathcal F}$$ and for some bounded family of operators $$(D_{s},0\leq s<\infty)\subset L({\mathcal F})$$ the infinitesimal operator $$A_{s}$$ of the semigroup $$(P_{st})$$ has the form $$A_{s}=A+D_{s}$$ on $${\mathcal F}_0$$. If $$\| P_{st}\|\to0$$, as $$t\to\infty$$, $$\forall s\geq0$$, then a semigroup $$(P_{st})$$ is called uniformly transient. One of the presented results is the following. Let $$(P_{st})$$ be a quasi-homogeneous bounded semigroup, $$T\leq\infty$$. If the corresponding basic homogeneous semigroup $$(Q_{t-s})$$ is uniformly transient, and for some $$\beta\geq0$$: $\varepsilon_{T}(\beta)\equiv\sup\limits_{t<T}\int_{0}^{t}\exp(\beta(t-u))\| D_{u}Q_{t-u}\| \,du<1,$ then $$(P_{st})$$ is uniformly transient and the following inequality holds $\sup\limits_{0\leq s\leq t<T}\exp(\beta(t-s))\| P_{st}-Q_{t-s}\|\leq\varepsilon_{T}(\beta){q_{T}(\beta)\over 1-\varepsilon_{T}(\beta)},$ where $$q_{T}(\beta)\equiv \sup\limits_{t<T}\exp(\beta t)\| Q_{t}\|$$.

### MSC:

 60J45 Probabilistic potential theory 60A05 Axioms; other general questions in probability
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