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

Kartashov, M. V.

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 \(q_{T}(\beta)\equiv \sup\limits_{t<T}\exp(\beta t)\| Q_{t}\|\).

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

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

Keywords:

stability; transient quasi-homogeneous Markov semigroups; estimate of ruin probability; infinitesimal operator; uniformly transient semigroup