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Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps. (English) Zbl 1186.93070
Summary: A strong solutions approximation approach for mild solutions of stochastic functional differential equations with Markovian switching driven by Lévy martingales in Hilbert spaces is considered. The Razumikhin-Lyapunov type function methods and comparison principles are studied in pursuit of sufficient conditions for the moment exponential stability and almost sure exponential stability of equations in which we are interested. The results of [A.V. Swishchuk and Yu. I. Kazmerchuk [Theory Probab. Math. Stat. 64, 167–178 (2001); translation from Teor. Jmovirn. Mat. Stat. 64, 141–151 (2001; Zbl 0998.60059)] are generalized and improved as a special case of our theory.

MSC:
93E03 Stochastic systems in control theory (general)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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