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Khasminskii-type theorems for stochastic functional differential equations
1. | Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China |
2. | Department of Applied Mathematics, Donghua Univerisity, Shanghai 201600, China |
3. | Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH |
4. | School of Electronic Information and Control Engineering, Beijing University of Technology, Beijing 100124, China |
References:
[1] |
A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, J. Math. Anal. Appl., 292 (2004), 364-380.
doi: 10.1016/j.jmaa.2003.12.004. |
[2] |
A. Bahar and X. Mao, Stochastic delay population dynamics, International J. Pure and Applied Math., 11 (2004), 377-400. |
[3] |
T. C. Gard, "Introduction to Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 114, Marcel Dekker, Inc., New York, 1988. |
[4] |
R. Z. Has'minskiĭ, "Stochastic Stability of Differential Equations," Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980. |
[5] |
V. Kolmanovskiĭ and A. Myshkis, "Applied Theory of Functional-Differential Equations," Mathematics and its Applications (Soviet Series), 85, Kluwer Academic Publishers Group, Dordrecht, 1992. |
[6] |
G. S. Ladde and V. Lakshmikantham, "Random Differential Inequalities," Mathematics in Science and Engineering, 150, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. |
[7] |
M. Loève, "Probability Theory," Third edition, D. Van Nostrand Company, Inc., Princeton, N. J.-Toronto, Ont.-London, 1963. |
[8] |
Q. Luo, X. Mao and Y. Shen, Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations, Automatica J. IFAC, 47 (2011), 2075-2081.
doi: 10.1016/j.automatica.2011.06.014. |
[9] |
X. Mao, "Stability of Stochastic Differential Equations with Respect to Semimartingales," Pitman Research Notes in Mathematics Series, 251, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. |
[10] |
X. Mao, "Exponential Stability of Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 182, Marcel Dekker, Inc., New York, 1994. |
[11] |
X. Mao, "Stochastic Differential Equations and Applications," Second edition, Horwood Publishing Limited, Chichester, 2008. |
[12] |
X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268 (2002), 125-142.
doi: 10.1006/jmaa.2001.7803. |
[13] |
X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations, J. Sto. Anal. Appl., 23 (2005), 1045-1069.
doi: 10.1080/07362990500118637. |
[14] |
X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching," Imperial College Press, London, 2006. |
[15] |
X. Mao, C. Yuan and J. Zou, Stochastic differential delay equations in population dynamics, J. Math. Anal. Appl., 304 (2005), 296-320.
doi: 10.1016/j.jmaa.2004.09.027. |
[16] |
S. E. A. Mohammed, "Stochastic Functional Differential Equations," Research Notes in Mathematics, 99, Pitman (Advanced Publishing Program), Boston, MA, 1984. |
show all references
References:
[1] |
A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, J. Math. Anal. Appl., 292 (2004), 364-380.
doi: 10.1016/j.jmaa.2003.12.004. |
[2] |
A. Bahar and X. Mao, Stochastic delay population dynamics, International J. Pure and Applied Math., 11 (2004), 377-400. |
[3] |
T. C. Gard, "Introduction to Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 114, Marcel Dekker, Inc., New York, 1988. |
[4] |
R. Z. Has'minskiĭ, "Stochastic Stability of Differential Equations," Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics and Analysis, 7, Sijthoff & Noordhoff, Alphen aan den Rijn-Germantown, Md., 1980. |
[5] |
V. Kolmanovskiĭ and A. Myshkis, "Applied Theory of Functional-Differential Equations," Mathematics and its Applications (Soviet Series), 85, Kluwer Academic Publishers Group, Dordrecht, 1992. |
[6] |
G. S. Ladde and V. Lakshmikantham, "Random Differential Inequalities," Mathematics in Science and Engineering, 150, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. |
[7] |
M. Loève, "Probability Theory," Third edition, D. Van Nostrand Company, Inc., Princeton, N. J.-Toronto, Ont.-London, 1963. |
[8] |
Q. Luo, X. Mao and Y. Shen, Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations, Automatica J. IFAC, 47 (2011), 2075-2081.
doi: 10.1016/j.automatica.2011.06.014. |
[9] |
X. Mao, "Stability of Stochastic Differential Equations with Respect to Semimartingales," Pitman Research Notes in Mathematics Series, 251, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. |
[10] |
X. Mao, "Exponential Stability of Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 182, Marcel Dekker, Inc., New York, 1994. |
[11] |
X. Mao, "Stochastic Differential Equations and Applications," Second edition, Horwood Publishing Limited, Chichester, 2008. |
[12] |
X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268 (2002), 125-142.
doi: 10.1006/jmaa.2001.7803. |
[13] |
X. Mao and M. J. Rassias, Khasminskii-type theorems for stochastic differential delay equations, J. Sto. Anal. Appl., 23 (2005), 1045-1069.
doi: 10.1080/07362990500118637. |
[14] |
X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching," Imperial College Press, London, 2006. |
[15] |
X. Mao, C. Yuan and J. Zou, Stochastic differential delay equations in population dynamics, J. Math. Anal. Appl., 304 (2005), 296-320.
doi: 10.1016/j.jmaa.2004.09.027. |
[16] |
S. E. A. Mohammed, "Stochastic Functional Differential Equations," Research Notes in Mathematics, 99, Pitman (Advanced Publishing Program), Boston, MA, 1984. |
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