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August  2013, 18(6): 1697-1714. doi: 10.3934/dcdsb.2013.18.1697

Khasminskii-type theorems for stochastic functional differential equations

Minghui Song 1, , Liangjian Hu 2, , Xuerong Mao 3, and  Liguo Zhang 4,

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

Received  October 2011 Revised  December 2011 Published  March 2013

For a stochastic functional differential equation (SFDE) to have a unique global solution it is in general required that the coefficients of the SFDE obey the local Lipschitz condition and the linear growth condition. However, there are many SFDEs in practice which do not obey the linear growth condition. The main aim of this paper is to establish existence-and-uniqueness theorems for SFDEs where the linear growth condition is replaced by more general Khasminskii-type conditions in terms of a pair of Laypunov-type functions.
Keywords: stochastic functional differential equations., Khasminskii-type condition, Brownian motion, Itô's formula, Khasminskii-test.
Mathematics Subject Classification: 34K50, 60H10, 34F05, 93E0.
Citation: Minghui Song, Liangjian Hu, Xuerong Mao, Liguo Zhang. Khasminskii-type theorems for stochastic functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1697-1714. doi: 10.3934/dcdsb.2013.18.1697
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.  Google Scholar

[2]

A. Bahar and X. Mao, Stochastic delay population dynamics, International J. Pure and Applied Math., 11 (2004), 377-400.  Google Scholar

[3]

T. C. Gard, "Introduction to Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 114, Marcel Dekker, Inc., New York, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

M. Loève, "Probability Theory," Third edition, D. Van Nostrand Company, Inc., Princeton, N. J.-Toronto, Ont.-London, 1963.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

X. Mao, "Exponential Stability of Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 182, Marcel Dekker, Inc., New York, 1994.  Google Scholar

[11]

X. Mao, "Stochastic Differential Equations and Applications," Second edition, Horwood Publishing Limited, Chichester, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching," Imperial College Press, London, 2006.  Google Scholar

[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.  Google Scholar

[16]

S. E. A. Mohammed, "Stochastic Functional Differential Equations," Research Notes in Mathematics, 99, Pitman (Advanced Publishing Program), Boston, MA, 1984.  Google Scholar

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.  Google Scholar

[2]

A. Bahar and X. Mao, Stochastic delay population dynamics, International J. Pure and Applied Math., 11 (2004), 377-400.  Google Scholar

[3]

T. C. Gard, "Introduction to Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 114, Marcel Dekker, Inc., New York, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

M. Loève, "Probability Theory," Third edition, D. Van Nostrand Company, Inc., Princeton, N. J.-Toronto, Ont.-London, 1963.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

X. Mao, "Exponential Stability of Stochastic Differential Equations," Monographs and Textbooks in Pure and Applied Mathematics, 182, Marcel Dekker, Inc., New York, 1994.  Google Scholar

[11]

X. Mao, "Stochastic Differential Equations and Applications," Second edition, Horwood Publishing Limited, Chichester, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

X. Mao and C. Yuan, "Stochastic Differential Equations with Markovian Switching," Imperial College Press, London, 2006.  Google Scholar

[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.  Google Scholar

[16]

S. E. A. Mohammed, "Stochastic Functional Differential Equations," Research Notes in Mathematics, 99, Pitman (Advanced Publishing Program), Boston, MA, 1984.  Google Scholar

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