Stability Criteria of Solutions for Stochastic Set Differential Equations

Abstract

The existence and uniqueness results on solutions of set stochastic differential equation were studied in [1]. In this paper, we present the stability criteria for solutions of stochastic set differential equation.

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H. Vu, N. Phung, N. Hoa and N. Phu, "Stability Criteria of Solutions for Stochastic Set Differential Equations," Applied Mathematics, Vol. 3 No. 4, 2012, pp. 354-359. doi: 10.4236/am.2012.34055.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] M. T. Malinowski and M. Michta, “Stochastic Set Differential Equations,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 72, No. 3-4, 2010, pp 1247-1256. doi:10.1016/j.na.2009.08.015
[2] F. Hiai and H. Umegaki, “Integrals, Conditional Expectations and Martingales for Multivalued Functions,” Journal of Multivariate Analysis, Vol. 7, No. 1, 1977, pp. 147182.
[3] J. P. Aubin and G. Da Prato, “The Viability Theorem for Stochastic Differential Inclusion,” Stochastic Analysis and Applications, Vol. 15, No. 5, 1997, pp. 783-800.
[4] J. Motyl, “Stochastic Functional Inclusion Driven by Semimartingle,” Stochastic Analysis and Applications, Vol. 16, No. 3, 1998, pp. 517-532. doi:10.1080/07362999808809546
[5] M. T. Malinowski and M. Michta, “Stochastic Fuzzy Differential Equations with an Application,” Kybernetika, Vol. 47, No. 1, 2011, pp. 123-143.
[6] M. Michta, “On Set-Valued Stochastic Integrals and Fuzzy Stochastic Equations,” Fuzzy Set and Systems, Vol. 177, No. 1, 2011, pp. 1-19.
[7] E. J. Jung and J. H. Kim, “On Set-Valued Stochastic Integrals,” Stochastic Analysis and Applications, Vol. 21, No. 2, 2003, pp. 401-418. doi:10.1081/SAP-120019292
[8] J. Zhang, “Set-Valued Stochastic Integrals with Respect to a Real Valued Martingales,” In: D. Dubois, M. A. Lubiano, H. Prade, et al., Eds., Soft Method for Handling Variability and Imprecision, Springer, Berlin, 2008, pp. 253-259.
[9] J. Li and S. Li, “Set-Valued Stochastic Lebesgue Integral and Representation Theorems,” International Journal of Computational Intelligence Systems, Vol. 1, No. 2, 2008, pp. 177-187. doi:10.2991/ijcis.2008.1.2.8
[10] J. Zhang, S. Li, I. Mitoma and Y. Okazaki, “On Set-Valued Stochastic Integrals in an M-Type 2 Banach Space,” Journal of Mathematical Analysis and Applications, Vol. 350, No. 1, 2009, pp. 216-233. doi:10.1016/j.jmaa.2008.09.017
[11] N. D. Phu, N. Van Hoa and N. M. Hai, “Some Kinds of Controls for Boundedness Properties of Stochastic Set Solutions with Selectors,” International Journal of Evolution Equations, Vol. 5, No. 4, 2011, pp. 54-69.
[12] N. D. Phu, N. Van Hoa and H. Vu, “On the Stability Properties by Quasi-Expectation of Stochastic Set Solutions with Selectors,” Journal of Nonlinear Evolution Equations and Applications, Vol. 3, 2011, pp. 57-71.
[13] N. D. Phu, N. Van Hoa, N. M. Triet and H. Vu, “Boundedness Properties of Solutions to Stochastic Set Differential Equations with Selectors,” International Journal of Evolution Equations, Vol. 6, No. 2, 2012.
[14] V. Lakshmikantham, T. G. Bhaskar and D. J. Vasundhara, “Theory of Set Differential Equations in Metric Spaces,” Cambridge Scientific Publisher, Cambridge, 2006.

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