On solutions set of a multivalued stochastic differential equation

Malinowski, Marek T.   ; Agarwal, Ravi P.

Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 11-28

Abstract (VO)

We analyse multivalued stochastic differential equations driven by semimartingales. Such equations are understood as the corresponding multivalued stochastic integral equations. Under suitable conditions, it is shown that the considered multivalued stochastic differential equation admits at least one solution. Then we prove that the set of all solutions is closed and bounded.

Détail

DOI : 10.21136/CMJ.2017.0072-15

Classification : 26E25, 60G20, 60H05, 60H10, 60H20, 93C41, 93E03
Keywords: multivalued stochastic differential equation; Covitz-Nadler fixed point theorem; multivalued stochastic process

Bibliographie

[1] Agarwal, R. P., O'Regan, D.: Existence for set differential equations via multivalued operator equations. Differential Equations and Applications 5 1-5 Nova Science Publishers, New York (2007). | MR

[2] Ahmad, B., Sivasundaram, S.: $\phi_0$-stability of impulsive hybrid setvalued differential equations with delay by perturbing Lyapunov functions. Commun. Appl. Anal. 12 (2008), 137-145. | MR | JFM

[3] Anguraj, A., Vinodkumar, A., Chang, Y. K.: Existence results on impulsive stochastic functional differential inclusions with delays. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 20 (2013), 301-318. | MR | JFM

[4] Aubin, J.-P., Prato, G. Da: The viability theorem for stochastic differential inclusions. Stochastic Anal. Appl. 16 (1998), 1-15. | DOI | MR | JFM

[5] Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Modern Birkhäuser Classics. Birkhäuser, Boston (2009). | DOI | MR | JFM

[6] Balasubramaniam, P., Ntouyas, S. K.: Controllability for neutral stochastic functional differential inclusions with infinite delay in abstract space. J. Math. Anal. Appl. 324 (2006), 161-176. | DOI | MR | JFM

[7] Bhaskar, T. G., Lakshmikantham, V., Devi, J. Vasundhara: Nonlinear variation of parameters formula for set differential equations in a metric space. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 63 (2005), 735-744. | DOI | MR | JFM

[8] Bouchen, A., Arni, A. El, Ouknine, Y.: Multivalued stochastic integration and stochastic differential inclusions. Stochastics Stochastics Rep. 68 (2000), 297-327. | DOI | MR | JFM

[9] Burachik, R. S., Iusem, A. N.: Set-Valued Mappings and Enlargements of Monotone Operators. Springer Optimization and Its Applications. Springer, Berlin (2008). | DOI | MR

[10] Chung, K. L., Williams, R. J.: Introduction to Stochastic Integration. Progress in Probability and Statistics 4. Birkhäuser, Boston (1983). | DOI | MR | JFM

[11] H. Covitz, S. B. Nadler, Jr.: Multi-valued contraction mappings in generalized metric spaces. Isr. J. Math. 8 (1970), 5-11. | DOI | MR | JFM

[12] Prato, G. Da, Frankowska, H.: A stochastic Filippov theorem. Stochastic Anal. Appl. 12 (1994), 409-426. | DOI | MR | JFM

[13] Blasi, F. S. De, Iervolino, F.: Equazioni differenziali con soluzioni a valore compatto convesso. Boll. Unione Mat. Ital., IV. Ser., 2 (1969), 491-501 Errata corrige ibid. 4 1969 699. | MR | JFM

[14] Dunford, N., Schwartz, J. T.: Linear Operators. I. General Theory. Pure and Applied Mathematics 7. Interscience Publishers, New York (1958). | MR | JFM

[15] Hiai, F., Umegaki, H.: Integrals, conditional expectations, and martingales of multivalued functions. J. Multivariate Anal. 7 (1977), 149-182. | DOI | MR | JFM

[16] Hu, S., Papageorgiou, N. S.: Handbook of Multivalued Analysis. Volume I: Theory. Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht (1997). | MR | JFM

[17] Jiang, J., Li, C. F., Chen, H. T.: Existence of solutions for set differential equations involving causal operator with memory in Banach space. J. Appl. Math. Comput. 41 (2013), 183-196. | DOI | MR | JFM

[18] Kuratowski, K., Ryll-Nardzewski, C.: A general theorem on selectors. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 13 (1965), 397-403. | MR | JFM

[19] Lakshmikantham, V., Bhaskar, T. G., Devi, J. Vasundhara: Theory of Set Differential Equations in a Metric Spaces. Cambridge Scientific Publishers, Cambridge (2006). | MR | JFM

[20] Malinowski, M. T.: On set differential equations in Banach spaces---a second type Hukuhara differentiability approach. Appl. Math. Comput. 219 (2012), 289-305. | DOI | MR | JFM

[21] Malinowski, M. T.: Second type Hukuhara differentiable solutions to the delay set-valued differential equations. Appl. Math. Comput. 218 (2012), 9427-9437. | DOI | MR | JFM

[22] Malinowski, M. T.: On a new set-valued stochastic integral with respect to semimartingales and its applications. J. Math. Anal. Appl. 408 (2013), 669-680. | DOI | MR | JFM

[23] Malinowski, M. T.: Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition. Open. Math. (electronic only) 13 (2015), 106-134. | DOI | MR | JFM

[24] Park, J. Y., Jeong, J. U.: Existence results for impulsive neutral stochastic functional integro-differential inclusions with infinite delays. Adv. Difference Equ. (electronic only) 2014 (2014), Article ID 17, 17 pages. | DOI | MR | JFM

[25] Protter, P.: Stochastic Integration and Differential Equations. A New Approach. Applications of Mathematics 21. Springer, Berlin (1990). | DOI | MR | JFM

[26] Wang, P., Sun, W.: Practical stability in terms of two measures for set differential equations on time scales. Sci. World J. (2014), (2014), Article ID 241034, 7 pages. | DOI

[27] Yun, Y. S.: On the estimation of approximate solution for SDI. Korean Annals Math. 20 (2003), 63-69.

[28] Yun, Y. S.: The boundedness of solutions for stochastic differential inclusions. Bull. Korean Math. Soc. 40 (2003), 159-165. | DOI | MR | JFM

[29] Yun, Y. S.: The closed property of set of solutions for stochastic differential inclusions. Commun. Korean Math. Soc. 20 (2005), 135-144. | DOI | MR | JFM