On solutions set of a multivalued stochastic differential equation

Marek T. Malinowski; Ravi P. Agarwal

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 1, page 11-28
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Malinowski, Marek T., and Agarwal, Ravi P.. "On solutions set of a multivalued stochastic differential equation." Czechoslovak Mathematical Journal 67.1 (2017): 11-28. <http://eudml.org/doc/287879>.

@article{Malinowski2017,
abstract = {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.},
author = {Malinowski, Marek T., Agarwal, Ravi P.},
journal = {Czechoslovak Mathematical Journal},
keywords = {multivalued stochastic differential equation; Covitz-Nadler fixed point theorem; multivalued stochastic process},
language = {eng},
number = {1},
pages = {11-28},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On solutions set of a multivalued stochastic differential equation},
url = {http://eudml.org/doc/287879},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Malinowski, Marek T.
AU - Agarwal, Ravi P.
TI - On solutions set of a multivalued stochastic differential equation
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 1
SP - 11
EP - 28
AB - 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.
LA - eng
KW - multivalued stochastic differential equation; Covitz-Nadler fixed point theorem; multivalued stochastic process
UR - http://eudml.org/doc/287879
ER -

References

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  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). (2007) MR2353574
  2. Ahmad, B., Sivasundaram, S., φ 0 -stability of impulsive hybrid setvalued differential equations with delay by perturbing Lyapunov functions, Commun. Appl. Anal. 12 (2008), 137-145. (2008) Zbl1185.34102MR2191489
  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. (2013) Zbl1268.34164MR3098454
  4. Aubin, J.-P., Prato, G. Da, 10.1080/07362999808809512, Stochastic Anal. Appl. 16 (1998), 1-15. (1998) Zbl0931.60059MR1603852DOI10.1080/07362999808809512
  5. Aubin, J.-P., Frankowska, H., 10.1007/978-0-8176-4848-0, Modern Birkhäuser Classics. Birkhäuser, Boston (2009). (2009) Zbl1168.49014MR2458436DOI10.1007/978-0-8176-4848-0
  6. Balasubramaniam, P., Ntouyas, S. K., 10.1016/j.jmaa.2005.12.005, J. Math. Anal. Appl. 324 (2006), 161-176. (2006) Zbl1118.93007MR2262463DOI10.1016/j.jmaa.2005.12.005
  7. Bhaskar, T. G., Lakshmikantham, V., Devi, J. Vasundhara, 10.1016/j.na.2005.02.036, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 63 (2005), 735-744. (2005) Zbl1153.34313MR2188146DOI10.1016/j.na.2005.02.036
  8. Bouchen, A., Arni, A. El, Ouknine, Y., 10.1080/17442500008834227, Stochastics Stochastics Rep. 68 (2000), 297-327. (2000) Zbl0957.60069MR1746184DOI10.1080/17442500008834227
  9. Burachik, R. S., Iusem, A. N., 10.1007/978-0-387-69757-4, Springer Optimization and Its Applications. Springer, Berlin (2008). (2008) MR2353163DOI10.1007/978-0-387-69757-4
  10. Chung, K. L., Williams, R. J., 10.1007/978-1-4757-9174-7, Progress in Probability and Statistics 4. Birkhäuser, Boston (1983). (1983) Zbl0527.60058MR0711774DOI10.1007/978-1-4757-9174-7
  11. H. Covitz, S. B. Nadler, Jr., 10.1007/BF02771543, Isr. J. Math. 8 (1970), 5-11. (1970) Zbl0192.59802MR0263062DOI10.1007/BF02771543
  12. Prato, G. Da, Frankowska, H., 10.1080/07362999408809361, Stochastic Anal. Appl. 12 (1994), 409-426. (1994) Zbl0810.60059MR1285803DOI10.1080/07362999408809361
  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. (1969) Zbl0195.38501MR0265653
  14. Dunford, N., Schwartz, J. T., Linear Operators. I. General Theory, Pure and Applied Mathematics 7. Interscience Publishers, New York (1958). (1958) Zbl0084.10402MR0117523
  15. Hiai, F., Umegaki, H., 10.1016/0047-259X(77)90037-9, J. Multivariate Anal. 7 (1977), 149-182. (1977) Zbl0368.60006MR0507504DOI10.1016/0047-259X(77)90037-9
  16. Hu, S., Papageorgiou, N. S., Handbook of Multivalued Analysis. Volume I: Theory, Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht (1997). (1997) Zbl0887.47001MR1485775
  17. Jiang, J., Li, C. F., Chen, H. T., 10.1007/s12190-012-0604-6, J. Appl. Math. Comput. 41 (2013), 183-196. (2013) Zbl1302.34114MR3017116DOI10.1007/s12190-012-0604-6
  18. Kuratowski, K., Ryll-Nardzewski, C., A general theorem on selectors, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 13 (1965), 397-403. (1965) Zbl0152.21403MR0188994
  19. Lakshmikantham, V., Bhaskar, T. G., Devi, J. Vasundhara, Theory of Set Differential Equations in a Metric Spaces, Cambridge Scientific Publishers, Cambridge (2006). (2006) Zbl1156.34003MR2438229
  20. Malinowski, M. T., 10.1016/j.amc.2012.06.019, Appl. Math. Comput. 219 (2012), 289-305. (2012) Zbl1297.34073MR2949593DOI10.1016/j.amc.2012.06.019
  21. Malinowski, M. T., 10.1016/j.amc.2012.03.027, Appl. Math. Comput. 218 (2012), 9427-9437. (2012) Zbl1252.34071MR2923039DOI10.1016/j.amc.2012.03.027
  22. Malinowski, M. T., 10.1016/j.jmaa.2013.06.054, J. Math. Anal. Appl. 408 (2013), 669-680. (2013) Zbl1306.60062MR3085061DOI10.1016/j.jmaa.2013.06.054
  23. Malinowski, M. T., 10.1515/math-2015-0011, Open. Math. (electronic only) 13 (2015), 106-134. (2015) Zbl1307.93381MR3314167DOI10.1515/math-2015-0011
  24. Park, J. Y., Jeong, J. U., 10.1186/1687-1847-2014-17, Adv. Difference Equ. (electronic only) 2014 (2014), Article ID 17, 17 pages. (2014) Zbl1343.93017MR3213919DOI10.1186/1687-1847-2014-17
  25. Protter, P., 10.1007/978-3-662-02619-9, Applications of Mathematics 21. Springer, Berlin (1990). (1990) Zbl0694.60047MR1037262DOI10.1007/978-3-662-02619-9
  26. Wang, P., Sun, W., 10.1155/2014/241034, Sci. World J. (2014), (2014), Article ID 241034, 7 pages. (2014) DOI10.1155/2014/241034
  27. Yun, Y. S., On the estimation of approximate solution for SDI, Korean Annals Math. 20 (2003), 63-69. (2003) 
  28. Yun, Y. S., 10.4134/BKMS.2003.40.1.159, Bull. Korean Math. Soc. 40 (2003), 159-165. (2003) Zbl1034.60009MR1958233DOI10.4134/BKMS.2003.40.1.159
  29. Yun, Y. S., 10.4134/CKMS.2005.20.1.135, Commun. Korean Math. Soc. 20 (2005), 135-144. (2005) Zbl1093.60047MR2167083DOI10.4134/CKMS.2005.20.1.135

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