123
Views
0
CrossRef citations to date
0
Altmetric
Research Article

A representation theorem for set-valued submartingales

ORCID Icon & ORCID Icon
Pages 487-498 | Received 13 Feb 2023, Accepted 06 Dec 2023, Published online: 28 Dec 2023

References

  • Hall, P., Christopher, C. H. (2014). Martingale Limit Theory and its Application. New York: Academic Press.
  • Hiai, F., Umegaki, H. (1977). Integrals, conditional expectations, and martingales of multivalued functions. J. Multivariate Anal. 7(1):149–182. doi:10.1016/0047-259X(77)90037-9.
  • Molchanov, I. S. (2017). Theory of random sets. In Volume of Probability Theory and Stochastic Modelling, 2nd ed., Vol. 87, London: Springer-Verlag.
  • Li, S., Ogura, Y., Kreinovich, V. (2002). Limit theorems and applications of set-valued and fuzzy set-valued random variables. In Theory and Decision Library B, Vol. 43. Springer.
  • Arutyunov, A. V., Obukhovskii, V. (2016). Convex and Set-Valued Analysis. Berlin, Germany: De Gruyter.
  • Aubin, J.-P., Frankowska, H. (2000). Introduction: Set-valued analysis in control theory. Set-Valued Anal. 8:1–9.
  • Kisielewicz, M., ( 2020). Set-Valued Stochastic Integrals and Applications. Cham, Switzerland: Springer.
  • Peng, Z.-Y., Chen, X.-J., Zhao, Y.-B., Li, X.-B. (2022). Painlevé-kuratowski convergence of minimal solutions for set-valued optimization problems via improvement sets. J. Glob. Optim. 87: 759–781.
  • Peng, Z.-Y., Wang, J.-J., Fai, Ka., Yiu, C., Zhao, Y.-B. (2022). Technical note on the existence of solutions for generalized symmetric set-valued quasi-equilibrium problems utilizing improvement set. Optimization. 72(7):1707–1028.
  • Aumann, R. J. (1965). Integrals of set-valued functions. J. Math. Anal. Appl. 12(1):1–12. doi:10.1016/0022-247X(65)90049-1.
  • Hukuhara, M. (1967). Integration des applications mesurables dont la valeur est un compact convexe. Funkcialaj Ekvacioj. 10(3):205–223.
  • Debreu, G., ( 1967). Integration of correspondences. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2. Berkeley, CA: University of California Press, p. 351–372.
  • Wu, C., Gong, Z. (2000). On Henstock integrals of interval-valued functions and fuzzy-valued functions. Fuzzy Sets Syst. 115(3):377–391. doi:10.1016/S0165-0114(98)00277-2.
  • Wu, W.-Z., Zhang, W.-X., Wang, R.-M. (2001). Set valued Bartle integrals. J. Math. Anal. Appl. 255(1):1–20. doi:10.1006/jmaa.2000.6976.
  • Kisielewicz, M. (1997). Set-valued stochastic intergrals and stochastic inclutions 1. Stoch. Anal. Appl. 15(5):783–800. doi:10.1080/07362999708809507.
  • Kisielewicz, M. (2012). Some properties of set-valued stochastic integrals. J. Math. Anal. Appl. 388(2):984–995. doi:10.1016/j.jmaa.2011.10.050.
  • Kisielewicz, M. (2014). Martingale representation theorem for set-valued martingales. J. Math. Anal. Appl. 409(1):111–118. doi:10.1016/j.jmaa.2013.06.066.
  • Kim, Y. K., Ghil, B. M. (1997). Integrals of fuzzy-number-valued functions. Fuzzy Sets Syst. 86(2):213–222. doi:10.1016/0165-0114(95)00400-9.
  • Kim, B. K., Kim, J. H. (1999). Stochastic integrals of set-valued processes and fuzzy processes. J. Math. Anal. Appl. 236(2):480–502. doi:10.1006/jmaa.1999.6461.
  • Jung, E. J., Kim, J. H. (2003). On set-valued stochastic integrals. Stoch. Anal. Appl. 21(2):401–418. doi:10.1081/SAP-120019292.
  • Zhang, J., Li, S., Mitoma, I., Okazaki, Y. (2009). On set-valued stochastic integrals in an M-type 2 Banach space. J. Math. Anal. Appl. 350(1):216–233. doi:10.1016/j.jmaa.2008.09.017.
  • Zhang, J., Mitoma, I., Okazaki, Y. (2021). Submartingale property of set-valued stochastic integration associated with poisson process and related integral equations on banach spaces. J. Nonlinear Convex Anal. 22(4):775–799.
  • Tuyen, L. T., Vuong, P. Q., Quynh, V. X., Dang, N. G. (2022). Weak set-valued martingale difference and its applications. Int. J. Appl. Math. 35(3):397–422.
  • Zhang, J., Yano, K. (2023). Remarks on martingale representation theorem for set-valued martingales. In International Conference on Soft Methods in Probability and Statistics. Cham, Switzerland: Springer, p. 398–405.
  • Assani, I., Klei, H. A. (1982). Parties décomposables compactes de l1e. CR Acad. Sci. Paris, Série I, 294:533–536.
  • Stefanini, L., Bede, B. (2019). A New gH-difference for multi-dimensional convex sets and convex fuzzy sets. Axioms. 8(2):48–68. doi:10.3390/axioms8020048.
  • Ararat, C., Ma, J., Wu, W. (2023). Set-valued backward stochastic differential equations. Ann. Appl. Probab. 33(5):3418–3448.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.