Abstract
In this paper, we firstly introduce the concept of set-valued square integrable martingales. Secondly, we give the definition of stochastic integral of a stochastic process with respect to a set-valued square integrable martingale, and then prove the representation theorem of this kind of integral processes. Finally, we show that the stochastic integral process is a set-valued sub-martingale.
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Ahmed, N.U.: Nonlinear Stochastic differential inclusions on Banach space. Stoch. Anal. Appl. 12, 1–10 (1994)
Aubin, J.P., Prato, G.D.: The viability theorem for stochastic differenrial inclusions. Stoch. Anal. Appl. 16, 1–15 (1998)
Aumann, R.: Integrals of set valued functions. J. Math Anal. Appl. 12, 1–12 (1965)
Bagchi, S.: On a.s. convergence of classes of multivalued asymptotic martingales. Ann. Inst. H. Poincaré Probab. Statist. 21, 313–321 (1985)
Castaing, C., Valadier, M.: Convex analysis and measurable multifunctions. Lecture Notes in Math., vol. 580. Springer, Berlin (1977)
Da Prato, G., Frankowska, H.: A stochastic Filippov theorem. Stoch. Anal. Appl. 12, 409–426 (1994)
Van. Cutsem, B.: Martingales de multiapplications à valeurs convexes compactes. C. R. Math. Acad. Sci. Paris 269, 429–432 (1969)
Jung, E.J., Kim, J.H.: On set-valued stochastic integrals. Stoch. Anal. Appl. 21, 401–418 (2003)
Hess, C.: On multivalued martingales whose values be unbounded: martingale selectors and Mosco convergence. J. Multivariate Anal. 39, 175–201 (1991)
Hiai, F., Umegaki, H.: Integrals, conditional expectations and martingales of multivalued functions. J. Multivariate Anal. 7, 149–182 (1977)
Hu, S., Papageorgiou, N.S.: Handbook of Multivalued Analysis. Kluwer Academic Publishers, Dordrecht (1997)
de Korvin, A., Kleyle, R.: A convergence theorem for convex set valued supermartingales. Stoch. Anal. Appl. 3, 433–445 (1985)
Karatzas, I.: Lectures on the mathematics of finance. American Mathematical Society, Providence (1997)
Kim, B.K., Kim, J.H.: Stochastic integrals of set-valued processes and fuzzy processes. J. Math. Anal. Appl. 236, 480–502 (1999)
Kisielewicz, M.: Set valued stochastic integrals and stochastic inclusions. Discuss. Math. 13, 119–126 (1993)
Kisielewicz, M.: Set-valued stochastic integrals and stochastic inclusions. Stoch. Anal. Appl. 15, 783–800 (1997)
Kisielewicz, M.: Weak compactness of solution sets to stochastic differential inclusions with non-convex right-hand sides. Stoch. Anal. Appl. 23, 871–901 (2005)
Kisielewicz, M., Michta, M., Motyl, J.: Set valued approach to stochastic control, part I: existence and regularity properties. Dynam. Systems Appl. 12, 405–432 (2003)
Kisielewicz, M., Michta, M., Motyl, J.: Set valued approach to stochastic control, part II: viability and semimartingale issues. Dynam. Systems Appl. 12, 433–466 (2003)
Li, J., Li, S.: Set-valued stochastic Lebesgue integral and representation theorems. Int. J. Comput. Intell. Syst. 1, 177–187 (2008)
Li, J., Li, S., Ogura, Y.: Strong solution of Itô type set-valued stochastic differential equation. Acta Math. Sinica (to appear, 2010)
Li, S., Li, J., Li, X.: Stochastic integral with respect to set-valued square integrable martingales. J. Math. Anal. Appl. (2010), doi:10.1016/j.jmaa.2010.04.040
Li, S., Ogura, Y.: Convergence of set valued sub- and super-martingales in the Kuratowski-Mosco sense. Ann. Probab. 26, 1384–1402 (1998)
Li, S., Ogura, Y.: Convergence of set valued and fuzzy valued martingales. Fuzzy Sets Syst. 101, 453–461 (1999)
Li, S., Ogura, Y.: A convergence theorem of fuzzy-valued martingales in the extended Hausdorff metric H ∞ . Fuzzy Sets Syst. 135, 391–399 (2003)
Li, S., Ogura, Y., Kreinovich, V.: Limit theorems and applications of set-valuded and fuzzy sets-valued random variables. Kluwer Academic Publishers, Dordrecht (2002)
Li, S., Ren, A.: Representation theorems, set-valued and fuzzy set-valued Itô intergal. Fuzzy Sets Syst. 158, 949–962 (2007)
Luu, D.Q.: Representations and regularity of multivalued martingales. Acta Math. Vietnam. 6, 29–40 (1981)
Luu, D.Q.: Applications of set-valued Radon-Nikodym theorms to convergence of multivalued L 1-amarts. Math. Scand. 54, 101–114 (1984)
Molchanov, I.: Theory of Random Sets. Springer, London (2005)
Oksendal, B.: Stochastic Differential Equations. Springer, London (1995)
Papageorgiou, N.S.: On the theory of Banach space valued multifunctions. 1. integration and conditional expectation. J. Multivariate Anal. 17, 185–206 (1985)
Papageorgiou, N.S.: A convergence theorem for set valued multifunctions. 2. set valued martingales and set valued measures. J. Multivariate Anal. 17, 207–227 (1985)
Papageorgiou, N.S.: On the conditional expectation and convergence properties of random sets. Trans. Amer. Math. Soc. 347, 2495–2515 (1995)
Qi, Y., Wang, R.: Set-valued stochastic integral of bounded predictable processes w.r.t. square integrable martingale. Comm. Appl. Math. Comput. 2, 77–81 (1998)
Shreve, S.E.: Stochastic Calculus for Finance. Springer, London (2004)
Wang, Z.P., Xue, X.: On convergence and closedness of multivalued martingales. Trans. Amer. Math. Soc. 341, 807–827 (1994)
Zhang, J., Li, S., Mitoma, I., Okazaki, Y.: On set-valued stochastic integrals in an M-type 2 Banach space. J. Math. Anal. Appl. 350, 216–233 (2009)
Zhang, J., Li, S., Mitoma, I., Okazaki, Y.: On the solution of set-valued stochastic differential equation in M-type 2 Banach space. Tohoku Math. J. 61, 417–440 (2009)
Zhang, W., Gao, Y.: A convergence theorem and Riesz decomposition for set valued supermartingales. Acta Math. Sinica 35, 112–120 (1992)
Zhang, W., Li, S., Wang, Z., Gao, Y.: An introduction of set-valued stochastic processes. Science Press, Beijing (2007)
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Li, S. (2010). Set-Valued Square Integrable Martingales and Stochastic Integral. In: Borgelt, C., et al. Combining Soft Computing and Statistical Methods in Data Analysis. Advances in Intelligent and Soft Computing, vol 77. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14746-3_51
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DOI: https://doi.org/10.1007/978-3-642-14746-3_51
Publisher Name: Springer, Berlin, Heidelberg
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