Abstract
In this chapter we present the definition and properties of Itô set-valued integrals of square integrable non-anticipative matrix-valued stochastic processes. We begin with the definition and properties of Itô set-valued functional integrals of subsets of the space \({\mathbb {L}}^2(\mathbb {R}^+\times \Omega ,\Sigma _{\mathbb {F}},\mathbb {R}^{d\times m})\).
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References
Bocşan, G.: On Wiener stochastic integrals of multifunctions. Univ. Tim. FSN 87, 1–4 (1987)
Hiai, F.: Multivalued Stochastic Integrals and Stochastic Inclusions. Division of Appl. Math., Research Institute of Appl. Electricity Sapporo, Japan (not published)
Hu, Sh., Papageorgiou, N.S.: Handbook of Multivalued Analysis I. Kluwer Academic Publishers, Dordecht (1997)
Jung, E.J., Kim, J.H.: On set-valued stochastic integrals. Stoch. Anal. Appl. 21, 401–418 (2003)
Kisielewicz, M.: Set-valued stochastic integrals and stochastic inclusions. Disc. Math. 13, 119–126 (1993)
Kisielewicz, M.: Properties of solutions set of stochastic inclusions. J. Appl. Math. Stoch. Anal. 6, 217–236 (1993)
Kisielewicz, M.: Existence theorem for nonconvex stochastic inclusions. J. Appl. Math. Stoch. Anal. 7, 151–15 (1994)
Kisielewicz, M.: Set-valued stochastic integrals and stochastic inclusions. Stoch. Anal. Appl. 15(5), 783–800 (1997)
Kisielewicz, M.: Some properties of set-valued stochastic integrals. J. Math. Anal. Appl. 388, 984–995 (2012)
Kisielewicz, M.: Stochastic Differential Inclusions and Applications. Springer, Berlin (2013)
Kisielewicz, M.: Properties of generalized set-valued stochastic integrals. Disc. Math. DICO 34, 131–144 (2014)
Kisielewicz, M.: Martingale representation theorem for set-valued martingales. J. Math. Anal. Appl. 409, 111–118 (2014)
Kisielewicz, M.: Boundedness of set-valued stochastic integrals. Disc. Math. DICO 35, 197–207 (2015)
Kisielewicz, M.: Approximation theorems for set-valued stochastic integrals. Stoch. Anal. Appl. 36(3), 495–520 (2018)
Kisielewicz, M.: Integrable boundedness of set-valued stochastic integrals. J. Math. Anal. Appl. 481, 123441 (2020). https://doi.org/10.1016/j.jmma.2019.123441
Kisielewicz, M., Michta, M.: Properties of stochastic differential equations. Optimization 65(12), 2153–2169 (2016)
Kisielewicz, M., Motyl, J.: Selection theorems for set-valued stochastic integrals. Stoch. Anal. Appl. 37(2), 243–270 (2019)
Michta, M.: Note on the selection properties of set-valued semimartingales. Disc. Math. Differ. Incl. 16, 161–169 (1996)
Michta, M.: On solutions to stochastic differential inclusions. Discr. Cont. Dynam. Syst. 140, 618–622 (2004)
Michta, M.: On weak solutions to stochastic differential inclusions driven by semimartingales. Stoch. Anal. Appl. 22(5), 1341–1361 (2004)
Michta, M.: Remarks on unboundedness of set-valued stochastic integrals. J. Math. Anal. Appl. 424, 651–663 (2015)
Mitocha, I., Okazaki, Y., Zhang, J.: Set-valued stochastic integrals with respect to Poisson process in Banach space. Int. J. Approx. Reas. 54, 404–417 (2013)
Motyl, J.: Note on strong solutions of a stochastic inclusions. J. Math. Anal. Appl. 8(3), 291–294 (1995)
Motyl, J.: Stochastic functional inclusion driven by semimartingale. Stoch. Anal. Appl. 16(3), 517–532 (1998)
Øksendal, B.: Stochastionc Differential Equations. Springer, Berlin (1998)
Zhang, J.: Set-valued stochastic integrals with respect to a real valued martingale. In: Advances in Soft Computing, vol. 48, pp. 253–259. Springer, Berlin (2008)
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Kisielewicz, M. (2020). Itô Set-Valued Integrals. In: Set-Valued Stochastic Integrals and Applications. Springer Optimization and Its Applications, vol 157. Springer, Cham. https://doi.org/10.1007/978-3-030-40329-4_5
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DOI: https://doi.org/10.1007/978-3-030-40329-4_5
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