Set-Valued Stochastic Integrals with Respect to Finite Variation Processes

Affiliation(s)

Department of Mathematics and Physics, North China Electric Power University, Beijing, China.

Department of Mathematics and Physics, North China Electric Power University, Beijing, China.

ABSTRACT

In a Euclidean space *R*^{d}, the Lebesgue-Stieltjes integral of set-valued stochastic processes with respect to real valued finite variation process is defined directly by employing all integrably bounded selections instead of taking the decomposable closure appearing in some existed references. We shall show that this kind of integral is measurable, continuous in *t* under the Hausdorff metric and *L*^{2}-bounded.

Cite this paper

J. Zhang and J. Qi, "Set-Valued Stochastic Integrals with Respect to Finite Variation Processes,"*Advances in Pure Mathematics*, Vol. 3 No. 9, 2013, pp. 15-19. doi: 10.4236/apm.2013.39A1003.

J. Zhang and J. Qi, "Set-Valued Stochastic Integrals with Respect to Finite Variation Processes,"

References

[1] M. Kisielewicz, “Set-Valued Stochastic Integrals and Stochastic Inclusions,” Discussiones Mathematicae, Vol. 13, 1993, pp. 119-126.

[2] B. K. Kim and J. H. Kim, “Stochastic Integrals of Set-Valued Processes and Fuzzy Processes,” Journal of Mathematical Analysis and Applications, Vol. 236, No. 2, 1999, pp. 480-502.

http://dx.doi.org/10.1006/jmaa.1999.6461

[3] R. J. Aumann, “Intgrals of Set-Valued Functions,” Journal of Mathematical Analysis Applications, Vol. 12, No. 1, 1965, pp. 1-12.

http://dx.doi.org/10.1016/0022-247X(65)90049-1

[4] E. J. Jung and J. H. Kim, “On Set-Valued Stochastic Integrals,” Stochastic Analysis and Applications, Vol. 21, No. 2, 2003, pp. 401-408.

http://dx.doi.org/10.1081/SAP-120019292

[5] S. Li and A. Ren, “Representation Theorems, Set-Valued and Fuzzy Set-Valued Ito Integral,” Fuzzy Sets and Systems, Vol. 158, No. 9, 2007, pp. 949-962.

http://dx.doi.org/10.1016/j.fss.2006.12.004

[6] J. Zhang, “Set-Valued Stochastic Integrals with Respect to a Real Valued Maringale,” Soft Method for Handling Vaelability and Imprecision ASC 48, Spinger-Verlag, Berlin Herdelberg, 2008.

[7] J. Zhang, S. Li, I. Mitoma and Y. Okazaki, “On Set-Valued Stochastic Integrals in M-Type 2 Banach Space,” Journal of Mathematical Analysis and Applications, Vol. 350, No. 1, 2009, pp. 216-233.

http://dx.doi.org/10.1016/j.jmaa.2008.09.017

[8] J. Zhang, S. Li, I. Mitoma and Y. Okazaki, “On the Solution of Set-Valued Stochastic Differential Equations in M-Type 2 Banach Space,” Tohoku Mathematical Journal, Vol. 61, No. 3, 2009, pp. 417-440.

http://dx.doi.org/10.2748/tmj/1255700202

[9] J. Zhang, “Integrals and Stochastic Differential Equations for Set-Valued Stochastic Processes,” Ph.D. Thesis, Saga University, Saga, 2009.

[10] I. Mitoma, Y. Okazaki and J. Zhang, “Set-Valued Stochastic Differential Equations in M-Type 2 Banach Space,” Communications on Stochastic Analysis, Vol. 4, No. 2, 2010, pp. 215-237.

[11] J. G. Li, S. Li and Y. Ogura, “Strong Solutions of Ito Type Set-Valued Stochastic Differential Equation,” Acta Mathematica Sinica, English Series, Vol. 26, No. 9, 2010, pp. 1739-1748.

http://dx.doi.org/10.1007/s10114-010-8298-x

[12] M. Malinowski and M. Michta, “Set-Valued Stochastic Integral Equations Driven by Martingales,” Journal of Mathematical Analysis and Applications, Vol. 394, No. 12, 2012, pp. 30-47.

http://dx.doi.org/10.1016/j.jmaa.2012.04.042

[13] J. Zhang, I. Mitoma and Y. Okazaki, “Set-Valued Stochastic Integrals with Respect to Poisson Processes in a Banach Space,” International Journal of Approximate Reasoning, Vol. 54, No. 3, 2013, pp. 404-417.

http://dx.doi.org/10.1016/j.ijar.2012.06.001

[14] Z. Wang and R. Wang, “Set-Valued Lebesgue-Stieltjes Integrals,” Journal of Applied Probability and Statistics, Vol. 13, No. 3, 1997, pp. 303-316.

[15] S. Li, Y. Ogura and Y. Kreinovich, “Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables,” 43rd Edition, Kluwer Academic Publishers, Dordrecht, 2002.

http://dx.doi.org/10.1007/978-94-015-9932-0

[16] W. Zhang, S. Li, Z. Wang and Y. Gao, “An Introduction about Set-Valued Stochastic Process,” Science Press, Beijing, 2007.

[17] L. Wang and H. Xue, “Set-Valued Lebesgue-Stieltjes Integrals,” Basic Sciences Journal of Textile Universities, Vol. 16, No. 4, 2004, pp. 317-320.

[1] M. Kisielewicz, “Set-Valued Stochastic Integrals and Stochastic Inclusions,” Discussiones Mathematicae, Vol. 13, 1993, pp. 119-126.

[2] B. K. Kim and J. H. Kim, “Stochastic Integrals of Set-Valued Processes and Fuzzy Processes,” Journal of Mathematical Analysis and Applications, Vol. 236, No. 2, 1999, pp. 480-502.

http://dx.doi.org/10.1006/jmaa.1999.6461

[3] R. J. Aumann, “Intgrals of Set-Valued Functions,” Journal of Mathematical Analysis Applications, Vol. 12, No. 1, 1965, pp. 1-12.

http://dx.doi.org/10.1016/0022-247X(65)90049-1

[4] E. J. Jung and J. H. Kim, “On Set-Valued Stochastic Integrals,” Stochastic Analysis and Applications, Vol. 21, No. 2, 2003, pp. 401-408.

http://dx.doi.org/10.1081/SAP-120019292

[5] S. Li and A. Ren, “Representation Theorems, Set-Valued and Fuzzy Set-Valued Ito Integral,” Fuzzy Sets and Systems, Vol. 158, No. 9, 2007, pp. 949-962.

http://dx.doi.org/10.1016/j.fss.2006.12.004

[6] J. Zhang, “Set-Valued Stochastic Integrals with Respect to a Real Valued Maringale,” Soft Method for Handling Vaelability and Imprecision ASC 48, Spinger-Verlag, Berlin Herdelberg, 2008.

[7] J. Zhang, S. Li, I. Mitoma and Y. Okazaki, “On Set-Valued Stochastic Integrals in M-Type 2 Banach Space,” Journal of Mathematical Analysis and Applications, Vol. 350, No. 1, 2009, pp. 216-233.

http://dx.doi.org/10.1016/j.jmaa.2008.09.017

[8] J. Zhang, S. Li, I. Mitoma and Y. Okazaki, “On the Solution of Set-Valued Stochastic Differential Equations in M-Type 2 Banach Space,” Tohoku Mathematical Journal, Vol. 61, No. 3, 2009, pp. 417-440.

http://dx.doi.org/10.2748/tmj/1255700202

[9] J. Zhang, “Integrals and Stochastic Differential Equations for Set-Valued Stochastic Processes,” Ph.D. Thesis, Saga University, Saga, 2009.

[10] I. Mitoma, Y. Okazaki and J. Zhang, “Set-Valued Stochastic Differential Equations in M-Type 2 Banach Space,” Communications on Stochastic Analysis, Vol. 4, No. 2, 2010, pp. 215-237.

[11] J. G. Li, S. Li and Y. Ogura, “Strong Solutions of Ito Type Set-Valued Stochastic Differential Equation,” Acta Mathematica Sinica, English Series, Vol. 26, No. 9, 2010, pp. 1739-1748.

http://dx.doi.org/10.1007/s10114-010-8298-x

[12] M. Malinowski and M. Michta, “Set-Valued Stochastic Integral Equations Driven by Martingales,” Journal of Mathematical Analysis and Applications, Vol. 394, No. 12, 2012, pp. 30-47.

http://dx.doi.org/10.1016/j.jmaa.2012.04.042

[13] J. Zhang, I. Mitoma and Y. Okazaki, “Set-Valued Stochastic Integrals with Respect to Poisson Processes in a Banach Space,” International Journal of Approximate Reasoning, Vol. 54, No. 3, 2013, pp. 404-417.

http://dx.doi.org/10.1016/j.ijar.2012.06.001

[14] Z. Wang and R. Wang, “Set-Valued Lebesgue-Stieltjes Integrals,” Journal of Applied Probability and Statistics, Vol. 13, No. 3, 1997, pp. 303-316.

[15] S. Li, Y. Ogura and Y. Kreinovich, “Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables,” 43rd Edition, Kluwer Academic Publishers, Dordrecht, 2002.

http://dx.doi.org/10.1007/978-94-015-9932-0

[16] W. Zhang, S. Li, Z. Wang and Y. Gao, “An Introduction about Set-Valued Stochastic Process,” Science Press, Beijing, 2007.

[17] L. Wang and H. Xue, “Set-Valued Lebesgue-Stieltjes Integrals,” Basic Sciences Journal of Textile Universities, Vol. 16, No. 4, 2004, pp. 317-320.