Jump to ContentJump to Main Navigation
Show Summary Details

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year

IMPACT FACTOR 2015: 0.512

SCImago Journal Rank (SJR) 2015: 0.521
Source Normalized Impact per Paper (SNIP) 2015: 1.233
Impact per Publication (IPP) 2015: 0.546

Mathematical Citation Quotient (MCQ) 2015: 0.39

Open Access
See all formats and pricing
Volume 13, Issue 1 (Jan 2015)


Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition

Marek T. Malinowski
  • Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-516 Zielona Góra, Poland
  • Email:
Published Online: 2014-10-28 | DOI: https://doi.org/10.1515/math-2015-0011


We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L2 (consisting of square integrable random vectors). The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect to data of the equation is also presented. We consider equations driven by semimartingale Z and equations driven by processes A;M from decomposition of Z, where A is a process of finite variation and M is a local martingale. These equations are not equivalent. Finally, we show that the analysis of the set-valued stochastic integral equations can be extended to a case of fuzzy stochastic integral equations driven by semimartingales under Osgood type condition. To obtain our results we use the set-valued and fuzzy Maruyama type approximations and Bihari’s inequality.

Keywords : Set-valued stochastic integral equation; Set-valued stochastic integrals; Fuzzy stochastic integral equation; Fuzzy stochastic differential equation; Semimartingale; Maruyama approximation; Existence and uniqueness of solution; Osgood’s condition; Bihari’s inequality


  • [1] Ahmad B., Sivasundaram S., Dynamics and stability of impulsive hybrid setvalued integro-differential equations with delay, Nonlinear Anal., 2006, 65(11), 2082-2093

  • [2] Ahmad B., Sivasundaram S., The monotone iterative technique for impulsive hybrid set valued integro-differential equations, Nonlinear Anal., 2006, 65(12), 2260-2276

  • [3] Ahmad B., Sivasundaram S., Setvalued perturbed hybrid integro-differential equations and stability in term of two measures, Dynam. Systems Appl., 2007, 16(2), 299-310

  • [4] Ahmad B., Sivasundaram S., Stability in terms of two measures of setvalued perturbed impulsive delay differential equations, Commun. Appl. Anal., 12(1), 2008, 57-68

  • [5] Ahmed N.U., Nonlinear stochastic differential inclusions on Banach spaces, Stoch. Anal. Appl., 1994, 12(1), 1-10

  • [6] Aubin J.-P., Fuzzy differential inclusions, Problems Control Inform. Theory, 1990, 19(1), 55-67.

  • [7] Aubin J.-P., Cellina A., Differential Inclusions, Set-Valued Maps and Viability Theory, Springer, Berlin, 1984

  • [8] Aubin J.-P., Da Prato G., The viability theorem for stochastic differential inclusions, Stoch. Anal. Appl., 1998, 16(1), 1-15

  • [9] Aubin J.-P., Da Prato G., Frankowska H., Stochastic invariance for differential inclusions, Set-Valued Anal., 2000, 8(1-2), 181-201

  • [10] Aubin J.-P., Frankowska H., Set-Valued Analysis, Birkhäuser, Basel, 1990

  • [11] Balasubramaniam P., Ntouyas S.K., Controllability for neutral stochastic functional differential inclusions with infinite delay in abstract space, J. Math. Anal. Appl., 2006, 324(1), 161-176

  • [12] Balasubramaniam P., Vinayagam D., Existence of solutions of nonlinear neutral stochastic differential inclusions in a Hilbert space, Stoch. Anal. Appl., 2005, 23(1), 137-151

  • [13] Bihari I., A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar., 1956, 7, 81-94

  • [14] Bouchen A., El Arni A., Ouknine Y., Integration stochastique multivoque et inclusions differentielles stochastiques, Stochastics Stochastics Rep., 2000, 68(3-4), 297-327 (in French)

  • [15] Brandão Lopes Pinto A.I., De Blasi F.S., Iervolino F., Uniqueness and existence theorems for differential equations with convex valued solutions, Boll. Unione Mat. Ital., 1970, 3(4), 47-54

  • [16] Chung K.L., Williams R.J., Introduction to Stochastic Integration, Birkhäuser, Boston, 1983

  • [17] Da Prato G., Frankowska H., A stochastic Filippov theorem, Stoch. Anal. Appl., 1994, 12(4), 409-426

  • [18] De Blasi F.S., Iervolino F., Equazioni differenziali con soluzioni a valore compatto convesso, Boll. Unione Mat. Ital., 1969, 2(4), 194-501 (in Italian)

  • [19] Deimling K., Multivalued Differential Equations, Walter de Gruyter, Berlin, 1992

  • [20] Ekhaguere G.O.S., Quantum stochastic differential inclusions of hypermaximal monotone type, Internat. J. Theoret. Phys., 1995, 34(3), 323-353 [Crossref]

  • [21] Galanis G.N., Gnana Bhaskar T., Lakshmikantham V., Palamides P.K., Set valued functions in Frechet spaces: continuity, Hukuhara differentiability and applications to set differential equations, Nonlinear Anal., 2005, 61(4), 559-575

  • [22] Hagen K., Multivalued Fields in Condensed Matter, Electrodynamics and Gravitation, World Scientific, Singapore, 2008

  • [23] Hiai F., Umegaki H., Integrals, conditional expectation, and martingales of multivalued functions, J. Multivar. Anal., 1977, 7(1), 149-182 [Crossref]

  • [24] Hong S., Differentiability of multivalued functions on time scales and applications to multivalued dynamic equations, Nonlinear Anal., 2009, 71(9), 622-3637

  • [25] Hong S., Liu J., Phase spaces and periodic solutions of set functional dynamic equations with infinite delay, Nonlinear Anal., 2011, 74(9), 2966-2984

  • [26] Hu S., Papageorgiou, N., Handbook of Multivalued Analysis, Vol. I: Theory, Kluwer Academic Publishers, Boston, 1997

  • [27] Hu S., Papageorgiou N., Handbook of Multivalued Analysis, Vol. II: Applications, Kluwer Academic Publishers, Dordrecht, 2000

  • [28] Jakubowski A, Kamenski˘ı M., Raynaud de Fitte P., Existence of weak solutions to stochastic evolution inclusions, Stoch. Anal. Appl., 2005, 23(4), 723-749

  • [29] Kisielewicz M., Differential Inclusions and Optimal Control, Kluwer Academic Publishers, Dordrecht, 1991

  • [30] Kisielewicz M., Properties of solution set of stochastic inclusions, J. Appl. Math. Stochastic Anal., 1993, 6(3), 217-236

  • [31] Lakshmikantham V., Gnana Bhaskar T., Vasundhara Devi J., Theory of Set Differential Equations in a Metric Space, Cambridge Scientific Publ., Cambridge, 2006

  • [32] Lakshmikantham V., Mohapatra R.N., Theory of Fuzzy Differential Equations and Inclusions, Taylor and Francis Publishers, London, 2003

  • [33] Lakshmikantham V., Tolstonogov A.A., Existence and interrelation between set and fuzzy differential equations, Nonlinear Anal., 2003, 55(3), 255-268

  • [34] Li J., Li S., Ogura Y., Strong solutions of Itô type set-valued stochastic differential equations, Acta Math. Sin. (Engl. Ser.), 2010, 26(9), 1739-1748 [Crossref]

  • [35] Malinowski M.T., On set differential equations in Banach spaces - a second type Hukuhara differentiability approach, Appl. Math. Comput., 2012, 219(1), 289-305 [Crossref]

  • [36] Malinowski M.T., Interval Cauchy problem with a second type Hukuhara derivative, Inform. Sci., 2012, 213, 94-105

  • [37] Malinowski M.T., Second type Hukuhara differentiable solutions to the delay set-valued differential equations, Appl. Math. Comput., 2012, 218(18), 9427-9437 [Crossref]

  • [38] Malinowski M.T., On equations with a fuzzy stochastic integral with respect to semimartingales, Advan. Intell. Syst. Comput., 2013, 190, 93-101

  • [39] Malinowski M.T., On a new set-valued stochastic integral with respect to semimartingales and its applications, J. Math. Anal. Appl., 2013, 408(2), 669-680

  • [40] Malinowski M.T., Approximation schemes for fuzzy stochastic integral equations, Appl. Math. Comput., 2013, 219(24), 11278-11290 [Crossref]

  • [41] Malinowski M.T., Michta M., Set-valued stochastic integral equations driven by martingales, J. Math. Anal. Appl., 2012, 394(1), 30-47

  • [42] Malinowski M.T., Michta M., Sobolewska J., Set-valued and fuzzy stochastic differential equations driven by semimartingales, Nonlinear Anal., 2013, 79, 204-220

  • [43] Mao X., Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients, Stochastic Process. Appl., 1995, 58(2), 281-292 [Crossref]

  • [44] Mitoma I., Okazaki Y., Zhang J., Set-valued stochastic differential equation in M-type 2 Banach space, Comm. Stoch. Anal., 2010, 4(2), 215-237

  • [45] Motyl J., Stochastic functional inclusion driven by semimartingale, Stoch. Anal. Appl., 1998, 16(3), 717-732

  • [46] Osgood W.F., Beweis der Existenz einer Lösung der Differentialgleichung dy dx D f.x; y/ ohne Hinzunahme der Cauchy- Lipschitz’schen Bedingung, Monatsh. Math. Phys., 1898, 9(1), 331-345 (in German)

  • [47] Øksendal B., Stochastic Differential Equations: An Introduction and Applications, Springer Verlag, Berlin, 2003

  • [48] Protter P., Stochastic Integration and Differential Equations: A New Approach, Springer Verlag, New York, 1990

  • [49] Ren J., Wu J., Zhang X., Exponential ergodicity of non-Lipschitz multivalued stochastic differential equations, Bull. Sci. Math., 2010, 134(4), 391-404

  • [50] Ren J., Xu S., Zhang X., Large deviations for multivalued stochastic differential equations, J. Theoret. Probab., 2010, 23(4), 1142-1156

  • [51] Ren J., Wu J., On regularity of invariant measures of multivalued stochastic differential equations, Stochastic Process. Appl., 2012, 122(1), 93-105 [Crossref]

  • [52] Tolstonogov A.A., Differential Inclusions in a Banach Space, Kluwer Acad. Publ., Dordrecht, 2000

  • [53] Truong-Van B., Truong X.D.H., Existence results for viability problem associated to nonconvex stochastic differentiable inclusions, Stoch. Anal. Appl., 1999, 17(4), 667-685

  • [54] Tsokos C.P., Padgett W.J., Random Integral Equations with Applications to Life Sciences and Engineering, Academic Press, New York, 1974

  • [55] Tu N.N., Tung T.T., Stability of set differential equations and applications, Nonlinear Anal., 2009, 71(5-6), 1526-1533

  • [56] Wu J., Uniform large deviations for multivalued stochastic differential equations with Poisson jumps, Kyoto J. Math., 2011, 51(3), 535-559

  • [57] Xu S., Multivalued stochastic differential equations with non-Lipschitz coefficients, Chinese Ann. Math. Ser. B, 2009, 30(3), 321-332 [Crossref]

  • [58] Yun Y.S., Ryu S.U., Boundedness and continuity of solutions for stochastic differential inclusions on infinite dimensional space, Bull. Korean Math. Soc., 2007, 44(4), 807-816

  • [59] Zhang J., Li S., Mitoma I., Okazaki Y., On the solutions of set-valued stochastic differential equations in M-type 2 Banach spaces, Tohoku Math. J., 2009, 61(2), 417-440

About the article

Received: 2013-09-02

Accepted: 2014-06-10

Published Online: 2014-10-28

Published in Print: 2015-01-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0011. Export Citation

© 2015 Marek T. Malinowski. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Mariusz Michta
Mathematical Problems in Engineering, 2015, Volume 2015, Page 1

Comments (0)

Please log in or register to comment.
Log in