Stochastic integrals and stochastic equations in set-valued and fuzzy-valued frameworks. (English) Zbl 1454.34007
In this paper, the authors study stochastic integrals and stochastic equations in set-valued and fuzzy-valued frameworks. The authors first recall the relevant material from stochastic, set valued and fuzzy-valued analysis. The notions of set-valued and fuzzy stochastic integrals of Ammann type and Ito type are defined. Some useful properties of these integrals are proved. Generalized versions of stochastic fuzzy and set-valued differential equations are studied.
Reviewer: Seenith Sivasundaram (Daytona Beach)
MSC:
| 34A07 | Fuzzy ordinary differential equations |
| 34A60 | Ordinary differential inclusions |
| 34F05 | Ordinary differential equations and systems with randomness |
| 60H20 | Stochastic integral equations |
| 26E25 | Set-valued functions |
| 28B20 | Set-valued set functions and measures; integration of set-valued functions; measurable selections |
Keywords:
set-valued function; fuzzy set; set-valued stochastic integral; fuzzy stochastic integral; set-valued stochastic differential equation; fuzzy stochastic differential equationReferences:
| [1] | Ahmad, B. and Sivasundaram, S., Dynamics and stability of impulsive hybrid set-valued integro-differential equations with delay, Nonlinear Anal.65 (2006) 2082-2093. · Zbl 1114.34062 |
| [2] | Ahmad, B. and Sivasundaram, S., The monotone iterative technique for impulsive hybrid set-valued integro-differential equations, Nonlinear Anal.65 (2006) 2260-2276. · Zbl 1111.45006 |
| [3] | Artstein, Z., Parametrized integration of multifunctions with applications to control and optimization, SIAM J. Control Optim.27 (1989) 1369-1380. · Zbl 0698.28004 |
| [4] | Artstein, Z., A calculus for set-valued maps and set-valued evolution equations, Set-Valued Anal.3 (1995) 216-261. · Zbl 0856.49016 |
| [5] | Bede, B. and Gal, S. G., Solutions of fuzzy differential equations based on generalized differentiability, Commun. Math. Anal.9 (2010) 22-41. · Zbl 1200.34003 |
| [6] | Bede, B., Rudas, I. J. and Benesick, A. L., First order linear fuzzy differential equations under generalized differentiability, Inform. Sci.177 (2007) 1648-1662. · Zbl 1119.34003 |
| [7] | Pinto, A. I. Brandao Lopes, De Blasi, F. S. and Iervolino, F., Uniqueness and existence theorems for differential equations with convex valued solutions, Boll. Unione Mat. Ital.4 (1970) 1-12. |
| [8] | Chalco-Cano, Y. and Roman-Flores, H., Comparison between some approaches to solve fuzzy differential equations, Fuzzy Sets Syst.160 (2009) 1517-1527. · Zbl 1198.34005 |
| [9] | Chung, K. L. and Williams, R. J., Introduction to Stochastic Integration (Birkhauser, 1983). · Zbl 0527.60058 |
| [10] | Colubi, A., Domínguez-Menchero, J. S., López-Díaz, M. and Ralescu, D. A., A \(D_E [0, 1]\) representation of random upper semicontinuous functions, Proc. Amer. Math. Soc.130 (2002) 3237-3242. · Zbl 1005.28003 |
| [11] | De Blasi, F. S., Pinto, A. I. Brandao Lopez and Iervolino, F., Uniqueness and existence theorems for differential equations with compact convex solutions, Boll. Unione Mat. Ital.2 (1970) 491-501. |
| [12] | De Blasi, F. S. and Iervolino, F., Equazioni differenziali con soluzioni a valore compatto convesso, Boll. Unione Mat. Ital.4 (1969) 194-501. · Zbl 0195.38501 |
| [13] | Denkowski, Z., Migórski, S. and Papageorgiou, N. S., An Introduction to Nonlinear Analysis and Its Applications. Part I (Kluwer Academic Publishers, 2003). · Zbl 1054.47001 |
| [14] | Diamond, P. and Kloeden, P., Metric Spaces of Fuzzy Sets: Theory and Applications (World Scientific, 1994). · Zbl 0873.54019 |
| [15] | Ding, X. and Wu, R., A new proof for comparison theorems for stochastic differential inequalities with respect to semimartingales, Stoch. Proc. Appl.78 (1998) 155-171. · Zbl 0934.60053 |
| [16] | Fei, W., Liu, H. and Zhang, W., On solutions to fuzzy stochastic differential equations with local martingales, Syst. Control Lett.65 (2014) 96-105. · Zbl 1285.93088 |
| [17] | Fryszkowski, A., Fixed Point Theory for Decomposable Sets (Kluwer Academic Publishers, 2004). · Zbl 1086.47004 |
| [18] | Galanis, G. N., Bhaskar, T. Gnana, Lakshmikantham, V. and Palamides, P. K., Set valued functions in Frechet spaces: Continuity, Hukuhara differentiability and applications to set differential equations, Nonlinear Anal.61 (2005) 559-575. · Zbl 1190.49025 |
| [19] | Bhaskar, T. Gnana, Lakshmikantham, V. and Devi, J. V., Nonlinear variation of parameters formula for set differential equations in a metric space, Nonlinear Anal.63 (2005) 735-744. · Zbl 1153.34313 |
| [20] | Hess, C., On the parametrized integral of a multifunction: The unbounded case, Set-Valued Anal.15 (2007) 1-20. · Zbl 1127.28009 |
| [21] | Hiai, F. and Umegaki, H., Integrals, conditional expectations and martingales for multivalued functions, J. Multivar. Anal.7 (1977) 147-182. · Zbl 0368.60006 |
| [22] | Himmelberg, C. J., Measurable relations, Fund. Math.87 (1975) 53-72. · Zbl 0296.28003 |
| [23] | Hu, S. and Papageorgiou, N., Handbook of Multivalued Analysis. Vol. 1. Theory (Kluwer Academic Publishers, 1997). · Zbl 0887.47001 |
| [24] | Jung, E. J. and Kim, J. H., On set-valued stochastic integrals, Stoch. Anal. Appl.21 (2003) 401-418. · Zbl 1049.60048 |
| [25] | Kaleva, O., Fuzzy differential equations, Fuzzy Sets Syst.24 (1987) 301-317. · Zbl 0646.34019 |
| [26] | Kaleva, O., The Cauchy problem for fuzzy differential equations, Fuzzy Sets Syst.35 (1990) 389-396. · Zbl 0696.34005 |
| [27] | Kaleva, O., A note on fuzzy differential equations, Nonlinear Anal.64 (2006) 895-900. · Zbl 1100.34500 |
| [28] | Kallenberg, O., Foundations of Modern Probability (Springer, 1997). · Zbl 0892.60001 |
| [29] | Kandilakis, D. A. and Papageorgiou, N. S., On the existence of solutions for random differential inclusions in a Banach space, J. Math. Anal. Appl.126 (1987) 11-23. · Zbl 0629.60075 |
| [30] | Kisielewicz, M., Description of a class of differential equations with set-valued solutions, Atti Acad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.8 (1975) 158-162. · Zbl 0346.34001 |
| [31] | Kisielewicz, M., Method of averaging for differential equations with compact convex valued solutions, Rend. Mat.6 (1976) 397-408. · Zbl 0364.34017 |
| [32] | Kisielewicz, M., Set-valued stochastic integrals and stochastic inclusions, Stoch. Anal. Appl.15 (1997) 783-800. · Zbl 0891.93070 |
| [33] | Kisielewicz, M., Stochastic Differential Inclusions and Applications (Springer, 2013). · Zbl 1277.93002 |
| [34] | Kisielewicz, M., Properties of generalized set-valued stochastic integrals, Discuss. Math. DICO34 (2014) 131-147. · Zbl 1329.60163 |
| [35] | Kisielewicz, M. and Michta, M., Properties of set-valued stochastic differential equations, Optimization65 (2016) 2153-2169. · Zbl 1353.60054 |
| [36] | Kisielewicz, M. and Michta, M., Integrably bounded set-valued stochastic integrals, J. Math. Anal. Appl.449 (2017) 1892-1910. · Zbl 1359.60069 |
| [37] | Kisielewicz, M. and Michta, M., Weak solutions of set-valued stochastic differential equations, J. Math. Anal. Appl.473 (2019) 1026-1052. · Zbl 1426.60073 |
| [38] | Lakshmikantham, V., Set differential equations versus fuzzy differential equations, Appl. Math. Comp.164 (2005) 277-294. · Zbl 1079.34005 |
| [39] | Lakshmikantham, V., Bhaskar, T. Gnana and Devi, J. Vasundhara, Theory of Set Differential Equations in a Metric Space (Cambridge Scientific Publishers, 2006). · Zbl 1156.34003 |
| [40] | Lakshmikantham, V., Leela, S. and Vatsala, A., Interconnection between set and fuzzy differential equations, Nonlinear Anal.54 (2003) 351-360. · Zbl 1029.34049 |
| [41] | Lakshmikantham, V. and Mohapatra, R. N., Theory of Fuzzy Differential Equations and Inclusions (Taylor and Francis, 2003). · Zbl 1072.34001 |
| [42] | Lakshmikantham, V. and Tolstonogov, A. A., Existence and interrelation between set and fuzzy differential equations, Nonlinear Anal.55 (2003) 255-268. · Zbl 1035.34064 |
| [43] | Li, J. and Li, S., Set-valued Lebesgue integral and representation theorems, Int. J. Comp. Intell. Syst.1 (2008) 177-187. |
| [44] | Li, J. and Li, S., Aumann type set-valued Lebesgue integral and representation theorem, Int. J. Comp. Intelligence Syst.2 (2009) 83-90. |
| [45] | Li, J., Li, S. and Ogura, Y., Strong solution of Itô type set-valued stochastic differential equation, Acta Math. Sinica26 (2010) 1739-1748. · Zbl 1202.60089 |
| [46] | Lupulescu, V., On a class of fuzzy functional differential equations, Fuzzy Sets Syst.160 (2009) 1547-1562. · Zbl 1183.34124 |
| [47] | Malinowski, M. T., Strong solutions to stochastic fuzzy differential equations of Itô type, Math. Comput. Modelling55 (2012) 918-928. · Zbl 1255.34004 |
| [48] | Malinowski, M. T., Itô type stochastic fuzzy differential equations with delay, Syst. Control Lett.61 (2012) 692-701. · Zbl 1250.93116 |
| [49] | Malinowski, M. T., Fuzzy stochastic differential equations with local Lipschitz condition, IEEE Trans. Fuzzy Syst.23 (2015) 1891-1898. |
| [50] | Malinowski, M. T. and Agarwal, R. P., On solutions to set-valued and fuzzy stochastic differential equations, J. Franklin Inst.352 (2015) 3014-3043. · Zbl 1395.34027 |
| [51] | Malinowski, M. T. and Michta, M., Fuzzy stochastic integral equations, Dynam. Syst. Appl.19 (2010) 473-494. · Zbl 1215.60044 |
| [52] | Malinowski, M. T. and Michta, M., Stochastic fuzzy differential equations with an application, Kybernetika47 (2011) 123-143. · Zbl 1213.60102 |
| [53] | Malinowski, M. T. and Michta, M., Set-valued stochastic integral equations driven by martingales, J. Math. Anal. Appl.394 (2012) 30-47. · Zbl 1246.60090 |
| [54] | Malinowski, M. T. and Michta, M., The interrelation between stochastic differential inclusions and set-valued stochastic differential equations, J. Math. Anal. Appl.408 (2013) 733-743. · Zbl 1317.34141 |
| [55] | Malinowski, M. T., Michta, M. and Sobolewska, J., Set-valued and fuzzy stochastic differential equations driven by semimartingales, Nonlinear Anal.79 (2013) 204-220. · Zbl 1260.60010 |
| [56] | Michta, M., Set-valued stochastic integrals and fuzzy stochastic equations, Fuzzy Sets Syst.177 (2011) 1-19. · Zbl 1255.60006 |
| [57] | Michta, M., Remarks on unboundedness of set-valued Itô stochastic integrals, J. Math. Anal. Appl.424 (2015) 651-663. · Zbl 1303.60044 |
| [58] | Michta, M., On connections between stochastic differential inclusions and set-valued stochastic differential equations driven by semimartingales, J. Differential Equations262 (2017) 2106-2134. · Zbl 1405.60080 |
| [59] | Michta, M. and Świa̧tek, K., Stochastic inclusions and set-valued stochastic equations driven by a two-parameter Wiener process, Stoch. Dynam.18 (2018), 36 pp. · Zbl 1417.60073 |
| [60] | Mitoma, I., Okazaki, Y. and Zhang, J., Set-valued stochastic differential equations in \(M\)-type 2 Banach space, Commun. Stoch. Anal.4 (2010) 215-237. · Zbl 1331.60098 |
| [61] | Mizukoshi, M. T., Barros, L. C., Chalco-Cano, Y., Roman-Flores, H. and Bassanezi, R. C., Fuzzy differential equations and the extension principle, Inform. Sci.177 (2007) 3627-3635. · Zbl 1147.34311 |
| [62] | Nieto, J. J., Rodriguez-Lopez, R. and Georgiou, D. N., Fuzzy differential systems under generalized metric spaces approach, Dynam. Syst. Appl.17 (2008) 1-24. · Zbl 1168.34005 |
| [63] | Ogura, Y., On stochastic differential equations with set coefficients and Black-Scholes model, in Proc. 8th Int. Conf. Intelligent Technologies, Sydney, (2007), pp. 300-304. |
| [64] | Papageorgiou, N., On Fatou’s lemma and parametric integrals for set-valued functions, Proc. Indian Acad. Sci. (Math. Sci.)103 (1993) 181-195. · Zbl 0790.28009 |
| [65] | Plotnikov, A. V. and Skripnik, N. V., Existence and uniqueness theorem for set integral equations, J. Adv. Res. Dyn. Control Syst.5 (2013) 65-72. |
| [66] | Plotnikov, A. V. and Skripnik, N. V., Existence and uniqueness theorem for set-valued Volterra integral equations, Amer. J. Appl. Math. Statist.1 (2013) 41-45. |
| [67] | Plotnikov, A. V. and Skripnik, N. V., Existence and uniqueness theorem for fuzzy integral equation, J. Math. Sci. Appl.1 (2013) 1-5. |
| [68] | Protter, P., Stochastic Integration and Differential Equations: A New Approach (Springer-Verlag, 1990). · Zbl 0694.60047 |
| [69] | Puri, M. L. and Ralescu, D. A., Differentials of fuzzy functions, J. Math. Anal. Appl.91 (1983) 552-558. · Zbl 0528.54009 |
| [70] | Puri, M. L. and Ralescu, D. A., Fuzzy random variables, J. Math. Anal. Appl.114 (1986) 409-422. · Zbl 0592.60004 |
| [71] | Qiu, D. and Shu, L., Supremum metric on the space of fuzzy sets and common fixed point theorems for fuzzy mappings, Inform. Sci.178 (2008) 3595-3604. · Zbl 1151.54010 |
| [72] | Saint-Pierre, J. and Sajid, S., Parametrized integral of multifunctions in Banach spaces, J. Math. Anal. Appl.239 (1999) 49-71. · Zbl 0980.28006 |
| [73] | Skripnik, N. V., Averaging of fuzzy integral equations, Appl. Math. Phys.1 (2013) 39-44. |
| [74] | Stojaković, M., Fuzzy conditional expectation, Fuzzy Sets Syst.52 (1992) 53-60. · Zbl 0782.60009 |
| [75] | Stojaković, M. and Stojaković, Z., Series of fuzzy sets, Fuzzy Sets Syst.160 (2009) 3115-3127. · Zbl 1183.03054 |
| [76] | Tolstonogov, A. A., Differential Inclusions in a Banach Space (Kluwer Academic Publishers, 2000). · Zbl 1021.34002 |
| [77] | Tu, N. N. and Tung, T. T., Stability of set differential equations and applications, Nonlinear Anal.71 (2009) 1526-1533. · Zbl 1188.34068 |
| [78] | Zadeh, L. A., Fuzzy sets, Inform. Control8 (1965) 338-353. · Zbl 0139.24606 |
| [79] | Zhang, J., Set-valued stochastic integrals with respect to a real valued martingale, in Soft Methods for Handling Variability and Imprecision, eds. Dubois, D., Lubiano, M. A., Prade, H., Gil, M. A. and Grzegorzewski, P., , Vol. 48 (Springr, 2008). · Zbl 1146.60004 |
| [80] | Zhang, J. and Qi, J., Set-valued stochastic integrals with respect to finite variation processes, Adv. Pure Math.3 (2013) 15-19. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
