We extend the definitions of the Gaussian Malliavin calculus operators to fuzzy stochastic processes. We consider Skorohod fuzzy stochastic differential equations, which the integrands of the stochastic integrals are not adapted to the filtration generated by a Wiener process. Such equations with randomness, fuzziness and non-adapted processes can be applied in financial models. We apply the fuzzy Malliavin derivative and related topics to discuss the existence and uniqueness of solutions.
ArmandaA., AllahviranloobT., AbbasbandyS. and GouyandehZ., Fractional relaxation-oscillation differential equations via fuzzy variational iteration method, Journal of Intelligent and Fuzzy Systems32(1) (2017), 363–371.
2.
BedeB. and GalS.G., Generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations, Fuzzy Set Syst151 (2005), 581–599.
3.
BedeB. and StefaniniL., Generalized differentiability of fuzzyvalued functions, Fuzzy Set Syst230 (2013), 119–141.
4.
BuckdahnR. and NualartD., Linear stochastic differential equations and Wick product, Prob Theory Rel Fields99 (1994), 501–525.
5.
ColubiA., Dominguez-MencheroJ.S., Lopez-DiazM. and RalescuD.A., A DE[0; 1] representation of random upper semicontinuous functions, Proc Amer Math Soc130 (2002), 3237–3242.
6.
DunguN.C., Fractional stochastic differential equations with applications to finance, Journal of Mathematical Analysis and Applications397(1) (2013), 334–348.
7.
FeiW.Y., Existence and uniqueness for solutions to fuzzy stochastic differential equations driven by local martingales under the non-Lipschitzian condition, Nonlinear Anal TMA76 (2013), 202–214.
8.
FeiW.Y., LiuH. and ZhangW., On solutions to fuzzy stochastic differential equations with local martingales, Systems and Control Letters65 (2014), 96–105.
9.
GomesL.T. and BarrosL.C., A note on the generalized difference and the generalized differentiability, Fuzzy Set Syst280 (2015), 142–145.
10.
HiaiF. and UmegakiH., Integrals, conditional expectation, and martingales of multivalued functions, J Multivariate Anal7 (1977), 149–182.
11.
KalevaO., Fuzzy differential equations, Fuzzy Set Syst24 (1987), 301–317.
12.
KimJ.H., On fuzzy stochastic differential equations, J Korean Math Soc42 (2005), 153–169.
MaX., ZhanJ. and AliM.I., A survey of decision making methods based on two classes of hybrid soft set models, Artificial Intelligence Review49(4) (2018), 511–529.
15.
MichtaM., Set-valued stochastic integrals and fuzzy stochastic equations, Fuzzy Set Syst177 (2011), 1–19.
16.
MalinowskiM.T., Stochastic fuzzy differential equations with an application, Kybernetika47 (2011), 123–143.
17.
MalinowskiM.T., Strong solutions to stochastic fuzzy differential equations of Itô type, Math Comput Modell55 (2012), 918–928.
18.
MalinowskiM.T., Itô type stochastic fuzzy differential equations with delay, Syst Control Lett61 (2012), 692–701.
19.
MalinowskiM.T., Some properties of strong solutions to stochastic fuzzy differential equations, Information Sciences252 (2013), 62–80.
20.
MichtaM., Fuzzy Stochastic Differential Equations Driven by Semimartingales-Different Approaches, Mathematical Problems in Engineering, 2015, Article ID 794607.
21.
MalinowskiM.T. and MichtaM., Set-valued stochastic integral equations driven by martingales, J Math Anal Appl394 (2012), 30–47.
22.
MilletA., NualartD., SanzM., Small Perturbations for Quasilinear Anticipating Stochastic Differential Equations, International Series of Numerical Mathematics, Birkhauser VerlagBasel, 1991.
23.
NegoitaC.V., RalescuD.A., Applications of Fuzzy Sets to Systems Analysis, Wiley, New York, NY, USA, 1975.
24.
NualartD., The Malliavin Calculus and Related Topics, 2, Springer, 2006.
25.
Di NunnoG. and ØksendalB., Malliavin Calculus for Levy Processes with Applications to Finance, Springer, 2009.
26.
PawlakZ., Rough sets, Int J Comput Inf Sci11(5) (1982), 341–356.
27.
PuriM.L. and RalescuD.A., Differentials of fuzzy functions, J Math Anal Appl91 (1983), 552–558.
28.
PuriM.L. and RalescuD.A., Fuzzy random variables, J Math Anal Appl114 (1986), 409–422.
29.
Sanz-SoléM., EPFL-CRC, Malliavin calculus with applications to stochastic partial differential equations, 2005.
30.
SeyaD.K., MakengoR.M., RemonM. and RebeccaW.O., Fuzzy itô integral driven by a fuzzy brownian motion, Journal of Fuzzy Set Valued Analysis3 (2015), 232–244.
31.
StefaniniL.A generalization of Hukuhara difference, Soft Methods for Handling Variability and Imprecision, in: Series on Advances in Soft Comuting, vol. 48, Springer, 2008.
32.
StefaniniL., A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Set Syst161 (2010), 1564–1584.
33.
StefaniniL. and GuerraM.L., On fuzzy arithmetic operations: Some properties and distributive approximations, International Journal of Applied Mathematics19 (2006), 171–199.
34.
ZhanJ. and AlcantudJ.C.R., A novel type o f soft rough covering and its application to multicriteria group decision making, Artificial Intelligence Review (2018). 10.1007/s10462-018-9617-3.
