Iterative method for non-adapted fuzzy stochastic differential equations
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2021), pp. 30-42.

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In this paper, an anticipating stochastic differential equation is considered, that the integrand processes are not adapted to the filtration generated by a Wiener process. Using the correspondence between the Skorohod integral and Itô-Skorohod integral, the equations can be solved by using standard iterative techniques. Then, the existence and uniqueness of strong solutions to these equations are discussed. Such equations with non-adapted, fuzziness, and randomness processes can be applied in financial models.
Mots-clés : Malliavin calculus
Keywords: fuzzy stochastic process, fuzzy stochastic integral, Skorohod integral.
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H. Jafari. Iterative method for non-adapted fuzzy stochastic differential equations. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2021), pp. 30-42. http://geodesic.mathdoc.fr/item/IVM_2021_7_a3/

[1] Kim J. H., “On fuzzy stochastic differential equations”, J. Korean Math. Soc., 42 (2005), 153–169 | DOI | MR | Zbl

[2] Malinowski M. T., “Stochastic fuzzy differential equations with an application”, Kybernetika, 47 (2011), 123–143 | MR | Zbl

[3] Malinowski M. T., “Strong solutions to stochastic fuzzy differential equations of Itô type”, Math. Comput. Modell., 55 (2012), 918–928 | DOI | MR | Zbl

[4] Malinowski M. T., “Itô type stochastic fuzzy differential equations with delay”, Syst. Control Lett., 61 (2012), 692–701 | DOI | MR | Zbl

[5] Malinowski M. T., “Some properties of strong solutions to stochastic fuzzy differential equations”, Inform. Sci., 252 (2013), 62–80 | DOI | MR | Zbl

[6] Fei W. Y., “Existence and uniqueness for solutions to fuzzy stochastic differential equations driven by lomathcal martingales under the non-Lipschitzian condition”, Nonlinear Anal. TMA, 76 (2013), 202–214 | DOI | MR | Zbl

[7] Buckdahn R., Nualart D., “Linear stochastic differential equations and Wick product”, Prob. Theory Rel. Fields, 99 (1994), 501–525 | DOI | MR

[8] Jafari H., Paripour M., Farahan H., “Fuzzy Malliavin derivative and linear Skorohod fuzzy stochastic differential equation”, J. of Intelligent Fuzzy Syst., 35:2 (2018), 2447–2458 | DOI

[9] Tudor C. A., “Martingale-type stochastic mathcalculus for anticipating integral processes”, Bernoulli, 10:2 (2004), 313–325 | DOI | MR | Zbl

[10] Nualart D., The Malliavin Calculus and Related Topics, Second Edition, Springer, 2006 | MR | Zbl

[11] Tudor C. A., “Itô-Skorohod stochastic equations and applications to finance”, J. of Appl. Math. and Stochastic Anal., 4 (2004), 359–369 | DOI | MR | Zbl

[12] Diamond P., Kloeden P. E., Metric Spaces of Fuzzy Sets: Theory and Applications, World Scientific, Singapore, 1994 | MR | Zbl

[13] Lakshmikantham V., Mohapatra R. N., Theory of Fuzzy Differential Equations and Inclusions, Taylor and Francis, London, 2003 | MR | Zbl

[14] Hiai F., Umegaki H., “Integrals, conditional expectation, and martingales of multivalued functions”, J. Multivariate Anal., 7 (1977), 149–182 | DOI | MR | Zbl

[15] Puri M. L., Ralescu D. A., “Fuzzy random variables”, J. Math. Anal. Appl., 114 (1986), 409–422 | DOI | MR | Zbl

[16] Colubi A., Dominguez-Menchero J. S., Lopez-Diaz M., Ralescu D. A., “A $D_{E} [0, 1]$ representation of random upper semi continuous functions”, Proc. Amer. Math. Soc., 130 (2002), 3237–3242 | DOI | MR | Zbl

[17] Kisielewicz M., Differential Inclusions and Optimal Control, Kluwer Academic Publishers, Dordrecht, 1991 | MR | Zbl

[18] Allen E., Modeling with Itô Stochastic Differential Equations, Springer, 2007 | MR | Zbl