Abstract
Random Fuzzy Differential Equation(RFDE) describes the phenomena not only with randomness but also with fuzziness. It is widely used in fuzzy control and artificial intelligence etc. In this paper, we shall discuss RFDE as follows:
dF̃(t)=f̃(t, F̃(t))dt+g(t, F̃(t))dB t ,
where f̃(t, F̃(t))dt is related to fuzzy set-valued stochastic Lebesgue integral, g(t, F̃(t))dB t is related to Itô integral. Firstly we shall give some basic results about set-valued and fuzzy set-valued stochastic processes. Secondly, we shall discuss the Lebesgue integral of a fuzzy set-valued stochastic process with respect to time t, especially the Lebesgue integral is a fuzzy set-valued stochastic process. Finally by martingale moment inequality, we shall prove a theorem of existence and uniqueness of solution of random fuzzy differential equation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Fei, W.: Existence and uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients. Infor. Sci. 177, 4329–4337 (2007)
Feng, Y.: Mean-square integral and differential of fuzzy stochastic processes. Fuzzy Sets and Syst. 102, 271–280 (1999)
Feng, Y.: Fuzzy stochastic differential systems. Fuzzy Sets and Syst. 115, 351–363 (2000)
Feng, Y.: The solutions of linear fuzzy stochastic differential systems. Fuzzy Sets and Syst. 140, 341–354 (2003)
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland, Kodansha (1981)
Li, J., Li, S.: Set-valued stochastic Lebesgue integral and representation theorems. Int. J. Comput. Intell. Syst. 1, 177–187 (2008)
Li, J., Li, S.: Aumann type set-valued Lebesgue integral and representation theorem. Int. J. Comput. Intell. Syst. 2, 83–90 (2009)
Li, J., Li, S., Xie, Y.: The space of fuzzy set-valued square integrable martingales. In: IEEE Int. Conf. Fuzzy Syst., Korea, pp. 872–876 (2009)
Li, J., Li, S., Ogura, Y.: Strong solution of Itô type set-valued stochastic differential equation. Acta Mathematica Sinica 26, 1739–1748 (2010)
Li, J., Wang, J.: Fuzzy set-valued stochastic Lebesgue integral (unpublished)
Li, S., Ogura, Y., Kreinovich, V.: Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables. Kluwer Academic Publishers, Dordrecht (2002)
Li, S., Ren, A.: Representation theorems, set-valued and fuzzy set-valued Itô integral. Fuzzy Sets and Syst. 158, 949–962 (2007)
Malinowski, M.: On random fuzzy differential equation. Fuzzy Sets and Syst. 160, 3152–3165 (2009)
Malinowski, M.: Existence theorems for solutions to random fuzzy differential equations. Nonlinear Analysis 73, 1515–1532 (2010)
Puri, M., Ralescu, D.: Differentials of fuzzy functions. J. Math. Anal. Appl. 91, 552–558 (1983)
Puri, M., Ralescu, D.: Fuzzy random variables. J. Math. Anal. Appl. 114, 409–422 (1986)
Steele, J.: Stochastic Calculus and Financial Applications. Springer, New York (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Li, J., Wang, J. (2011). Solution of Random Fuzzy Differential Equation. In: Li, S., Wang, X., Okazaki, Y., Kawabe, J., Murofushi, T., Guan, L. (eds) Nonlinear Mathematics for Uncertainty and its Applications. Advances in Intelligent and Soft Computing, vol 100. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22833-9_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-22833-9_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22832-2
Online ISBN: 978-3-642-22833-9
eBook Packages: EngineeringEngineering (R0)