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.
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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
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DOI: https://doi.org/10.1007/978-3-642-22833-9_19
Publisher Name: Springer, Berlin, Heidelberg
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