Fuzzy Stochastic Differential Equations Of It(?)-type

The complex phenomenon in the real world often involves both randomness and fuzziness.The theoretical framework of stochastic differential equations has already been systematic for the problem without fuzziness.However,it still has some limitations for fuzzy problem.Great achievements have been made in the study of fuzzy number space and fuzzy random variable with the last 20 years'development,and have provided a foundation for the research of fuzzy stochastic differential equations.Recently,there are some research on fuzzy stochastic differential equations in the mean-square sense.However,the existence and uniqueness of fuzzy stochastic differential equations of It(?)-type and estimation of the approximate solutions are still need to be studied systematically.We will focus on both aspects in this paper.This paper studies fuzzy stochastic differential equations and mainly includes two parts:Discussion on existence and uniqueness of solutions to fuzzy stochastic differential equations of It(?)-type and the corresponding approximate solutions.In the first part,we introduce the definition of solutions to fuzzy stochastic differential equations of It(?)-type and study the existence and uniqueness of solutions under global Lipschitz conditions,or alternatively local Lipschitz conditions.In addition,we give the error bound of Picard approximation.In the second part we study the approximate solutions to fuzzy stochastic differential equations of It(?)-type.Usually,it is rarely pos- sible to obtain the explicit solutions for fuzzy stochastic differential equations,so the research on approximate solutions is imperative.There are two methods of approximate solutions discussed in this paper:Caratheodory's approximate solutions and Cauchy-Maruyama's approximate solutions.Finally,an example of numerical solutions is given for fuzzy stochastic differential equations of It(?)-type.In the study of fuzzy stochastic differential equations of It(?)-type,the objects of study are both fuzzy and random variables.Consequently we have to deal with new concepts and properties,and the approach is correspondingly different.Unlike the existing references,the solutions we discussed are in a larger space.Moreover,in the proof of existence and uniqueness of solutions under local Lipschitz conditions,we use new methods to deal with the new weaken conditions.The estimate of solution is actually to examine how the iterations converge to the unique solution.Both the proof of existence and uniqueness of solutions and the estimation are theoretical.Forturnately,the following two approximate methods to obtain the solutions are practical,and more general.These methods save a lot of time and calculation,and are used widely in reality.