Set-valued And Fuzzy Set-valued Research

In the previous study of probability,the random variables often referred to are single-valued random variables,its value is single value.But in the real life,the value of a random variable is often not a specific numerical value.Sometimes,it is also possible that the result of value is a collection of many values,known as the set-value.Even sometimes,we can’t use clear numerical value to represent the variable,but a fuzzy numerical value,namely fuzzy set-value.In the field of finance or control,there will be some limitations if we only use single-valued random variables to describe an event.In this case,we need to use the set-valued and fuzzy set-valued theory.At present,the related research of fuzzy set-valued theory still needs to be enriched.In this paper,we discuss the fuzzy set-valued theory on the basis of the set-valued ralated theory.Firstly,we briefly introduce the fuzzy set-valued martingale and fuzzy set-valued square integrable martingale,then prove the fuzzy set-valued square integrable martingale representation theorem.Aiming at stochastic integral of the predictable stochastic process with respect to a fuzzy set-valued square integrable martingale,we give its definition.At the same time,we also give the representation theorem of the this stochastic integral.It is proved that the stochastic integral of the fuzzy set-valued square integrable martingale is a set-valued submartingale.Some properties of set-valued random integrals are discussed.In particular,some properties of set-valued submartingales are also discussed.In the content of third part,on the basis of the stochastic differential equation of the set-valued square integrable martingale,we make a little change that set-valued square integrable martingale is repalced by fuzzy set-valued square integrable martingale.Then we research the fuzzy set-valued stochastic differential equations,by using linear bounded conditions and Lipschitz continuity conditions,fuzzy set-valued integral inequality and some properties of Hausdorff distance,the existence and uniqueness theorem of solution of the stochastic differential equation with fuzzy set-valued square integrable martingale is proved by successive approximation method.