It?-Henstock Integral Of The Fuzzy Stochastic Process And Its Numerical Calculation

It is well known that fuzzy measure and fuzzy integral theory,as an important branch of fuzzy analysis has been investigated and applied widely in other fields Stochastic integral,as a significant component of stochastic analysis,is extensively used in the fields of stochastic control and mathematical finance.Aumann is an easy way to define and study the It? integral of set-valued random variables(even fuzzy random variables).So the Aumann-It? integral of set-value random variables or fuzzy value random variables has been researched,but it is not convenient to calculate the Aumann-It? integral of fuzzy value random variables.We note that in the classical real analysis,the method of Riemann has a distinctive advantage in integral numerical calculation.Henstock integral,as a generalization of Riemann integral,can well deal with the integral of some“highly oscillating”functions,e-specially the numerical calculation.This paper tries to define and discuss the It? integral of set-valued functions(even fuzzy-number-valued functions)by means of the method of Riemann.We summarize the main results of the paper as followsFirst of all,based on It?-Henstock integral and It?-McShane integrals for a adapted real-valued stochastic process with respect to a Brownian motion,combining the integrability of the adapted fuzzy stochastic process with respect to a Brownian motion,the fuzzy It?-Henstock integral and the fuzzy It?-McShane integrals for adapted fuzzy stochastic process with respect to a Brownian motion are defined and their properties are discussed in detail.In addition,some examples are given to illustrate the propertiesThen,the interrelation of the fuzzy It?-Henstock integral and the fuzzy It?-McShane integral is discussed.The result shows that the fuzzy It?-Henstock integral is equivalent to the fuzzy It?-McShane integral when its primitive of fuzzy It?-Henstock integral is It? absolutely continuousFinally,considering that the fuzzy It?-Henstock integral and the fuzzy It?-McShane integral are characterized by Riemann type,the advantage of which is numerical calculation of the fuzzy stochastic process It? integral.With the modu-lus of oscillatory of the fuzzy-number-value functions,the quadrature rules of the fuzzy It?-Henstock integral are discussed and three quadrature rules,such as the It? midpoint-type rule,It? trapezoidal-type rule,It? Simpson's rule and ?-fine delay quadrature rule of the It? integral are given.