Study On Stochastic Functional Differential Equations Driven By Fractional Brownian Motion

The fractional Brownian motion BH = {BH(t),t?0} of Hurst parameter 0<H<1 is a centered Gaussian process with zero mean.When H =1/2,BH becomes the standard Brownian motion,and BH neither is a semimartingale nor a Markov process if H?1/2.But,the pathwise of the fractional Brownian motion is ?-order Holder continuous for any 0<?<H;Besides,the fractional Browni-an motion is H-self similar and has the stationary increments and it's increment process has long-range dependence when Hurst parameter 1/2<H<1;Fur-thermore,the increment of the fractional Brownian motion with Hurst parameter 1/2<H<1 is positively correlated,and the increment of the fractional Brownian motion with Hurst parameter 0<H<1/2 is negatively correlated.These spe-cial properties make it more reasonable and effective to use fractional Brownian motion as random noise in stochastic models such as mathematical finance,net-work communication and population dynamic system.And in fact many systems have different time delays,that is,the change of the system is not only related to the current state of the system but also depends on the past state of the system,which makes it more reasonable to simulate these systems with functional differen-tial equations.So,it has important theoretical significance and application value to explore stochastic functional differential equations driven by fractional Brown-ian motion by using the techniques on stochastic analysis of fractional Brownian motion.This paper mainly deals with three problems of stochastic functional dif-ferential equations driven by fractional Brownian motion.The main results are as follows:1.By approximation arguments and a comparison theorem,we prove the existence of strong solutions to a class of stochastic differential equations driven by fractional Brownian motion with Hurst parameter 1/2<H<1 just under the linear growth condition,and we study the continuous dependence of the solution with respect to the initial data.By using the relation of the integral representation with respect to the fractional Brownian motion with different Hurst parameter,we prove the existence of weak solutions to a class of stochastic differential equations with a time-dependent diffusion driven by a fractional Brownian motion with Hurst parameter 1/2<H<1 just under the linear growth condition,but the drift can be discontinuous.By using pathwise Riemann-Stieltjes approach and fixed point theorem,we prove a global existence and uniqueness result of the mild solution for a class of stochastic functional differential equations in Hilbert spaces driven by a fractional Brownian motion with Hurst parameter 1/2<H<1 considered just under some local Lipschitz conditions.2.Using the techniques of stochastic analysis and distance functions,we give some necessary and sufficient conditions for the viability of a closed convex set K in Rn for weak solutions to a class of stochastic functional differential equation.By establishing some new estimates,using pathwise Riemann-Stieltjes approach and stochastic tangency cone approach,we give some equivalent conditions for the viability of the mild solution for a class of stochastic functional differential equations in Hilbert spaces driven by a fractional Brownian motion with Hurst parameter 1/2<H<1.3.By establishing some new integral estimation of fraction Brownian motion of Hurst parameter 0<H<1/2,using delay integral inequalities,we consider the global attracting sets and the exponential decay in the p-th moment of the mild solution of a class of neutral stochastic functional differential equations driven by fractional Brownian motion with Hurst parameter 0<H<1/2 in the Hilbert space.