Asymptotic Properties Of Solutions Of Stochastic Functional Differential Equation Of Stratonovich Type With Finite Delay

This work is aimed to investigate asymptotic properties of solution of stochastic functional differential equation of Stratonovich type with finite delay as follows Firstly,by the relationship of Ito and Stratonovich integrals,we can prove the pth moment estimates theorem and the pth moment continuity theorem of the solution of equation when p≥2 with the help of the Lipschitz condition,the linear growth condition,Ito formula and Burkholder-Davis-Gundy inequality and other tools;Next,we can get the conclusion of the pth moment estimates and the pth moment continuity of the solution of equation when 0<p<2 by using the Holder inequality.Then we will use Ito formula,the Exponential martingale inequality,the Gronwall inequality and the Borel-Cantelli lemma to prove the almost surely asymptotic estimates of the sample Lyapunov exponent of the solution of equation;Finally,we will give a special case i.e.equation(4.11),and we will get three kinds of the asymptotic properties of the solution when the three different conditions are applied to the drift coefficient with delay of the equation(4.11);Furthermore,it is proved that the almost surely asymptotic estimates of equation(4.4)is feasible.