Asymptotic Properties Of Stochastic Differential Equations With Unbounded Delays

Asymptotic properties of stochastic differential equations with unbounded delays are investigated in this thesis, which include the existence and uniqueness of the global so-lution, moment estimation, pathwise estimation, stability and the inferior limit of growth, etc. Lyapunov function idea runs through the thesis. By introducing a class ofΨ-type functions which have rich connotation and using the semi-martingale convergence theorem, exponential martingale inequality and Razumikhin method, the existence and uniqueness of the global solution, moment boundedness, moment boundedness average in time, path-wise boundedness, pathwise superior limit of growth, pth momentψ-stability, almost sureψ-stability, pth moment inferior limit of growth and pathwise inferior limit of growth are obtained.The existence of the global solution is the premise of investigating the asymptotic prop-erties of equations. This thesis establishes some general conditions for the existence and uniqueness of the global solution in Rn and R×n without the restriction of the linear growth condition. As one of the main features in the thesis, the conditions which imposed on guaranteeing asymptotic properties also ensure the existence and uniqueness of the global solution.Using the semi-martingale convergence theorem, the boundedness of the solution is studied. By choosing proper Lyapunov function V(x) and establishing the growth control-lable conditions on LV(t, x, y), the existence and uniqueness of the global solution, moment boundedness, moment boundedness average in time and pathwise boundedness are obtained simultaneously. By imposing polynomial growth conditions on f and g, general controllable conditions on CV(t, x, y) are specified, and then the main results are derived.By using exponential martingale inequality and Borel-Cantelli Lemma, the pathwise growth problem is investigated. Under the polynomial growth conditions, the thesis gets that the global solution of the equation growth with no more than some rate, and the results also guarantee the existence and uniqueness of the global solution.Stability is the central problem of dynamic system theory. Two methods, the semi-martingale convergence theorem and Razumikhin method, are chosen to study the stability of the equation. Both pth momentψ-stability and almost sureψ-stability are obtained. The procedure adopts "two phases mode":firstly, general theorems are established; secondly, by imposing polynomial growth conditions on the coefficients of the equation to specify the general condition, the main results are obtained. Razumikhin theorems give both pth moment stability and almost sure stability in the thesis.At last, the thesis discusses the inferior limit of growth of the global solution. By using Razumikhin method and exponential martingale inequality, pth moment inferior limit of the rate of growth and almost sure pathwise inferior limit of the rate of growth are obtained, respectively. A "reversible" Razumikhin theorem is established. And the inferior limit of growth of a linear system is discussed in great detail.The research method is applicable to both bounded delay and unbounded delay. For unbounded delay, the decay factorψ-ε(δ(t)) is introduced to overcome the infinite memory of the system. For bounded delay, the decay factorψ-ε(δ(t)) can be deleted without influencing the results. Without the linear growth condition and under the polynomial growth conditions on the coefficients, the existence and uniqueness of the global solution and asymptotic properties can be obtained simultaneously, which extends application field of the equation. M-matrix tool is used many times in the thesis. By making full use of M-matrix technique, the form of results is simpler and more convenient in application. The thesis introduces theΨ-type function which includes exponential function and polynomial function as a general reference function to survey the rate of growth and decay of the global solution, which makes the results more abundant.