Stability Of Numerical Solutions For Stochastic Delay Differential Equations

Without the linear growth condition and the global Lipschitz condition, this thesis deals with analytical and numerical stability properties of stochastic differential equations with fixed delay and stochastic differential equations with unbounded delay. Using Lyapunov function idea, continuous-type and discrete-type semi-martingale convergence theorem, we obtained stability of the trivial solution, including exponential stability andφ-stability in pth moment sense and in almost sure sense; And obtained Stability of numerical solutions, including asymptotic stability, exponential stability andφ-stability in pth moment sense and in almost sure sense.Using continuous-type semi-martingale convergence theorem, this thesis obtained the pth moment exponential stability and almost sure exponential stability of SDEs with fixed delay, under monotone-type condition. By introducing a class of Ψ-type functions and at-tenuation factor φ-ε(δ(t)), this thesis obtained the pth moment φ-stability and almost sure φ-stability of SDEs with unbounded delay, under monotone-type condition.Using discrete-type semi-martingale convergence theorem, this thesis investigated sta-bility of Backward Euler-Maruyama (BEM) and stochastic θ method. In the fixed delay situation, under monotone-type condition, this thesis establishes moment asymptotic stabil-ity and almost sure asymptotic stability of the above two numerical solutions for SDEs. As one of the main features in the thesis, using strong monotone-type condition in place of the above condition, we obtained that both numerical approximation are exponential stability in moment sense and almost sure sense. It’s worth mentioning that a counterexample in this thesis shows that the stochastic8method cannot reproduce the stability of the exact solution for θ∈[0,0.5); But for θ∈(0.5,1] the method can. In the unbounded delay sit-uation, by the attenuation factor φ-ε(δ(t)), this thesis establishes asymptotic stability and^-stability of the above two numerical solutions under monotone-type condition and strong monotone-type conditionThe procedure adopts "two phases mode":firstly, general theorems are established under monotone-type condition; secondly, by imposing growth conditions on the coeffi-cients f and g of the equation to specify the general condition, obtained convenient condi-tion which only include system parameter. In this thesis we focus on the following specific conditions, the one-side linear condition and polynomial growth condition. In the polyno-mial growth condition, the diffusion coefficient g obey polynomial growth, then the result be able to cover more real system, for example the Lotka-Volterra system whose diffusion coefficient is not linear. The above specific conditions not only include in the monotone-type condition but also include in the strong monotone-type condition. This implies that the strong monotone-type condition has widely range of application, and be close to the monotone-type condition.In this thesis,the only restriction on step size is very weak which is used to ensure the implicit numerical approximate is well define. In other word, under monotone-type condition, if the numerical approximate is well define, BEM and stochastic θ method can can reproduce stability of SDEs And we given a feasible method to compute the speed of decay γ or γ(△) In all situations. The speed of decay of numerical solution γ(△) is relevant to the step size, but it is sufficiently close to which of the exact solution decay to trivial solution γ, when the step size is sufficiently small.