The Convergence And Stability Of Multi-stage Methods For Stochastic Delay Differential Equations

Stochastic delay differential equations(SDDEs) have been widely used in economics, biol-ogy, medicine, ecology and other sciences. As usually analytic solutions for stochastic delaydifferential equations(SDDEs) are hardly available, numerical method is an important approachto investigate the differential equations.In the past decades, some kinds of numerical methods have been researched for the stochas-tic delay differential equations such as Euler method, θ method, Milstein method,split stepmethod.The Euler type schemes for solving SDDEs have proposed, and it has strong convergenceorder1/2in mean-square sense. Further, the Milstein-type scheme achieves a strong order ofconvergence higher than that of the Euler-type scheme in mean-square sense.In this thesis, we introduce some multi-stage methods for the stochastic delay differentialequations. In Chapter2, a new splitting method designed for the numerical solutions of stochasticdelay Hopfield neural networks is introduced and analysed. Under Lipschitz and linear growthconditions, this split-step θ-Milstein method is proved to have a strong convergence of order1in mean-square sense, which is higher than that of existing split-step θ-method. Further,mean-square stability of the proposed method is investigated. Numerical experiments and com-parisons with existing methods illustrate the computational efficiency of our method. In Chapter3, an explicit Runge-Kutta type method is introduced for Stratonovich stochastic delay differen-tial equation. In general, the mean-square convergence order of this method achieves1/2at least.Meanwhile, the order can reach1when the diffusion function is independent of time delay or themagnitude of noise is the same as the square root of time step size of the discrete scheme. Anddelay-dependent numerical mean-square stability is considered for a linear scalar test equation.Some illustrative numerical examples are provided to confirm the theoretical results.