Numerrical Methods For Two Classes Of Stochastic Delay Differential Equations

Recently, stochastic delay differential equations have been developedextensively. When we construct the model, they are often involved the delay and theenvironment noise term. So stochastic delay differential equations were applied tothe field of physics, biology and medicine, etc. However, we should usually usenonlinear stochastic delay differential equations when the models were building. Itis difficult to solve their explicit solution. So it is very important to use thenumerical methods to investigate the properties of the solutions of stochastic delaydifferential equations.When we are constructing the numerical methods for stochastic delaydifferential equations, we usually investigate the convergence and stability of themethods. Due to them, we discuss the convergence and stability of two kinds ofnumerical methods for stochastic delay differential equations in this paper.We first apply the split-step backward Euler method to solve stochastic delaydifferential equations. We construct a martingale when we are researching its almostsurely exponential stability. By using the continuous semi-martingale convergencetheorem, we then conclude the boundedness of the numerical solution of the methodwhen the step size is sufficiently small. In the proof of the almost surely exponentialstability, we use the boundedness of the delay term. When we construct thenumerical method of stochastic pantograph equation, the difficulty is how to dealwith the pantograph term. So we take the advantage of the method with linearinterpolation in order to obtain the precise numerical solution. By using this idea, weconstruct the split-step backward Euler method with linear interpolation for solvingstochastic pantograph equations. Under the Lipschitz condition and linear growthcondition, we obtain that the method is mean-square convergence and the order ofconvergence is0.5. We also lead in linear stochastic pantograph equations withconstant coefficients to discuss the mean-square stability of the method. Theconditions that guarantee the mean-square stability and general mean-squarestability of the method are obtained. In the proof, we use Lyapunov method. Weverify the conclusions with numerical examples for every numerical method.