Study On Stability Of Numerical Methods For Stochastic Differential Delay Equations

Stochastic differential delay equations are the promotion of stochasticdifferential delay equations and delay differential equations and have beenapplied widely in many areas，such as finance, microelectronics and biology. For thereason that it is difficult to find the analytic solutions of most of stochastic differentialdelay equations, it is significant to develop applicable numerical methods and discussthe stability of the numerical solution.In this thesis, Chapter1and Chapter2introduce the research background, basicconcept and the research status for main numerical methods and stability of stochasticdifferential delay equations.Chapter3studies the T-stability of the Heun method for solving stochasticdifferential delay equations. By applying the Heun method on two kinds of linearstochastic differential delay equations and approximating the Wiener increment with adiscrete random variable with two-point distribution，we give the T-stability conditionsfor the Heun methods in this two cases. The results of numerical simulations verify theconclusion in this thesis.Chapter4applies the balanced method on two kinds of linear stochastic differentialdelay equations and approximating the Wiener increment with a discrete randomvariable with two-point distribution. The T-stability conditions for the balanced methodsin the two cases are given. The results of numerical simulations show that the balancedmethods are T-stable if the step size satisfies the limit condition in this thesis.Chapter5constructs the composite Milstein method for nonlinear stochasticdifferential delay equations and the mean square stability of this method is analyzed.The step size limit conditions under which the composite Milstein method isMS-stability and GMS-stability are obtained. Some numerical tests verify thetheoretical results proposed.Chapter6gives a conclusion of the thesis and presents some thinking for furtherresearch about the numerical methods and stability of stochastic differential delayequations.